Control of PDE systems - Uni Kiel

Control of PDE systemsLecture Notes

Kiel University, TU Graz

Winter Term 2020/21

Control of PDE systems

Lecture Notes, Winter Term 2020/21

Kiel University, TU Graz

kDr. Alexander Schaum, [email protected]–kiel.dem http://www.control.tf.uni–kiel.de

Acknowledgement

These notes are based on the following previous ones:

Regelung verteilt–parametrischer Systeme - Skriptum zur Vorlesung, Thomas Meurer, Kiel University,2018.

Control of PDEs - Lecture Notes for the Elgersburg School 2016, Thomas Meurer, Alexander Schaum, KielUniversity, 2016.

Further sources of inspiration are referenced in the text.

iii

iv

Contents

1 Some preliminaries 1

1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Function spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Introduction 3

3 Modelling with partial differential equations 5

3.1 Transition from discrete to continuously distributed parameters . . . . . . . . . . . . . . . . . . . . 5

3.1.1 Multi-body oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3.1.2 Electrical transmission line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.1.3 Heat conduction and diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.1.4 Diffusion–Convection–Reaction–Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.1.5 Flexible mechanical structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2 Classification of second order PDEs and the method of characteristics . . . . . . . . . . . . . . . . 13

3.2.1 Semilinear PDEs in 2 coordinates z and t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2.2 Types of partial differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.3 Numerical solution methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.4 Basic concepts in the control of PDE systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.4.1 Sensor and actuator equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.4.2 Early and late lumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4 State-space representation 25

4.1 Introduction to state space analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.1.1 Motivation and introductory examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.1.2 Uniformly continuous semigroups and infinitesimal generators . . . . . . . . . . . . . . . . 34

4.1.3 Riesz operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.2 Controllability and observability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.2.1 Controllability of linear DPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

v

4.2.2 Observability notions for linear DPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.2.3 Analysis of approximative controllability and observability . . . . . . . . . . . . . . . . . . . 48

5 Stability theory 59

5.1 Introduction to the stability theory of SVPn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.1.1 Stability of C0–semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.1.2 Exponential stability and the „spectrum determined growth assumption” . . . . . . . . . . 62

5.1.3 Lyapunov stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

6 Feedback control for PDE systems 79

6.1 Backstepping–based control design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6.1.1 Feedback control for a linear diffusion–reaction system using backstepping . . . . . . . . . 79

6.2 Spectral, or modal control design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6.2.1 Modal (or spectral) control design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

7 Frequency-domain analysis and design 95

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

7.2 Feedback–control design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

7.2.1 Mathematical fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

7.2.2 Input–output stability of the open loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

7.2.3 Input–output stability of the closed–loop system . . . . . . . . . . . . . . . . . . . . . . . . . 103

8 Flatness-based control 109

8.1 Finite–dimensional nonlinear control systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

8.2 Distributed parameter control systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

8.2.1 Trajectory planning for PDE systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

8.3 Operational calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

8.3.1 Flatness–based trajectory planning for the linear heat equation . . . . . . . . . . . . . . . . 112

8.3.2 Flatness–based trajectory planning for the linear wave equation . . . . . . . . . . . . . . . . 116

8.4 Riesz spectral operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

8.4.1 Flatness–based state and input parametrization . . . . . . . . . . . . . . . . . . . . . . . . . . 118

8.4.2 Application to the linear heat and wave equation with in–domain control . . . . . . . . . . 122

A Basic results from functional analysis 127

A.1 The Hille–Yosida theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

A.2 The Lumer–Phillips theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

vi

B Properties and correspondences of the Laplace transform 129

vii

viii

Chapter 1

Some preliminaries

1.1 Notation

In the following z = [z1 · · · zm]ᵀ ∈Ω denotes a vector pointing into the spatial domainΩ⊂Rm , an open set withboundary ∂Ω and closure Ω=Ω∪∂Ω. For one-dimensional spatial domains one has, e.g.,Ω= (a,b), ∂Ω= a,band Ω= [a,b]. In this case the spatial coordinate is just denoted by z instead of z1.

The partial derivatives of a function x : [a,b]× [0,∞) →R at a point (z, t ) ∈ [a,b]× [0,∞) are denoted by either ofthe following, equivalent notations

∂kz x(z, t ) = ∂k x

∂zk(z, t ), ∂k

t x(z, t ) = ∂k x

∂t k(z, t ), k ∈N0.

Thus, in particular ∂2z x = ∂2x

∂z2 . In the particular case of a function x : [0,∞) → R depending only on the time

t ∈ [0,∞) we also write ddt x = x and dx

dt = x. The explizit point of evaluation of functions is sometimes neglectedwhen it is clear from the context. A · indicates that no particular evaluation is considered, whereas when apoint (z, t ) is indicated it denotes the value of the function at this point. In consequence one has, e.g., that

x : [a,b]× [0,∞) →R

x( · , t ) : [a,b] →R for a given t ∈ [0,∞)

x(z, · ) : [0,∞) →R for a given z ∈ [a,b]

x(z, t ) ∈R for a given (z, t ) ∈ [a,b]× [0,∞).

1.2 Function spaces

The following function spaces play an important role in the following.

• Space of continuously differentiable functions C k (a,b): Let k ∈N and a,b ∈R. We denote by C k (a,b) theset of functions x : [a,b] →R that is k-times continuously differentiable, i.e., the function ∂r

t x : [a,b] →R

is differentiable with continuous differential ∂r+1t x : [a,b] → R for all r ≤ k − 1 and ∂k

t x : [a,b] → R iscontinous. The differential ∂t x of x at a point t ∈ (a,b) can be defined, e.g., through the limit of thedifference quotients (if it exists) by one of the following forms (Farlow, 1982; Dettman, 1988)

∂t x(t ) = limh→0

x(t +h)−x(t )

h(forward difference)


x(t )−x(t −h)

h(backward difference)

1


x(t +h)−x(t −h)

2h(central difference).

Higher order derivatives can be defined similarly by applying the above representations. In addition, e.g.,the second order derivative of a function x ∈C k (a,b) with k ≥ 2 can be defined as

∂2t x(t ) = lim

h→0

x(t +h)−2x(t )+x(t −h)

h2 .

Note that at the boundaries, i.e., for t ∈ a,b, some of the above difference quotients are not defined.Note that this also follows, e.g., considering Taylor series expansions.

• Space of Lebesgue measurable functions Lp (a,b): Let p ≥ 1 be a fixed integer and let a,b ∈R. We denote

by Lp (a,b) the set of Lebesgue measurable functions x : [a,b] →Rwith∫ b

a |x|p dt <∞ equipped with thenorm 1

‖x‖Lp =(∫ b

a|x|p dt

) 1p

. (1.1)

For p = 2 an inner product on L2(a,b) can be introduced by

⟨x, y⟩L2 =∫ b

ax ydt , ⟨x, x⟩L2 = ‖x‖2

L2 (1.2)

given x, y ∈ L2(a,b). If p =∞, then the norm is defined as

‖x‖L∞ = ess supt∈[a,b]

|x| (1.3)

provided, that esssupt∈[a,b] |x| <∞.

• Sobolev spaces H p (a,b): Let p ≥ 1 be a fixed integer and let a,b ∈R. The subspace of L2(a,b) defined by

H p (a,b) =

x ∈ L2(a,b) : ∂ jt x ∈ L2(a,b), j = 0,1, . . . , p

(1.4)

equipped with the inner product

⟨x, y⟩H p =p∑

j=0⟨∂ j

t x,∂ jt y⟩L2 (1.5)

is a Hilbert space. It can be shown, that H p (a,b) is the completion of C p (a,b) or C ∞(a,b) function withrespect to the norm (1.5). Note also the embedding H p+1(a,b) ⊂ H p (a,b). For details on Sobolev spacesand their properties the reader is referred to (Adams and Fournier, 2003).

References

Adams, R. and J. Fournier (2003). Sobolev Spaces. 2nd. Acadamic Press, Amsterdam (cit. on p. 2).Curtain, R. F. and H. Zwart (1995). An introduction to infinite-dimensional linear systems theory. Springer

(cit. on p. 2).Dettman, J. W. (1988). Mathematical Methods in Physics and Engineering. Dover Publications, McGraw-Hill

(cit. on p. 1).Farlow, S. J. (1982). Partial differential equations for scientists and eingeers. Dover, New York (cit. on p. 1).

1One actually needs to consider equivalence classes since for p > 1 the fact that ‖x‖Lp = 0 implies x = 0 only almost everywhere. Fordetails consult, e.g., (Curtain and Zwart, 1995).

2 Chapter 1 Some preliminaries

Chapter 2

Introduction

Partial differential equations (PDEs) can be considered as the fundamental mathematical description of manytechnical processes. In general, this distributed parameter description becomes an essential ingredient of themodeling and analysis process if the spatial or property–related distribution of the process variables can nolonger be neglected. Following the exposition in Meurer, 2013 some characteristic examples are summarizedbelow:

• chemical or biochemical reactors (Jakobsen, 2008) including three-way catalysts for exhaust gas after–treatment in automotive applications, (reactive) distillation and adsorption processes (Sundmacher andKienle, 2002), or activated sludge processes for wastewater treatment (Lee et al., 2006);

• thermal systems (Baehr and Stephan, 2006), LED heated silicon wafers (Kleindienst et al., 2018), or thereheating and cooling of metal slabs during the steel processing to achieve desired metallurgical changes(Unger and Tröltzsch, 2001);

• electrochemical systems including fuel cells (Sundmacher, Kienle, et al., 2007) and Li–ion or Li–polymerbattery devices for energy production and storage (Chen and Evans, 1996; Gu and Wang, 2000);

• smart materials, adaptive structures and resonant systems (Meirovitch, 1990; Banks et al., 1996; Preumont,2002);

• flexible structures in aerospace and mechanical applications such as adaptive or flapping wing structures(Stanewsky, 2001), micro–mechanic bending cantilevers in atomic force microscopes (Bining et al., 1986),or deformable mirrors in adaptive optics (Roddier(ed), 1999);

• fluid dynamical systems (Aamo and Krstic, 2003; Bewley, 2000), mixing processes and coupled fluid–structure interactions;

• wave propagation in optical fibers (Shaw, 2004) and traffic congestion (Whitham, 1999; Helbing, 2001);

• energy production in fusion reactors (Taylor, 1997; Ambrosino and Albanese, 2005).

The dynamic operation of these distributed parameter systems (DPSs) essentially relies on the incorporation ofsuitable control strategies to influence the system dynamics and to enlarge the operating range.

References

Aamo, O. and M. Krstic (2003). Flow Control by Feedback. Springer–Verlag, London (cit. on p. 3).Ambrosino, G. and R. Albanese (2005). „Magnetic control of plasma current, position, and shape in Tokamaks: a

survey of modeling and control approaches“. In: IEEE Contr Sys Magazine (25(5)), pp. 76–92 (cit. on p. 3).Baehr, H. and K. Stephan (2006). Heat and Mass Transfer. 2nd. Springer–Verlag, Berlin (cit. on p. 3).

3

Banks, H., R. Smith, and Y. Wang (1996). Smart Material Structures: Modeling, Estimation and Control. JohnWiley & Sons, Chichester (cit. on p. 3).

Bewley, T. (2000). „Flow control: New Challenges for a new Renaissance.“ In: Prog Aerosp Sci (37), pp. 21–58(cit. on p. 3).

Bining, G., C. Quate C., and Gerber (1986). „Atomic force microscope“. In: Phys Rev Letter (56(9)), pp. 930–933(cit. on p. 3).

Chen, Y. and J. Evans (1996). „Thermal analysis of lithium-ion batteries“. In: J Electrochem Soc (143(9)), pp. 2708–2712 (cit. on p. 3).

Gu, W. and C. Wang (2000). „Thermal–Electrochemical Modeling of Battery Systems“. In: J Electrochem Soc(147(8)), pp. 2910–2922 (cit. on p. 3).

Helbing, D. (2001). „Traffic and related self–driven many-particle systems“. In: Rev Mod Phys (73(4)), pp. 1067–1141 (cit. on p. 3).

Jakobsen, H. (2008). Chemical Reactor Modeling – Multiphase Reactive Flows. Springer Verlag, Berlin Heidelberg(cit. on p. 3).

Kleindienst, M., M. Reichhartinger, M. Horn, and F. Staudegger (2018). „Observer-based temperature control ofan LED heated silicon wafer“. English. In: Journal of Process Control 70, pp. 96–108 (cit. on p. 3).

Lee, T., F. Wang, and R. Newell (2006). „Advances in distributed parameter approach to the dynamics and controlof activated sludge processes for wastewater treatment“. In: Water Research (40(5)), pp. 853–869 (cit. onp. 3).

Meirovitch, L. (1990). Dynamics and Control of Structures. Wiley, New York (cit. on p. 3).Meurer, T. (2013). Control of Higher Dimensional PDEs. Communication and Control Engineering. Springer

(cit. on p. 3).Preumont, A. (2002). Vibration Control of Active Structures. 2nd. Kluwer Academic, Dordrecht (cit. on p. 3).Roddier(ed), F. (1999). Adaptive Optics in Astronomy. Cambridge University Press, Cambridge (cit. on p. 3).Shaw, J. (2004). Mathematical Principles of Optical Fiber Communications. SIAM, Philadelphia (cit. on p. 3).Stanewsky, E. (2001). „Adaptive wing and flow control technology“. In: Prog Aerosp Sci (37), pp. 583–667 (cit. on

p. 3).Sundmacher, K. and A. Kienle (2002). Reactive Distillation – Status and Future Directions. Wiley–VCH, Weinheim

(cit. on p. 3).Sundmacher, K., A. Kienle, H. Pesch, J. Berndt, and G. Huppmann (eds) (2007). Molten Carbonate Fuel Cells:

Modeling, Analysis, Simulation, and Control. Wiley–VCH Verlag GmbH, Weinheim (cit. on p. 3).Taylor, T. (1997). „Physics of advanced tokamaks“. In: Plasma Physics and Controlled Fusion (39(12B)), B47

(cit. on p. 3).Unger, A. and F. Tröltzsch (2001). „Fast Solution of Optimal Control Problems in the Selective Cooling of Steel“.

In: Z Angew Math Mech (81(7)), pp. 447–456 (cit. on p. 3).Whitham, G. (1999). Linear and Nonlinear Waves. John Wiley & Sons, New York (cit. on p. 3).

4 Chapter 2 Introduction

Chapter 3

Modelling with partial differentialequations

In this chapter a short overview of some forms partial differential equations (PDEs) is provided that typically arisein the mathematical description of technical systems. This also includes introducing analysis and classificationapproaches for distributed parameter systems (DPS) with regard to their analytical and numerical solution.

The following examples are intended as a first introduction to the mathematical description of DPS. For thispurpose, on the one hand, differences or connections to the description by concentrated parameter systems(CPS) and, on the other hand, a first insight into the system structures that are of interest for control engineeringare illustrated.

3.1 Transition from discrete to continuously distributed parameters

To show the transition from discrete to continuously distributed parameters in this section different importantapplication examples are considered.

3.1.1 Multi-body oscillator

3.1.1.1 Concentrated-parameter description

Consider a system consisting of n bodies with mass m coupled through springs in a linear row as shown inFigure 3.1 where x j (t) denotes the deviation of mass j , and F j (t) the external force acting on this mass. It is

cm

-x1, F1

c

. . .

cm

-x j , F j

c

. . .

cm

-xn , Fn

-scFL(t )

-xn+1

Figure 3.1: Multi-body oscillator with discrete distributed parameters.

considered that each spring has a stiffness c. The kinematic analysis and the application of Newton’s secondlaw directly yield the equations of motion for the system in form of

5

• n coupled second-order differential equations

mx j (t ) =−c[−x j+1(t )+2x j (t )−x j−1(t )

]+F j (t )

• 2n initial conditions (ICs)

x j (0) = x j ,0, x j (0) = x j ,1 j = 1,2, ...,n

• 2 boundary conditions (BCs)

j = 0 : x0 = 0

j = n +1 : FL(t )+ c(xn(t )−xn+1(t )) = 0.

The n second order equations can also be rewritten in form of 2n coupled first order ordinary differentialequations (ODEs). In consequence the system is denoted as a system with concentrated parameters of order 2n.The concentrated parameters are explicitely the masses m, stiffnesses c, forces F j and resulting deviations x j ,j = 1, . . . ,n.

3.1.1.2 Distributed parameter description

In the sequel we analysis how the above description changes when considering infinitessimal elements ofmass mdz as illustrated in Figure 3.2 interconnected by infinitessimal small springs of stiffness c/dz. At each

z = 0

cdz

mdz

-x(dz, t ), F dz

dz

cdz

. . .

cdz

mdz

-x(z, t ), F dz

z

cdz

. . .

cdz

mdz

-x(L, t ), F dz

L

-sc

dzFL(t )

Figure 3.2: Multi-body oscillator with continuously distributed parameters.

infinitessimal mass the load F associated to the force F dz is acting and additionally the external force FL isimposed at the right end of the oscillator chain. Newtons second law applied to the mass element at the positionz yields

• 1 second order partial differential equation for z ∈ (0,L), t > 0

mdz∂2t x(z, t ) =− c

dz[x(z, t )−x(z −dz, t )+x(z, t )−x(z +dz, t )]+ F dz.

Dividing by dz and taking the limit for dz → 0 implies

m∂2t x(z, t ) = c lim

dz→0

x(z +dz, t )−2x(z, t )+x(z −dz, t )

dz2︸︷︷︸=∂2

z x(z,t )

+F = c∂2z x(z, t )+ F

• 2 initial conditions (ICs)

x(z,0) = x0(z), z ∈ [0,L]

∂t x(z,0) = x1(z).

where the order of the distributed parameter system is given by ∞, according to the continuous indepen-dent variable z ∈ [0,L].

6 Chapter 3 Modelling with partial differential equations

• 2 boundary conditions (BCs)

The particular boundary conditions are depending on the specific constructive constraints, leading toone of the following types:

– Imposing the value of the function (Dirichlet–BC)

x(L, t ) = 0

– Imposing the slope (Neumann–BC)

limdz→0

− c

dz[x(L, t )−x(L+dz, t )] = c∂z x(L, t ) = FL(t )

– Mixed BC

cL[xL(t )−x(L, t )] = c∂z x(L, t )

cdz

mdz

-

x(L, t )

-scL

xL(t )

In difference to the n–body oscillator the consideration of infinitessimal mass elements yields a description inform of a partial differential equation, which is also called a wave equation. In particular it should be noticedthat due to the continuous independent variable z ∈ [0,L] the system has infinite order. In consequence systemswith distributed parameters are also referred to as infinite-dimensional systems.

3.1.2 Electrical transmission line

Consider the simplified electrical RC transmission line depicted in Figure 3.3 with supplied current i0(t) attime t ≥ 0. The end of the line is closed with a resistance R . At time t = 0 the transmission line is assumed to befree of charge. Application of Kirchoff’s laws to the differential line element yiels, in analogy to the mechanical

- e

e p p p p p pu u

p p p - - p p pu u e

e? ? ? ?

- - -

i0(t ) i (z, t )R dz

i (z+dz, t ) i (L, t )

x(0, t ) x(z, t ) C dz x(z+dz, t ) x(L, t ) R

0 z z+dz L

Figure 3.3: Differentiel RC –line element.

multi-body oscillator considered in the last section, the partial differential equations for the current i (z, t ) andthe voltage x(z, t ) at time t > 0 and z ∈ (0,L).

The current law implies that

i (z, t )− i (z +dz, t )︸︷︷︸=−∂z i (z, t )dz +O2(dz)

−C dz∂t x(z +dz, t ) = 0

and the voltage law that

Rdz i (z, t )+ x(z +dz, t )−x(z, t )︸︷︷︸= ∂z x(z, t )dz +O2(dz)

= 0

3.1 Transition from discrete to continuously distributed parameters 7

Dividing by dz and considering the limit dz → 0 yields the following system of first-order partial differentialequations

∂t x(z, t )+ 1

C∂z i (z, t ) = 0, z ∈ (0,L), t > 0 (3.1a)

∂z x(z, t )+ Ri (z, t ) = 0, z ∈ (0,L), t > 0. (3.1b)

From the consideration that the line is free of charge at t = 0 the initial condition x(z,0) = 0 follows. Theboundary conditions can be determined from the current source at z = 0 and the application of Ohm’s law atz = L, and read consequently

i (0, t ) = i0(t ), x(L, t ) = Ri (L, t ), t ≥ 0. (3.1c)

It should be mentioned at this place that the system of coupled PDEs (3.1) can be rewritten to a single secondorder PDE by eliminating i (z, t ) with inhomogeneous Dirichlet boundary condition at the left end and mixed(or Robin) boundary conditions at the right end

∂t x(z, t )− 1

RC∂2

z x(z, t ) = 0, z ∈ (0,L), t > 0 (3.2a)

x(0, t ) = Rio(t ), ∂z x(L, t ) =− R

Rx(L, t ), t > 0. (3.2b)

Interestingly this equation is equivalent to the heat equation that is derived in the following section.

3.1.3 Heat conduction and diffusion

One of the classical examples for the analysis of control design methods for PDEs is the heat conductionor diffusion equation. This is used in the following to determine the temperature distribution in a simplyconnected three-dimensional region. Here the heat conducting body with geometryΩ and arbitrary subvolumeV (shown in Figure 3.4) is considered. The outer normal in the considered area element dA is denoted by n. Tosolve the heat conduction problem it is necessary to solve the temperature distribution or the temperature fieldT (z , t ) with z = (z1, z2, z3) in its spatio-temporal dependency. The first law of thermodynamics states that the

V

dA

n

Figure 3.4: Region of a heat conducting body with volume V . Differential surface element dA with outward normal vectorn.

temporal change of internal energy U corresponds to supplied minus the sum of the dissipated heat flux, i.e.

d

dtU (t ) = Q(t )+P (t ), t > 0 (3.3)

where Q denotes the heat flux and P the (mechanical or electrical) power that is supplied to the body throughthe surface. For the considered incompressible body it results for the internal energy that

d

dtU (t ) = d

dt

∫VρudV = ρ

∫V

d

dtudV


with the specific internal energy u which is determined by the isochore caloric state equation du = c(T )dT withthe specific heat capacity c(T ). In consequence it holds that

d

dtU = ρ

∫V

c(T )∂t T dV.

The supplied heat flux Q results from inspecting the differential area element dA. Especially for the heat fluxentering the element it holds that dQ =−qndA with the heat flux density q . It is important to note here, thata heat flux is counted positive if entering the area. In this case the vector q points towards the interior of thebody whereas the corresponding normal vector n points outwards so that q ·n ≤ 0. In consequence, with thenegative sign it results that dQ > 0. The complete heat flux Q can be determined by integration of all differentialheat fluxs dQ, i.e.

Q =−∫

AqndA =−

∫V

divqdV ,

where the divergence theorem (Gauß integral identity) was used.

The supplied power P consists of one part PV causing a change in the volume and one part Pdiss, correspondingto dissipated energy. For the considered incompressible body it holds that PV = 0. Part of the dissipated energyPdiss is, e.g., the so-called Joule heating associated to loss of energy in electrically conducting materials inconsequence of the electric resistance. According to (Baehr and Stephan, 2006; Smith and van Ness, 1975) forthese irreversible energy transformations in the interior of the body the following approach is used

Pdiss(t ) =∫

VW (T, z , t )dV ,

where W (T, z , t ) denotes the so-called power density.

Substituting these results into the energy balance (3.3) yields∫V

(ρc(T )∂t T +divq −W (T, z , t )

)dV = 0

Given that the balance volume V is arbitrary, it necessarily follows that

ρc(T )∂t T =−divq +W (T, z , t ).

Using Fourier’s law of heat conduction (Baehr and Stephan, 2006; Smith and van Ness, 1975)

q =−λ(T )gradT,

with λ(T ) being the thermal conductivity it follows that

ρc(T (z , t ))∂t T (z , t ) = div[λ(T (z , t ))gradT (z , t )

]+W (T (z , t ), z , t ) (3.4)

what is the well-known heat equation in a three-dimensional domain.

In particular, for a one-dimensional spatial domain and constant parameters this equation simplifies to

∂t T (z, t ) = λ

ρc∂2

z T (z, t )+ W (T (z, t ), z, t )

ρc. (3.5a)

For the complete determination of the temperature field T (z , t ) one further requires an initial condition

T (z ,0) = T0(z), z ∈Ω (3.5b)

and boundary conditions. The boundary conditions again depend on the particular setup (cp. Section 3.1.1.2)and can be characterized as any of the following:

• Dirichlet–boundary condition T (z , t ) = TΣ(z , t ), z ∈Σ⊆ ∂Ω,

• Neumann–boundary condition −λ(T (z , t ))gradT (z , t ) = qΣ(z , t ), z ∈Σ⊆ ∂Ω,


• mixed (Robin) boundary condition −λ(T (z , t ))gradT (z , t ) =α [TΣ(z , t )−T (z , t )], z ∈Σ⊆ ∂Ω.

The last one described, e.g., the transfer with a body of another medium and temperature TΣ(z , t). For moredetailed derivations, particular parameter values and additional (physical) analysis the reader is referred, e.g.,to (Baehr and Stephan, 2006; Smith and van Ness, 1975).

3.1.4 Diffusion–Convection–Reaction–Systems

Fixed bed reactors represent the most important and widely used type of reactor in chemical industry. Asschematically represented in Figure 3.5 a fluid(e.g., a gas or a liquid) is passed through the reactor that is filledwith catalytic material. The reaction thus takes place not in the fluid phase but on the surface of the catalyst.

z z +dzz

Catalyst

Empty space

Cooling jacket

T, w j

p,ρ, v

Figure 3.5: Differential reaktor element.

Depending on the desired level of detail, models with an arbitrary complexity can be derived describing thechemical, thermal and fluid mechanical processes taking place in the reactor (see, e.g., (Jakobsen, 2008)). Forthis one has to take into account global and componentwise mass balances, energy or enthalpy balance, impulsebalances, and thermodynamics equations of state. Given that highly complex models are hardly accessiblefor control design, different simplifications are typically introduced related to fluid flow in the reactor (e.g.,plug flow assumption), dimension (e.g., one–dimensional axial-dispersion instead of three–dimensional spatialdispersion) as well as the type of the reaction rates.

To illustrate the main effects and basic nomenclature in the following strongly simplified model equations aregiven.

∂t x(z, t ) = D∂2z x(z, t )︸︷︷︸

Diffusion

+C∂z x(z, t )︸︷︷︸Convection

+D(u(z, t )−x(z, t ))︸︷︷︸Transfer

+R(z, t , x(z, t ))︸︷︷︸Reaction

, z ∈ (0,L), t > 0

R0∂z x(0, t ) = u0(t )−x(0, t ), t > 0

RL∂z x(L, t ) = 0, t > 0

x(z,0) = x0(z), z ∈ [0,L].

The state x comprises typically concentrations of the participating components as well as the reactor tempera-ture and cooling jacket temperature. A derivation of a one-dimensional axial-dispersion reactor is consideredin Exercise 1.


3.1.5 Flexible mechanical structures

A prototypical application area of distributed-parameter systems and control theory are given by flexiblemechanical and mechatronic structures. Examples are given by flexible and foldable aerospace structures,deformable mirrors adaptive optics, so-called "smart structures", which contain integrated embedded actuatorsand sensors, or supporting and crane structures.

The following is a short introduction to energy-based modeling of distributed parameter mechanical systemsusing the (extended) Hamilton’s principle (see e.g. (Reddy, 1984; Meirovitch, 1997)). To use this approach ashort digression into the calculus of variations is necessary.

In general terms, the (extended) Hamilton’s principle provides the equations of motion of a mechanical systembased on the corresponding kinetic energy Wk , the potential energy Wp and the virtual work of the externaland non-conservative forces Wnc . The Hamilton’s principle says that from the set of all permissible paths, i.e.the possible local and temporal evolutions of the dependent variables that define the geometric boundaryconditions, for the true path the following action integral

IH =∫ t2

t1

[Wk −Wp +Wnc

]dt (3.6)

for any t1 < t2, t1, t2 ∈ R becomes extremal. Here the term L = Wk −Wp provides the so-called associatedLagrange function.

An alternative and for the actual calculation directly accessible formulation results from the calculus of variation.Accordingly, the equations of motion of the system correspond to the solution for which at any time instancest1, t2 the variation of the above integral with respect to the dependent variables x j (z, t ), j = 1,2, ..., J is identicalto zero (Meirovitch, 1997). For this, consider a family of small deviations around the function x parametrizedby a scalar α, i.e., x +α∆x for some arbitrary (up to a certain regularity or smoothness) function ∆x with∆x( · , t1) =∆x( · , t2) = 0 and letting the first variation1 satisfy

δIH := dIH (x +α∆x)

dα

∣∣∣∣α=0

= 0. (3.7)

With the above definition of IH given in (3.6) this implies that

δIH =∫ t2

t1

[δWk −δWp +δWnc

]dt = 0 (3.8a)

with

δx j (z, t1) = δx j (z, t2) = 0, j = 1.2, ..., J . (3.8b)

The operator δ is called variation operator, while δx =α∆x describes the variation of x.

The basic procedure for deriving the equations of motion for distributed-parameter mechanical systems is nowillustrated for the example of a bending beam.

Example 3.1 (Bending vibration of a beam). Using the example of the transversal bending vibration of thebeam shown in Figure 3.6 the application of Hamilton’s principle is illustrated. It should be noted that inthe subsequent development the potential energy of the system is given under the classical Euler–Bernoulli-assumption. In general, however, results from continuum mechanics must be used to determine the potentialenergy (see, e.g., (Reddy, 1984; Meirovitch, 1997) and references therein).

The considered beam of length L, mass per unit length m(z), modulus of elasticity E and moment of inertiaI (z) is clamped at z = 0 and excited by a surface area load f (z, t ). The influence of gravity is neglected in thefollowing, but can be directly integrated into the formulation of the potential energy.

1In fact the first variation corresponds to the first Gateau derivative of the functional IH in the direction of ∆x.


zx(z, t )

f (z, t )

m(z), E I (z)

Figure 3.6: Bending vibration of a beam.

For the kinetic and potential energy of the beam it follows that (Meirovitch, 1997)

Wk (t ) = 1

2

∫ L

0m(z) (∂t x(z, t ))2 dz, Wp (t ) = 1

2

∫ L

0E I (z)

(∂2

z x(z, t ))2

dz.

The virtual work of the external and non-conservative force f (x, t ) results in

Wnc (t ) =∫ L

0f (z, t )x(z, t )dz.

For given t1, t2 ∈R this yields the action integral

IH =∫ t2

t1

∫ L

0

(1

2m(z) (∂t x(z, t ))2 − 1

2E I (z)

(∂2

z x(z, t ))2 + f (z, t )x(z, t )

)dz. (3.9)

Denoting the variation in ∂t x by δxt and in ∂2z x as δxzz , the first variation according to (3.8) reads

δIH =∫ t2

t1

∫ L

0

[m(z)∂t x(z, t )δxt (z, t )−E I (z)∂2

z x(z, t )δxzz (z, t )+ f (z, t )δx(z, t )]

dzdt .

Considering δxt = ∂tδx and δxzz = ∂2zδx, the partial integration of the first term regarding t , taking into

account that δx( · , t1) = δx( · , t2) = 0 yields∫ t2

t1

∫ L

0m(z)∂t x(z, t )δxt (z, t )dz =

∫ L

0m(z)

[∂t x(z, t )δx(z, t )

]t2

t1dz

−∫ t2

t1

∫ L

0m(z)∂2

t x(z, t )δx(z, t )dzdt

=−∫ t2

t1

∫ L

0m(z)∂2

t x(z, t )δx(z, t )dzdt

The double partial integration of the second term with respect to z

−∫ t2

t1

∫ L

0E I (z)∂2

z x(z, t )δxzz (z, t )dzdt =−∫ t2

t1

[E I (z)∂2

z x(z, t )δxz (z, t )]L

0dt

+∫ t2

t1

∫ L

0∂z

(E I (z)∂2

z x(z, t ))δxz (z, t )dzdt

=−∫ t2

t1

[E I (z)∂2

z x(z, t )δxz (z, t )]L

0dt

+∫ t2

t1

[∂z

(E I (z)∂2

z x(z, t ))δx(z, t )

]L

0dt

−∫ t2

t1

∫ L

0∂2

z

(E I (z)∂2

z x(z, t ))δx(z, t )dzdt


Summarizing one obtains for the first variation of the action integral

δIH =∫ t2

t1

∫ L

0

[−m(z)∂2t x(z, t )−∂2

z

(E I (z)∂2

z x(z, t ))+ f (z, t )

]δx(z, t )dzdt

−∫ t2

t1

[E I (z)∂2

z x(z, t )δxz (z, t )−∂z(E I (z)∂2

z x(z, t ))δx(z, t )

]L0 dt .

From the main theorem of the calculus of variations (given that δx is an arbitrary function, up to someregularity requirements) it follows that

m(z)∂2t x(z, t )+∂2

z

(E I (z)∂2

z x(z, t ))= f (z, t ), z ∈ (0,L), t > 0 (3.10a)

and the boundary conditions

either ∂z x = 0 or M = E I (z)∂2z x = 0 at z = 0 or z = L,

either x = 0 or Q =−∂z(E I (z)∂2

z x)= 0 at z = 0 or z = L.

Hereby M and Q indicate the moment and the lateral force. From the geometrical boundary conditions of theclamping, i.e.

x(0, t ) = 0, ∂z x(0, t ) = 0, t > 0 (3.10b)

thus follow the so-called natural boundary conditions of the system directly to

E I (L)∂2z x(L, t ) = 0, −∂z

(E I (L)∂2

z x(L, t ))= 0. t > 0 (3.10c)

confirming a well-known result from classical mechanics, namely that at the free boundary no moments andlateral forces are transmitted. The equation of motion of the beam thus results from the equation set (3.10).For a complete system description additionally consistent initial conditions

x(z,0) = x0(z), ∂t x(z,0) = x1(z), z ∈ [0,L] (3.11)

are required. Note that typically it is assumed that the initial conditions satisfy the boundary conditions. Itshould be mentioned that the partial differential equation (3.10a) of the bending beam are also known asbiharmonic differential equation.

3.2 Classification of second order PDEs and the method of characteris-tics

The classification of PDEs allows a characterization of the dynamic solution behavior, from which efficient(analytical) solution methods especially for so-called hyperbolic PDEs can be derived. Furthermore, from thenumerical point of view, the classification allows to determine adequate numerical solution methods, sincefor each of the types of PDEs different, specific numerical methods exist, or at least specific criteria have tobe taken into account (cp. Section 3.3). With regard to control applications the classification is also of greatimportance in order to account for the different dynamic behavior of solutions of the PDEs.

3.2.1 Semilinear PDEs in 2 coordinates z and t

To motivate and interpret the classification of PDEs in the following a linear or semilinear second order PDE inthe coordinates z and t is considered

a11∂2z x +2a12∂t∂z x +a22∂

2t x +b1∂z x +b2∂t x + cx = f (z, t , x), (3.12)

where ai j , b j , i , j = 1.2 are known functions in z and t . The so-called Cauchy problem consists of finding asolution x :Ω× [0,∞) of (3.12) which has consistent values of x, ∂z x, ∂t x on a curve Γ of the (z, t )-plane. TheCauchy problem is called well-posed if there exists a unique solution that continuously depends on the initial

3.2 Classification of second order PDEs and the method of characteristics 13

data on Γ. For a curve given in parameter form

Γ : (z, t ) = (α(ζ),β(ζ)), ζ ∈R

in the following we consider initial values according to

x|Γ = h(ζ), ∂z x|Γ =ψ(ζ), ∂t x|Γ =φ(ζ). (3.13)

Obviously it holds along the curve Γ that

dx

dζ

∣∣∣Γ= h′(ζ) = ∂z x|Γα′(ζ)+∂t x|Γβ′(ζ) =ψ(ζ)α′(ζ)+φ(ζ)β′(ζ), (3.14)

where ( · )′ represents the derivative with respect to ζ. This makes it easy to see that only two of the functions h,ψ, φ can be selected independent of each other. Equation (3.14) thus provides a consistency– or compatibilitycondition for the initial values. Similar conditions apply to the higher derivatives along Γ

d

dζ∂z x

∣∣∣Γ=ψ′(ζ) = ∂2

z x|Γα′(ζ)+∂z∂t x|Γβ′(ζ) (3.15)

d

dζ∂t x

∣∣∣Γ=φ′(ζ) = ∂z∂t x|Γα′(ζ)+∂2

t x|Γβ′(ζ) (3.16)

...

A solution x from (3.12), (3.13) therefore has to satisfy the system of equationsa11 2a12 a22

α′(ζ) β′(ζ) 0

0 α′(ζ) β′(ζ)

︸︷︷︸

=:Λ

∂2

z x|Γ∂t∂z x|Γ∂2

t x|Γ

=

f (α(ζ),β(ζ), x|Γ)− (b1∂z x|Γ+b2∂t x|Γ+ cx|Γ)

ψ′(ζ)

φ′(ζ)

(3.17)

which can be uniquely solved provided that

detΛ= a11β′2(ζ)−2a12α

′(ζ)β′(ζ)+a22α′2(ζ) 6= 0. (3.18)

The curve Γ is called a characteristic curve or characteristic direction if detΛ= 0 along Γ or non-characteristiccurve or non-characteristic direction if detΛ 6= 0 along Γ (John, 1982; Farlow, 1982).

Along any non-characteristic curve Γ all second order derivatives can be determined from (3.17) in termsof x,∂z x,∂t x along Γ. Thus in a similar way all derivatives of higher order (existence assumed) on Γ can bedetermined, so that in each point (z0, t0) of Γ the solution of the Cauchy problem (3.12), (3.13) can be obtainedin form of a formal power series in (z − z0), (t − t0). For analytical solutions x this results in a direct formulationof the power- or Taylor series expansion in the neighborhood of a each point (z0, t0) on Γ. However, this requiresanalytic initial values (3.13) and leads to the Cauchy–Kovalevskaya (existence) theorem (John, 1982).

If the initial data (3.13) is given along a characteristic curve Γ, the Cauchy problem (3.12) in general does nothave a unique solution. However, from detΛ= 0 further insights into the solution behavior and accordinglyinto the classification of the PDE and the existence of solutions can be obtained. In particular, because ofα′(ζ) = dz/dζ and β′(ζ) = dt/dζ equation (3.18) is equivalent to

a11dt 2 −2a12dtdz +a22dz2 = 0.

Considering dz > 0, after dividing the preceding equation by dz, this results in the equation for determining thecharacteristic curve Γ as solution of the ordinary differential equation

dt

dz=

a12 ±√

a212 −a11a22

a11. (3.19)


According to (John, 1982) this allows the classification2 of the PDE (3.12) in

• hyperbolic, if a212 −a11a22 > 0;

• elliptic, if a212 −a11a22 < 0;

• parabolic, if a212 −a11a22 = 0 with b2 6= 0.

For constant ai j , b j , i , j = 1,2 the solutions of (3.19) correspond to straight lines t (z) = mz+c in the (z, t )-plane

with slope m = (a12 ±√

a212 −a11a22)/a11.

-

6

z

t

hyperbolic

@@

@@@@

-

6

z

t

parabolic

heat equation

-

6

z

t

elliptic

Figure 3.7: Progression of the characteristics in the (z, t )-plane in case a hyperbolic, parabolic or elliptical PDEs of the 2ndorder

Furthermore in the hyperbolic case it can be shown that the Cauchy problem is well-posed, i.e., has a uniquesolution that continuously depends on the initial data, if the initial data are not given on a characteristic curve(see example 3.2 below).

It has to be emphasized here, that from a system theoretic or a control engineering point of view, only parabolicand hyperbolic PDEs are of interest.

Example 3.2 (Linear wave equation). In the following the results of the previous section is used to analyzethe linear wave equation.

The comparison of the wave equation

∂2t x(z, t ) = c2∂2

z x(z, t )

with the general form of the PDE (3.12) implies

a11 =−c2, a12 = 0, a22 = 1.

Because of a212 − a11a22 = c2 > 0 the wave equation is hyperbolic according to the above classification. As

shown in the following, this allows the determination of a normal form of the wave equation, through theintroduction of new coordinates. From the normal form one can directly create a closed solution of theassociated initial value problem.

• Determination of characteristics: With the parameters of the wave equation, the evaluation (3.19) yields

dt

dz=±1

c= const.,

2The name of the classification takes place according to the classification of conic sections.


so that with t (z0) = 0, z0 ∈R (arbitrary but fixed) the family of curves in the (z, t )-plane is given by

Γ1,2 : t =±1

c(z − z0).

-

6

z

t

@@

@@

@@

• PDE normal form of the wave equation: Based on Γ1,2 one can introduce new coordinates:

Γ1 : t = 1

c(z −η) ⇒ η= z − ct

Γ2 : t =−1

c(z −ζ) ⇒ ζ= z + ct .

In this new (η,ζ) coordinate system, the PDE of the wave equation has a particularly simple form. In thefollowing let x(z, t ) = x(η(z, t ),ζ(z, t )), so that

∂t x = ∂ηx∂tη+∂ζx∂tζ= c(∂ζx −∂ηx

), ∂z x = ∂ζx +∂ηx

The wave equation in new coordinates reads

0 = ∂2t x − c2∂2

z x = (∂t − c∂z ) (∂t + c∂z ) x =−4c2∂η∂ζx.

Due to its simple structure, the PDE on the right side of the last identity is called normal form. Thesolution can be directly obtained as x(η,ζ) = p(η)+q(ζ) with any 2-times continuously differentiablefunctions p and q. In the original coordinates this corresponds to the solution

x(z, t ) = p(z − ct )+q(z + ct ).

• Well posedness and solution of the initial value problem (method of characteristics):

For the choice of the initial conditions, two cases are considered in the following:

– The initial values are not given on Γ1,2. The initial values x(z,0) = x0(z) and ∂x/∂t (z,0) = x1(z) areconsidered. To determine the general solution x(z, t ) the functions p( · ) and q( · ) can be determinedfrom the conditions

p(z)+q(z) = x0(z), c(q ′(z)−p ′(z)

)= x1(z).

This leads to the so-called D’Alembertian solution of the initial value problem

x(z, t ) = 1

2

(x0(z − ct )+x0(z + ct )+ 1

c

∫ z+ct

z−ctx1(τ)dτ

).

– The initial values are given on Γ1,2. In this case, for simplicity, the initial values are consideredin (η,ζ)–coordinates at x(η,0) = 0 and ∂ζx(η,0) = 0. It immediately follows that both x(η,ζ) = 0and for example x(η,ζ) = g (ζ) with g (0) = 0 and dg /dζ(0) = 0 are solutions for the PDE normalform −4c2∂η∂ζx = 0. Thus no unique solution of the initial value problem can be determined.Accordingly, the Cauchy problem in this configuration, i.e. with initial values on the characteristiccurve, is not well-defined.


The above results for the classification of semilinear second order PDEs can in principle be extended to the caseof n > 2 coordinates zi and quasilinear second order PDEs (i.e., linear in the highest derivatives with coefficientsdepending on the state and its 1st order derivatives). However, the derivation requires a somewhat greatereffort, so that the reader is referred, e.g., to (John, 1982; Whitham, 1999). Furthermore, for quasilinear systemsfurther restrictions need to be considered, in particular to ensure the existence of solutions.

3.2.2 Types of partial differential equations

The summary below provides a compact overview with examples for the individual classes of partial differentialequations.

Elliptic PDEs

Physical model Equation Characteristics

Spatially two-dimensionalpotential distributions

∂z21 x +∂z2

2 x = f (x,u)Examples:

• stationary electric or magnetic field(Laplace or Poission equation)

• Stationary balance processes (e.g. 2-D heat equation)

• Gravitational field

dz1dz2

=±p−1(no pyhsical meaning)

Parabolic PDEs


Balance processes ∂t x −a∂2z x = f (x,u)

Examples:

• heat equation

• diffusion

• Electric RC-circuit(cp. Sect. 3.1.2)

Balance processes with trans-port

∂t x −a∂2z x −b∂z x = f (x,u)

Examples:

• diffusion-convection-reaction sys-tems

dt = 0(1 family)


Hyperbolic PDEs


Transport system ∂t x + v∂z x = f (x,u)Transport velocity vdelay L/v

dzdt = vz(t ) = z0 + v t(1 family)

Harmonic oscillationsWave propagation

∂2t x −av∂2

z x = f (x,u)Examples

• Electric LC linea = 1/(LC )

• pressure/sonic wavesSpeed of sound

pa = c

• Vibrating string

Damped oscillations and waves ∂2t x +a1∂z x +a2∂

2z x +a3∂z∂t x = f (x,u)

damping constant a2

dzdt =±paz(t ) = z0 +

pat

z(t ) = z0 −p

at(2 families)

Biharmonic PDEs


Biharmonic oscillation µ∂2t x +α1∂t x +∂2

z

(β∂2

z x)+α2∂t∂

4z x

−∂z(γ∂z x

)= f (x,u)Examples:

• Elastic beam (cp. ex. 3.1)mass/length µdamping constant αi

(i = 1 external, i = 2 internal)bending stiffness βlongitudinal force γ

3.3 Numerical solution methods

From a numerical point of view, the solution of the Initial Boundary Value Problems (IBVP) in general thefollowing approaches can be followed (see e.g. (Larsson and Tmoée, 2003; Farlow, 1982))

• discretization :– Semi-discretization (”method of lines”) regarding the spatial coordinate(s) and numerical solution

of the obtained system of ordinary differential equations (numerical stability conditions determinethe choice of space and time increment)

– full-discretization regarding the local and temporal coordinates and solution of the obtained systemof algebraic equations e.g. Crank–Nicholson procedure (often absolutely stable, i.e. independent ofthe choice of space and time step size, but more complex)

• Weighted residual methods and derived therefrom the Finite Element Methods (FEM)

Example 3.3 (Semi–discretization of the heat equation). For the benchmark example of the linear heatconduction equation the principle procedure for explicit semi-discretization using the finite difference methodis introduced. The temperature x(z, t ) in the heat conductor is described by the PDE with BCs and IC

∂t x(z, t ) = a∂2z x(z, t )+u(z, t ), z ∈ (0,1), t > 0 (3.20)


x

0 ∆z 2∆z (N −1)∆z 1 z

x0(t ) x1(t ) x2(t )

xN−1(t )xN (t )

Figure 3.8: Spatial discretization

∂z x(0, t ) = 0, x(1, t ) = v(t ), t > 0 (3.21)

x(z,0) = x0(z), z ∈ [0,1]. (3.22)

The basic procedure is shown in the figure 3.8.

Substituting the following approximations for the first and second order derivatives (motivated by the forwardand backward Taylor series expansion in z)

∂z x(z, t )|z=iδz =xi+1(t )−xi−1(t )

2δz+O2(δz) or

xi+1(t )−xi (t )

∆z+O (∆z)

∂2z x(z, t )

∣∣z=iδz =

xi+1(t )−2xi (t )+xi−1(t )

∆z2 +O2(∆z)

into the PDE (3.20) yields

d xi

d t(t ) = a

∆z2 (xi+1(t )−2xi (t )+xi−1(t ))+u(zi , t ), i = 0, . . . , N −1.

The corresponding evaluation of the BCs (3.21) reads

x0(t )−x−1(t )

∆z= 0 ⇒ x−1(t ) = x0(t )

xN (t ) = v(t )

This results in an N –dimensional system of ordinary differential equations for xi (t), i = 0, . . . , N −1 of theform

d x0

d t(t ) = a

∆z2 (x1(t )−x0(t ))+u(0, t )

d xi

d t(t ) = a

∆z2 (xi+1(t )−2xi (t )+xi−1(t ))+u(zi , t ), i = 1, . . . , N −2

d xN−1

d t(t ) = a

∆z2 (v(t )−2xN−1(t )+xN−2(t ))+u(zN−1, t )

for t > 0 with the ICs

xi (0) = x0(zi ), i = 0, . . . , N −1.

3.3 Numerical solution methods 19

Note that this system of ODEs can be compactly written as

x = Ax +Bu, x(0) = x0

with

x =

x0

...

xN−1

, u =

u0

...

uN−1

v

, A = a

∆z2

−1 1 0 · · · 0

1 −2 1. . .

...

0. . .

. . .. . . 0

.... . . 1 −2 1

0 · · · 0 1 −2

, B =

1 0 · · · 0 0

0. . .

. . ....

......

. . .. . . 0 0

0 · · · 0 1 a∆z2

The basic procedure of semi-discretization can thus be summarized as follows:

(i) Transfer of the PDE into a system of ODEs, whose numerical solution is a standard task of numerics andthere exist efficient standard solvers that can be used.

(ii) Appropriate choice of the time increment ∆t of the numerical integrators and the spatial step size ∆z.These are usually not independent of each other. In particular, the numerical stability for example3.3 requires the condition ∆t ≤ ∆z2/2 (Larsson and Tmoée, 2003; Farlow, 1982). In general the semi-discretization does not rely on an absolutely stable numerical procedure.

Furthermore, it should be noted that in addition to the above used difference quotients, also approximations ofhigher order can be employed. While this yields an increased accuracy, the advantageous three-band structureof the obtained system matrix A is lost.

Example 3.4 (Semi–discretization of a semilinear heat equation). Based on the designs in example 3.3 willbe the Semi-discretization of a semilinear PDE of the form

∂t x(z, t ) = a∂2z x(z, t )+ f (x(z, t ))+u(z, t ), z ∈ (0,1), t > 0 (3.23)

with BCs and IC analog to (3.21) and (3.22). An analogous procedure under consideration of of the locallyevaluated nonlinearity f (x(z, t )) lists

d x0

d t(t ) = a

∆z2 (x1(t )−x0(t ))+ f (x0(t ))+u(0, t )

d xi

d t(t ) = a

∆z2 (xi+1(t )−2xi (t )+xi−1(t ))+ f (xi (t ))+u(zi , t ), i = 1, . . . , N −2

d xN−1

d t(t ) = a

∆z2 (v(t )−2xN−1(t )+xN−2(t ))+ f (xN−1(t ))+u(zN−1, t )

for t > 0 with the IC

xi (0) = x0(zi ), i = 0, . . . , N −1.

or equivalently

x = Ax +Bu + f (x), x(0) = x0, fi (x) = f (xi ), t > 0.

As shown in the characterization of PDEs in Section 3.2, the (numeric) solution of hyperbolic PDEs requires amuch larger effort, eventhough in some cases it can be addressed using the methods of lines (semidiscretization)approach up to some limitations. This is mainly due to their dynamic properties which is characterized by thepropogation of characteristic wave phenomena. For linear hyperbolic PDEs especially the numerical resolutionof the propagation of discontinuities along characteristics is a difficulty. For the case of quasilinear andnonlinear hyperbolic PDEs new effects like shock waves and dilution waves, whose numerical approximationrequires special approaches, e.g., so called “upwind”– or high-resolution processes). For the case that the initialdata is not provided on a characteristic the above semi-discretization or full discretization methods can beemployed.


In addition to the many numerical solution packages available some more specific software tools can also beemployed as, e.g.,

• CENTPACK and CLAWPACK: for quasilinear systems of hyperbolic PDEs in 1–2(3) spatial dimensions, basedon Fortran/C++ with Matlab interface for graphic output, very efficient, but requires sufficient theoreticalknowledge

• MATMOL: Matlab–Toolbox, which has some implemented upwind– and high-resolution procedures,requires theoretical knowledge

Modern FEM-packages like FIREDRAKE, or ANSYS have further far-reaching numerical possibilities, making thesolution of certain quasilinear hyperbolic problems more reliable.

3.4 Basic concepts in the control of PDE systems

In this section some important general concepts for estimation and control of PDE systems are introduced.This includes some considerations about the types and modeling of sensors and actuators, as well as a basiccharacterization of design approaches.

3.4.1 Sensor and actuator equations

Given the spatial extension of the above considered distributed parameter systems it is clear that a wide varietyof sensor and actuator configurations can be considered. The following three different types of are the mostcommon ones:

• Boundary input and/ or boundary output

x(z, t )

-

dz0 L

- -

u(t ) y(t ) → y∗(t )

The practially important case of a boundary input or boundary output correspond to the situation wherethe control and measurement variables are locally concentrated at the boundarys of the local area. Typicalexamples of such systems are tubular reactors, where, e.g., the inlet temperature and composition areconsidered as inputs and the temperature or part of the composition is measured at the outlet (e.g., theλ-value at the catalytic converter in a car).

• Continuously distributed control and measured variable

x(z, t ) → w(z, t )-

z

0 L

? ? ? ? ? ? ? ? ?

??

? ? ??

?? ?

y(z, t )

u(z, t )

3.4 Basic concepts in the control of PDE systems 21

For this, in itself only theoretically interesting case with infinite-dimensional control or measured variableit is assumed that u and y are arbitrary and not separable in space and time. In the case that such aseparation needs to be assumed, the following type is more realistic.

• Point-wise distributed control and measurement variables

x(z, t )-

z

0 L

? ?

. . .u1(t ) um(t ) ⇒ u(t ) ∈Rm

? ?. . .

y1(t ) yp (t ) ⇒ y(t ) ∈Rp

In this case, the m manipulated variables and p measured variables can be mathematically described by

u(z, t ) =m∑

i=1bi (z)ui (t ) (3.24)

and

y j (t ) =∫ L

0c j (z)x(z, t )dz, j = 1, . . . , p. (3.25)

where the bi (z) and c j (z) represent the local characteristics of the actuators or measuring elements andare also called shape functions. Typical examples for this configuration are provided by chemical reactorswith cooling or heating jackets (see section 3.1.4) or mechanical structures actuated by piezo patches.From a mathematical point of view however, the difficulty arises that the description of point-wise actingcontrol and measurement variables for the occurrence of Dirac δ functions, i.e. u(z, t ) = δ(z− z)u(t ). Thisyiels to the topic of unbounded operators that will be considered at a later moment in Chapter 4. For themeasurement case this corresponds to the consideration of

y(t ) = x(z0, t ) =∫ 1

0δ(z − z0)x(z, t )dz. (3.26)

Regional sensors, that take, e.g., the mean value over a sensor patch yield measurement equations of theform

y(t ) =∫ z0+ε

z0−εx(z, t )dz (3.27)

corresponding to (3.25) with the shape function

c(z) =

12ε , z ∈ [z0 −ε, z0 +ε]

0, else..

It can be seen that for ε→ 0 the case (3.27) converges to (3.26).

3.4.2 Early and late lumping

In general, the control of PDE systems can be addressed using two different approaches. In the so called”early lumping” approach, on the one side, the PDE model is approximated using, e.g., finite-difference orGalerkin approaches, as illustrated in Section 3.3. This yields finite-dimensional models for which results andapproaches from finite-dimensional control and observer design theory can be employed to end up with afinite-dimensional control and observer. The main disadvantage of this method consists in the dependencyon the particular approximation method which makes it rather non-trivial to answer the question about the


performance when it is applied to the PDE model. Furthermore, the resulting models are typical of high-dimensionl, so that in most cases additional model reduction techniques must be employed. The so called”late lumping” approach, on the other side, exploits the properties and structural information contained in thePDE model without any prior approximation to design observer and control schemes for which a performancemeasure can be obtained direclty in the infinite-dimensional setup. The disadvantage of this approach isthat it implies a considerably higher mathematical complexity that has to be accounted for. For practicalimplementations the resulting schemes than typically have to be approximated, motivating the name latelumping. The different schemes are explained schematically in Figure 3.9.

Implementable fin.–dim.control and observer

Non-implemtable ∞–dim.control and observer

Implementable control andobserver

Design method for CPS

?

Design methods for DPS

?

Direct use of distributed param-eter (PDE) model

? ?

n-dim. CPS model

?

Approximation, e.g.• Galerkin/modal–method• Finite Difference/Element method• . . .

?

Distributed parameter system (DPS)

Figure 3.9: Schematic comparison of ”early lumping” and ”late lumping” approaches.

References

Baehr, H. and K. Stephan (2006). Heat and Mass Transfer. 2nd. Springer–Verlag, Berlin (cit. on pp. 9, 10).Farlow, S. J. (1982). Partial differential equations for scientists and eingeers. Dover, New York (cit. on pp. 14, 18,

20).Jakobsen, H. (2008). Chemical Reactor Modeling – Multiphase Reactive Flows. Springer, Berlin Heidelberg

(cit. on p. 10).John, F. (1982). Partial Differential Equations. 4th. Springer–Verlag, New York (cit. on pp. 14, 15, 17).Larsson, S. and V. Tmoée (2003). Partial Differential Equations with Numerical Methods. Springer- Verlag, Berlin

Heidelberg (cit. on pp. 18, 20).Meirovitch, L. (1997). Principles and Techniques of Vibrations. Prentice Hall, New Jersey (cit. on pp. 11, 12).Reddy, J. (1984). Energy and Variational Methods in Applied Mechanics. Wiley–Interscience, New York (cit. on

p. 11).Smith, J. M. and H. C. van Ness (1975). Introduction to chemical engineering thermodynamics. McGraw-Hill

(cit. on pp. 9, 10).Whitham, G. (1999). Linear and Nonlinear Waves. John Wiley & Sons, New York (cit. on p. 17).

REFERENCES 23


Chapter 4

State-space representation

In analogy to the finite-dimensional case, for DPS a far reaching and powerful analysis and design methodologyhas been developed based on the state space approach. This offers in particular the advantage to be able toanalyze a large class of DPS in a common funtional-analytic framework, and to provide a rigorous definitionof structural properties like controllability and observability. In addition, stability analysis of the open andclosed loop system can be carried out within a rather general, but in comparison to the finite-dimensional caseconsiderably more complex setting.

4.1 Introduction to state space analysis

In the following, linear inhomogenous PDEs are put into a mathematical framework that can be seen as ageneralization of finite-dimensional systems theory, for which the concept of state space has turned out tobe essential and extremely useful. In particular, in finite dimensions this allows to consider a general n-thorder ODE as an n-dimensional system of first order ODEs. For linear time-invariant systems this leads to arepresentation in the form

dx(t )

dt= Ax(t )+Bu(t ) t > 0, x(0) = x0 ∈Rn (4.1)

y(t ) =C x(t )+Du(t ) t ≥ 0 (4.2)

with the system matrix A ∈ Rn×n , the input matrix B ∈ Rn×p , the measurement matrix C ∈ Rq×n and thefeedthrough matrix D ∈Rq×p . Obviously the solution x : [0,∞) →Rn of the ODE (4.1) given by

x(t ) = e At x0 +∫ t

0e A(t−τ)Bu(τ)dτ. (4.3)

is contained in the linear n–dimensional vector space Rn which represents the state space X of the system forall t ≥ 0. The matrices A, B , C , D can be viewed as linear operators between linear finite-dimensional vectorspaces, e.g. A :Rn →Rn with the domain of definition D(A) =Rn .

The importance of the state space relies in particular on the fact that the structural properties of controlla-bility and observability can only be defined in the time domain. Furthermore, there exist a wide variety ofpowerful design approaches for feedback control (like, e.g., LQR, optimal control, . . . ) which have no (direct)correspondence in the frequency domain.

A transfer of the state space concept to systems described by linear PDEs thus seems promissing. This yields tothe consideration of differential equations in abstract linear infinite-dimensional function spaces (Curtainand Zwart, 1995).

25

4.1.1 Motivation and introductory examples

To introduce the abstract consideration of DPS in linear function spaces in the following the fundamentalconcepts are explained on the basis of examples.

Example 4.1 (Linear heat transfer with spatially distributed input). Considering a spatially pointwiseacting input u(z, t ) influencing the temperature distribution x(z, t ) yields the PDE

∂t x(z, t ) = ∂2z x(z, t )+b(z)u(t ), z ∈ (0,1), t > 0 (4.4)

with the BCs

x(0, t ) = x(1, t ) = 0, t > 0 (4.5)

and the IC

x(z,0) = x0(z), z ∈ [0,1]. (4.6)

For the spatial characteristic function b consider

b(z) = 1

2ε

(σ(z − (z0 −ε))−σ(z − (z0 +ε))

), 0 < z0 −ε< z0 +ε< 1. (4.7)

Due to the homogenous boundary conditions the solution of the DPS (4.4)–(4.6) can be easily obtained in formof a product x(z, t ) =α(t )ψ(z). This leads to spatial Fourier series with time-dependent coefficients

x(z, t ) =∞∑

k=1e−(kπ)2t x0

kψk (z)+∞∑

k=1bkψk (z)

∫ t

0e−(kπ)2(t−τ)u(τ)dτ (4.8)

with x0k = ∫ 1

0 x0(z)ψk (z)dz, bk = ∫ 10 b(z)ψk (z)dz =p

2/(kπε)sin(kπz0)sin(kπε) as well asψk (z) =p2sin(kπz)

for k ≥ 1. It should be noted that∫ 1

0 ψk (z)ψl (z)dz = δk,l holds true with the so-called Kronecker–Delta δk,l = 1if k = l and δk,l = 0 if k 6= l .

Exercise 4.1. Derive the solution (4.8).

Solution 4.1. In the following a solution of (4.4)–(4.6) will be derived using the product formula

x(z, t ) =α(t )ψ(z). (4.9)

First, the homogenous PDE (4.4) with b(z) ≡ 0 is considered. Substituting (4.9) into (4.4) and separatingvariables yields

dα(t )dt

α(t )=

d2ψ(z)dz2

ψ(z)=−µ2, (4.10)

where the last identy holds true due to the fact that the only possibility for a function of t to correspond to afunction of z for all t and z is that both are constants. This constant is assumed to be equal to −µ2. Using thisone obtains the following solutions

α(t ) = e−µ2tα0 (4.11)

ψ(z) = A sin(µz)+B cos(µz). (4.12)

Note that the temporal part corresponds to a decaying exponential function implying exponential stability.This is why the sign of the constant in (4.10) is assumed as negative, in correspondence with physical experience.In addition, evaluating the boundary conditions (4.5) with (4.12) yields

B = 0, A sin(µ) = 0.

The case A = 0 can be directly excluded, given that this implies a trivial solution ψ(z) ≡ 0 and thus x(z, t ) ≡ 0.Further, we consider only A 6= 0, implying that the BC can be satisfied only if µ= kπ with k ∈Z\ 0. The case

26 Chapter 4 State-space representation

µ= 0 again yields the trivial solution and thus is excluded in the following. The remaining parameter A, or Ak

gives a degree of freedom which cannot be determined using any of the conditions. For the further evaluationwe exploit the property that∫ 1

0sin(kπz)sin(lπz)dz = δk,l /2

with the Kronecker–Delta δk,l = 1 for k = l and δk,l = 0 for k 6= l . Obviously this implies that∫ 1

0ψ2

k (z)dz = A2k

2= 1,

∫ 1

0ψk (z)ψ−k (z)dz =− Ak A−k

2= 1

for all k ∈N. This motivates to set Ak =p2 and A−k =−p2, leading to

∫ 10 sin(kπz)sin(lπz)dz = δk,l für alle

k, l ∈Z\ 0. It is shown in the subsequent considerations in these notes that the preceding equality implies anorthonormality condition. It follows directly that the components ψk (z) are not independent of each otherbecause for all k ∈Z\ 0 it holds true that ψk (z) =ψ−k (z). According to (4.9) this implies that for all k ∈Z\ 0one has

xk (z, t ) = e−(kπ)2tα0k Ak sin(kπz).

Given the super position property of linear systems the complete solution can be written in form of the series

x(z, t ) = ∑k∈Z\0

e−(kπ)2tα0kψk (z) = ∑

k∈Ne−(kπ)2tβ0

kψk (z) (4.13)

with β0k =α0

k +α0−k . Given the orthogonality of the functions ψk (z)k∈N, in the case that b(z) ≡ 0 the constants

β0k can be directly determined for each l ∈N from the equation∫ 1

0x0(z)ψl (z)dz =

∞∑k=1

β0k

∫ 1

0ψk (z)ψl (z)dz =

∞∑k=1

β0kδk,l =β0

l (4.14)

At this place it should be noted that the set ψk (z)k∈N forms a basis of the Hilbert space L2(0,1). In the abovethe convergence of the (Fourier–)series (4.13) is assumed in order to allow for the change of summation andintegration. As known from Fourier analysis, this convergence is ensured, e.g., for all initial conditions inL2(0,1) and is absolute (i.e., considering absolute values of each element of the series) if the initial condition isan element of C 1(0,1) (Dettman, 1988).

For the case b(z) 6= 0 the solution of the inhomogenous PDE can be directly determined considerng the variationof constants formula (4.3) as shown next. Given

x(z, t ) = ∑k∈N

e−(kπ)2tβk (t )ψk (z) (4.15)

the evaluation of (4.4) yields

∑k∈N

e−(kπ)2t(

dβk (t )

dt−µ2

kβk (t )

)ψk (z) = ∑

k∈Ne−(kπ)2tβk (t )

d2ψk

dz2 (z)+b(z)u(t )

or, due to (4.10),

∑k∈N

e−(kπ)2t dβk (t )

dtψk (z) = b(z)u(t ).

In virtue of the orthnormality of the elements of ψk (z)k∈N this implies that∫ 1

0

∑k∈N

e−(kπ)2t dβk (t )

dtψk (z)ψl (z)dz =

∫ 1

0b(z)ψl (z)dzu(t )

4.1 Introduction to state space analysis 27

or equivalently

e−(lπ)2t dβl (t )

dt= bl u(t ), ∀l ∈N.

Obisouly this yields βl (t ) = bl∫ t

0 e(lπ)2τu(τ)dτ, so that with (4.15) the particular solution is given by

x(z, t ) = ∑k∈N

bkψk (z)∫ t

0e−(kπ)2(t−τ)u(τ)dτ. (4.16)

The complete solution of (4.4)–(4.5) is thus given by the (Fourier–)series (4.8).

Example 4.2 (Linear heat conduction (example 4.1 continued)). On the basis of the known solution (4.8)it is shown in the following how the DPS (4.4)–(4.6) can be represented in a state space form.

Note that ⟨φ(z),ψ(z)⟩ = ∫ 10 φ(z)ψ(z)dz with φ(z), ψ(z) ∈ L2(0,1) is the so called inner producta in the Hilbert

space L2(0,1). It follows from (4.8) that

x(z, t ) =∞∑

k=1e−(kπ)2t ⟨x0,ψk⟩ψk (z)+

∫ t

0

∞∑k=1

e−(kπ)2(t−τ) ⟨bu( · ,τ),ψk⟩ψk (z)dτ

This corresponds at least formally to the application of the operatorb

T (t ) =∞∑

k=1e−(kπ)2t ⟨ · ,ψk⟩ψk (4.17)

on the IC x(z,0) and the inhomogenity b(z)u(t ) according to

x(z, t ) = T (t )x0(z)+∫ t

0T (t −τ)b(z)u(τ)dτ. (4.18)

The comparison with (4.3) shows the formal relation between T (t) and the matrix exponential e At in thefinite-dimensional case. To make this more precise we will later consider the formal extension using anoperator A : D(A) ⊂ L2(0,1) → L2(0,1) with domain of definition D(A) (to be defined later). In particular, inconsequence of this relation it follows that (4.18) is the solution of the inhomogenous Cauchy–problem

dx

dt( · , t ) = Ax( · , t )+bu( · , t ), t > 0, x( · ,0) = x0 ∈D(A) (4.19)

with the operators Ax = d2xdz2 for x ∈D(A) and b = b(z). The definition of the domain D(A) of the operator A

depends on the definition of the state space that is addressed later in this chapter.

aA direct generalization of the scalar product in a finite-dimensional vector space, corresponding to a projection of a vector toanother one.

bAs this operator maps functions depending on the space variable to other such functions, the space dependency is typically notindicated but should be kept in mind.

Before proceeding with the introduction of an associated state space, consider the adaptation of the precedinganalysis for the linear wave equation.

Example 4.3 (Linear wave equation with spatially distributed input). Having the discussion of example4.1 as point of departure in the following the linear wave equation with spatially distributed input is analyzed.

∂2t x(z, t ) = ∂2

z x(z, t )+b(z)u(t ), z ∈ (0,1), t > 0 (4.20)

with the BCs

x(0, t ) = x(1, t ) = 0, t > 0 (4.21)

and the ICs

x(z,0) = x0(z), ∂t x(z,0) = x1(z) z ∈ [0,1]. (4.22)


The spatial characteristics b corresponds to (4.7). In analogy and given the homogenous BCs (4.21) the solutionof the hyperbolic IBVP (4.20)–(4.22) can be derived using a product approach. After a couple of intemediatecalculations, this yields the solution in terms of an infinite series

x(z, t ) =∞∑

k=1

[x0

k cos(ωk t )+ x1k

ωksin(ωk t )

]ψk (z)+

∫ t

0

∞∑k=1

[ bk

ωksin(ωk (t −τ))u(τ)

]ψk (z)dτ. (4.23)

Here it holds that ωk = kπ, x0k = ⟨x0,ψk⟩, x1

k = ⟨x1,ψk⟩ as well as bk = ⟨b,ψk⟩ =p

2/(kπε)sin(kπz0)sin(kπε)

and ψk (z) =p2sin(kπz) for k ≥ 1.

It should be recalled that for the state space description of finite-dimensional second order systems (oscillatorequations) often x und x are chosen as states. In the following this idea is applied for the DPS (4.20)–(4.22).

Let x(t ) = [x1(t ), x2(t )]T with x1(t ) = x( · , t ) and x2(t ) = ∂t x( · , t ). It follows that the series representation (4.23)of the formal application of the operator

T (t )x = ∑∞

k=1

[⟨x1,ψk⟩L2 cos(ωk t )+ 1

ωk⟨x2,ψk⟩L2 sin(ωk t )

]ψk (z)∑∞

k=1

[−ωk⟨x1,ψk⟩L2 sin(ωk t )+⟨x2,ψk⟩L2 cos(ωk t )]ψk (z)

(4.24)

on the IC x0 = [x01 , x0

2]T and the distributed input bu(t ) = [0,b(z)]T u(t ) yields

x(t ) = T (t )x0 +∫ t

0T (t −τ)bu(τ)dτ. (4.25)

In particular this implies that (4.25) is the solution of the inhomogenous Cauchy–problem

dx

dt(t ) = Ax(t )+bu(t ), t > 0, x(0) = x0 ∈D(A) (4.26)

with the operators

A = 0 1

∂2z 0

, b = 0

b(z)

. (4.27)

In analogy to the considerations in example 4.1 in the following the question about an adequate domain ofdefinition D(A) of the operator A has to be addressed. This is directly connected anyway to the choice of aproper state space.

Based on the above examples 4.1 and 4.3 important aspects for the choice of a state space X , and thus statesxi ( · , t ), i = 1, ...,n for a given problem are obtained. In particular the states and the state space have to be suchthat the solution in the state space satisfies the well-definiteness properties of Hadamard (Curtain and Zwart,1995; Evans, 2002):

a) a unique solution x exists with x( · , t ) ∈X for all t ≥ 0, and

b) the solution continuously depends on the initial condition (and the input).

For the analysis of the second aspect one can make use of the following result (cp. (Jüngel, 2001; Engel andR.Nagel, 2006):

Theorem 4.1: (Jüngel, 2001)

Let T : X →X be linear. Then the following statements are equivalent:

a) T is bounded, i.e., ∃K , 0 < K <∞, so that ‖T x‖ ≤ K ‖x‖ ∀x ∈X .

b) T is continuous.


c) ∀x0 ∈X : T is continuous in x0.

The importance of this result is illustrated in the following on the basis of the homogenous Cauchy–problems(4.4)–(4.6) and (4.20)–(4.22) with u ≡ 0 illustriert. For this the influence of the ICs x( · ,0) = x0( · ) (for example4.1) and x( · ,0) = [x0( · ), x1( · )]T =: x0 (for example 4.3) on the corresponding solution x( · , t) = T (t)x0( · ) andx( · , t ) = T (t )x0( · ) is analyzed, respectively.

Example 4.4 (Linear heat equation (example 4.1 continued)). Let X = L2(0,1) with inner product ⟨φ,ψ⟩L2 =∫ 10 φ(z)ψ(z)dz for φ, ψ ∈ L2(0,1) and induced norm ‖φ‖L2 =√⟨φ,φ⟩L2 . Let

x0(z) =∞∑

k=1x0

kψk (z), x0k = ⟨x0,ψk⟩ , ψk =p

2sin(kπz), k ∈N

be the (Fourier) series representation of the initial condition. Note that this series converges for all x0 ∈ L2(0,1).It further holds true that

x(z, t ) =∞∑

k=1e−µ

2k t x0

kψk (z) = T (t )x0

implying that

‖x( · , t )‖2L2 = ⟨x( · , t ), x( · , t )⟩ = ⟨

∞∑k=1

e−µ2k t x0

kψk ,∞∑

l=1e−µ

2k t x0

l ψl ⟩

=∫ 1

0

∞∑k=1

k∑l=1

e−µ2l t e−µ

2k−l t x0

l x0k−l sin(lπz)sin((k − l )πz)dz

=∞∑

k=1

k∑l=1

x0l x0

k−l e−µ2l t e−µ

2k−l t

∫ 1

0sin(lπz)sin((k − l )πz)dz

=∞∑

k=1

k∑l=1

x0l x0

k−l e−µ2l t e−µ

2k−l tδl ,(k−l )

=∞∑

k=1x2

k e−2µ2k t ≤ e−2µ2

1t∞∑

k=1

(x0

k

)2 ≤ ‖x0‖2L2

because the series converges for all x0 ∈ L2(0,1) and thus integration and (infinite) summation can be inter-changed. Equivalently, in compact form, this becomes

‖T (t )x0‖L2 ≤ ‖x0‖L2 .

This implies the boundedness of the linear operator T : L2(0,1) → L2(0,1), i.e., property a) in Theorem 4.1 withK = 1, and thus, in consequence of Theorem 4.1 c), its continuous dependency on x0. With X = L2(0,1) theCauchy–problem (4.4)–(4.6) is thus well-defined in the sense of Hadamard and one can write

dx

dt= Ax, x ∈D(A), x(0) = x0

with

A = d2

dz2 , D(A) =

x ∈X∣∣∣ d2x

dz2 ∈X with x(0) = x(1) = 0

.

It should be noticed that the boundary conditions are part of the definition of the domain D(A).

Example 4.5 (Linear wave equation (example 4.3 continued)). For the example of the linear wave equationthere excist different possibilities for the introduction of a suitable state space.

• Let X = L2(0,1)⊕L2(0,1) with inner product ⟨φ,ψ⟩ = ⟨φ1,ψ1⟩L2 + ⟨φ2,ψ2⟩L2 for φk , ψk ∈ L2(0,1),k = 1,2, and induced norm ‖φ‖X =√⟨φ,φ⟩.Consider the initial value x0 =ψm =p

2sin(mπz) and x1 = 0, with ψm(z) = sin(mπz) (cp. example 4.3)


so that according to (4.24)

x1(z, t ) =ψm(z)cos(ωm t )

x2(z, t ) = ∂t x1(z, t ) =−ωmψm(z)sin(ωm t ).

This implies that

‖x( · , t )‖2X = ‖T (t )x0‖X =

∫ 1

0

(x2

1(z, t )+x22(z, t )

)dz

= (cos(ωm t )−ωm sin(ωm t ))∫ 1

0ψm(z)2dz︸︷︷︸‖x0‖2

X

and thus

‖x( · , t )‖2X = (cos(ωm t )−ωm sin(ωm t ))‖x0‖2

X = (1+ [(mπ)2 −1]sin(mπt )

)‖x0‖2X

implying that the operator T (t) is unbounded, as m can go to infinity (corresponding to the infiniteseries of eigenfunctions). In consequence, according to 4.1 the operator is not continuous in x0, and thusthe problem (4.20)–(4.22) not well-defined in the sense of Hadamard with this inner product.

• Leta X = H 1(0,1)⊕L2(0,1) with inner product ⟨φ,ψ⟩ = ⟨dφ1dz , dψ1

dz ⟩L2 +⟨φ2,ψ2⟩L2 for φ1, ψ1 ∈ H 1(0,1)

and φ2, ψ2 ∈ L2(0,1) with induced norm ‖φ‖X =√⟨φ,φ⟩.For this choice of inner product one can easily show that

⟨ x( · , t )

∂t x( · , t )

,

x( · , t )

∂t x( · , t )

⟩X

=∥∥∥∥∥∥ x( · , t )

∂t x( · , t )

∥∥∥∥∥∥2

X

= ⟨∂z x( · , t ),∂z x( · , t )⟩L2 +⟨∂t x( · , t ),∂t x(z, t )⟩L2

=∫ 1

0

[(∂t x(z, t ))2 + (∂z x(z, t ))2]d z = 2Wk (t )+2Wp (t ). (4.28)

Writing (4.24) as

x(z, t ) =∞∑

k=1ψk (z)

a1,k (t )

a2,k (t )

, a1,k = a2,k , a1,k = x0,k cos(ωk t )+ x1,k

ωksin(ωk t ), ψk (z) = sin(ωk z)

where xi ,k = ⟨xi ,ψk⟩ , i = 0,1, it turns out that

d

dt

(2Wk (t )+2Wp (t )

)= ∞∑k=1

∫ 1

0

2a1,k (t )a1,k (t )(ψ′

k (z))2 +2 a2,k (t )︸︷︷︸

=a1,k (t )

a2,k (t )︸︷︷︸=a1,k (t )=−ωk a1,k (t )

(ψk (z)

)2

dz

= 2∞∑

k=1a1,k (t )a1,k (t )

∫ 1

0

[(ψ′

k (z))2 −ω2

k

(ψk (z)

)2]

dz = 0

implying that the norm remains constant over time. This inmediately implies that

‖x( · , t )‖X = ‖T (t )x0‖X = ‖x0‖X

so that the operator T (t ) is bounded for all t ≥ 0 and property a) of Theorem 4.1 is satisfied with K = 1,implying that T (t )x0 is continuous in x0 by Theorem 4.1 part c). Accordingly, problem (4.20)–(4.22) iswell-defined in the sense of Hadamard with this inner product b.

Note that with the chosen inner product the state space is a so-called Sobolev–space (cp. Section 1).


Furthermore the domain of definition D(A) ⊂X = H 1(0,1)⊕L2(0,1) of the operator

A = 0 1

∂2z 0

must be adequately defined. Given that by the BCs x(0, t ) = x(1, t ) = 0 must hold true one has

D(A) =D1(A)⊕D2(A) (4.29)

D1(A) =

x ∈ L2(0,1) : x,dx

dzabs. continuous,

d2x

dz2 ∈ L2(0,1), x(0) = x(1) = 0

and (4.30)

D2(A) =

x ∈ L2(0,1) : x,dx

dz∈ L2(0,1)

. (4.31)

aHere the Sobolev space Hm (a,b) =

x ∈ L2(a,b) :dαx

dzα∈ L2(a,b), α= 0,1, ...,m

is employed.

bNote that the uniqueness can be shown using the same energy function and following the energy method, i.e. considering a differentsolution v exists for the same initial, i.e. v( · ,0) = x0 and considering the energy stored in the difference x = x −v . Compare exercisesheet 2 for more details.

Remark 4.1

Choosing so-called energy–coordinates x1(z, t) = ∂z x(z, t) and x2(z, t) = ∂t x(z, t) the considered ho-mogenous Cauchy–problem (4.20)–(4.22) with u(t) ≡ 0 becomes well-defined in the (standard) norm‖x‖2 = ∫ 1

0 (x21 +x2

2)d z, i.e., in L2(0,1)⊕L2(0,1).

With these examples one can inmediately summarize the fundamental principle for the abstract representationof DPS in state space: by introducing a suitable state space (typically a Hilbert space) and the interpretationof the spatial differential operator A in this state space with a domain of definition D(A) which includes thehomogenous boundary conditions and an input operator B , or Bu, respectively, the DPS can be brought intothe state space form

dx

dt(t ) = Ax(t )+Bu(t ), t > 0, x(0) = x0 ∈D(A) (4.32)

Further one should notice that also the inhomogeneity f = Bu has to be contained in the appropriate spacefor all t ≥ 0. This can be, e.g., the space of continuous functions mapping the interval [0,τ], τ> 0 onto X , i.e.,f (t) ∈C ([0,τ],X ), or the space of Lebesque–integrable functions, mapping the interval [0,τ], τ> 0 onto X ,i.e., f (t ) ∈ Lp ([0,τ],X ). The first case leads to, e.g., the definition of classical solutions of abstract differentialequations, and the second one to so-called mild or weak solutions (Curtain and Zwart, 1995).

In addition, note that up to now only spatially distributed inputs have been considered. Boundary inputs, incontrary, require additional considerations, as motivated through the following example.

Example 4.6 (Linear heat conduction with boundary input). Let the spatio-temporal temperature distribu-tion x in a medium be governed by the following PDE with BCs and IC

∂t x(z, t )−∂2z x(z, t ) = u(z, t ), z ∈ (0,1), t > 0 (4.33)

x(0, t ) = u0(t ), x(1, t ) = u1(t ), t > 0 (4.34)

x(z,0) = x0(z), z ∈ [0,1]. (4.35)

Under the assumption that u0(t ) = u1(t ) = 0 the introduction of the operator

Aφ= d2φ

dz2 (4.36)

with domain

D(A) =φ ∈ L2(0,1) : φ,

dφ

dzabs. continuous,

d2φ

dz2 ∈ L2(0,1), φ(0) = 0 =φ(1)

(4.37)


enables to write the inhomogenous Cauchy–problem (4.33)–(4.35) in the form of the abstract differentialequation (4.32) with Bu(t) = u( · , t) t ≥ 0. It is obvious that both x(0, ·) = u0 as x(1, ·) = u1 can not beintegrated in the domain D(A) given their temporal dependency.

Oviously one needs additional steps in order to represent DPS with inhomogenous boundary conditions instate space. One possibility consists in homogenizing the boundary conditions. This is illustrated next for theexample 4.6.

Example 4.7 (Linear heat conduction with boundary input (example 4.6 continued)). As commented, thestate space representation of DPS requires homogenous boundary conditions. In the case of an inhomogenousBC homogenization of the BCs can be an option. The principle is shown below.

-

z

6x

0 1

x(z, t )

x(z, t )

6

6

6 6

?

?

x(z, t )@@@

The basic idea consists in the separation of the state x(z, t ) of the DPS in a part x(z, t ), which corresponds to ahomogenous BC, as well as a part x(z, t ), which takes into account the influence of the inhomogeneity. Forthe model of the heat conductor (4.33)–(4.35) this leads to

x(0, t ) = u0(t ) = x(0, t )︸︷︷︸=0

+x(0, t ) (4.38)

x(1, t ) = u1(t ) = x(1, t )︸︷︷︸=0

+x(1, t ). (4.39)

For example

x(z, t ) = u0(t )+ z[u1(t )−u0(t )] (4.40)

satisfies these conditions. Substitution of x(z, t ) = x(z, t )+ x(z, t ) into (4.33)–(4.35) yields the following inho-mogenous PDE for x with homogenous BCs

∂t x(z, t )−∂2z x(z, t ) = u(z, t )−∂t x(z, t )+∂2

z x(z, t )︸︷︷︸=0

, z ∈ (0,1), t > 0 (4.41)

x(0, t ) = 0, x(1, t ) = 0, t > 0 (4.42)

x(z,0) = x0(z)− x(z,0), z ∈ [0,1]. (4.43)

This analysis assumes of course the corresponding differentiability of x(z, t), where in particular it holds forthe boundary values u0(t ), u1(t ) ∈C 1([0,τ]), τ> 0. For the representation of (4.41)–(4.43) in state space for thepurpose of determining the associated solution for known u(z, t), u0(t) and u1(t), one can bring the DPS bysubstitution of u(z, t) = u(z, t)−∂t x(z, t)+∂2

z x(z, t) into a form corresponding to (4.32). For the special casethat u0(t ) or u1(t ) are control inputs one can bring (4.41)–(4.43) into an extended state space using appropriatemethods including u0(t ) or u1(t ) as additional states. This leads to the theory of so-called „Boundary Control


Systems” (Fattorini, 1968; Curtain and Zwart, 1995).

Yet a different approach to analyze systems with inhomogenous BCs is based on the introduction of operatorextensions (Ho and Russell, 1983; Tucsnak and Weiss, 2009), where the resulting system formulation in the form(4.32) does not allow for a pointwise fulfillment of the boundary conditions.

4.1.2 Uniformly continuous semigroups and infinitesimal generators

Based on the above examples the fundamental concept of so-called uniformly continuous, or strongly continu-ous semigroups, respectively, can be deduced (Engel and R.Nagel, 2006).

Definition 4.1: Uniformly and strongly continuous semigroup

Let X be a Banach–space and T (t ) : X →X , 0 ≤ t <∞ a family of linear bounded operators.

a) T (t ) is a semigoup of linear bounded operators on X , if

T (0) = I (4.44)

T (t + s) = T (t )T (s) ∀t , s ≥ 0 (4.45)

hold true.

b) A semigroup T (t ) of linear bounded operators on X is called uniformly continuous, if

limt→0+

‖T (t )− I‖ = 0. (4.46)

c) A semigroup T (t ) of linear bounded operators on X is called strongly continuous, if

limt→0+

T (t )x0 = x0 ∀x0 ∈X . (4.47)

A strongly continuous semigroup of linear bounded operators is also called C0–semigroup.

Remark 4.2

Obviously T (t ), as defined in (4.17) or (4.24) are both C0–semigroups in the corresponding state spacesX .

Theorem 4.2

Let T (t ) be a C0–semigroup in the Hilbert space X and let ω0 = inft>0( 1

t log‖T (t )‖). Then

ω0 = limt→∞

(1

tlog‖T (t )‖

)<∞

and for all ω≥ω0 there exists a constant Mω, so that for all 0 ≤ t <∞ it holds true that

‖T (t )‖ ≤ Mωeωt . (4.48)

In the preceding inequality ‖T ‖ represents the operator norm 1 of T . Obviously this theorem provides astatement on the growth or the growth bounds of C0–semigroups and leads to the following definition (Curtainand Zwart, 1995; Engel and R.Nagel, 2006).

Definition 4.2

Let T (t ) be a C0–semigroup with ‖T (t )‖ ≤ Meωt , M ,ω ∈R. Then T (t ) is called

1Recall that the norm of an operator T : X → Y can be defined as sup06=x∈D(T )‖T x‖Y‖x‖X

.


• uniformly bounded, if ω= 0, i.e., ‖T (t )‖ ≤ M , ∀t ≥ 0,

• C0–semigroup of contractions, if M = 1 and ω= 0, i.e., ‖T (t )‖ ≤ 1, ∀t ≥ 0.

It is obvious that the contractivity property of a C0–semigroup plays an important role in the analysis of stabilityof a given system.

Remark 4.3

The C0–semigroup T (t ) defined in (4.24) is contracting with ‖T (t )‖ ≤ 1.

In general it is anyway not always possible to determine explicitely the solution, i.e., the semigroup T for a giveninfinite-dimensional problem. Thus, it is of interest to analyze a system on the basis of the system operator, theso-called infinitesimal generator of a semigroup.

Definition 4.3

The linear operator A : D(A) →X defined by

D(A) =

x ∈X | limt→0+

1

t(T (t )x −x) exists

(4.49)

Ax = limt→0+

1

t(T (t )x −x) = d

d tT (t )x

∣∣∣∣t=0

(4.50)

for x ∈D(A), is called the infinitesimal generator of the semigroup T (t ).

In analogy to the linear finite-dimensional case in the following sections directly the analysis of stability,controllability and observability are based on the infinitesimal generator A as well as the input and outputoperators B and C , respectively.

Some important properties of C0–semigroups and their infinitesimal generators are summarized in the followingtheorem (Curtain and Zwart, 1995; Engel and R.Nagel, 2006).

Theorem 4.3

Let T (t ) be a C0–semigroup and A its infinitesimal generator. Then it holds true that

a) for all x ∈X :

limh→0

1

h

∫ t+h

tT (s)x d s = T (t )x . (4.51)

b) for all x ∈X :∫ t

0T (s)x d s ∈D(A) and A

(∫ t

0T (s)x d s

)= T (t )x −x . (4.52)

c) for all x ∈D(A):

T (t )x ∈D(A) andd

dtT (t )x = AT (t )x = T (t )Ax . (4.53)

d) for all x ∈D(A):

T (t )x −T (s)x =∫ t

sT (τ)Ax dτ=

∫ t

sAT (τ)x dτ. (4.54)

Based on these properties the following results can be shown (Engel and R.Nagel, 2006).

Corollary 4.1. Let A be the infinitesimal generator of a C0–semigroup T (t ). Then2 A is a linear closed operator

2The following concepts are used in this part:


with D(A) =X , i.e., D(A) is dense in X .

Further notions and properties of C0–semigroups and their infinitesimal generators will be introduced directlywith the subsequent considerations. Particular noteworthy results are the Hille–Yosida theorem and theLumer–Phillips theorem that are presented in appendix A. These two theorems provide criteria to conclude ifa given operator A is the infinitesimal generator of a C0–semigroup.

4.1.3 Riesz operators

Riesz spectral operators form a class of operators that are very important from an application point of view. Fortheir proper introduction some further functional analytic concepts are necessary, like the adjoint operator,compact operators and the eigenvalue problem for linear operators in Hilbert spaces. These will be shortlydiscussed in the following and illustrated on the basis of some examples.

4.1.3.1 Adjoint and self–adjoint operators

Adjoint operators can in principle be viewed as the generalization of the transposition of a matrix in linearalgebra 3.

Definition 4.4: Adjoint operator

Let A : D(A) ⊂X →X be a linear operator in the Hilbert space X with domain D(A) that is dense in X .Let D(A∗) be the set of all y ∈X that are such that there exists a y∗ ∈X with

⟨Ax , y⟩ = ⟨x , y∗⟩ ∀x ∈D(A).

Then for all y ∈D(A∗) the adjoint operator A∗ : D(A∗) ⊂X →X is defined as

A∗y = y∗.

An application of this definition is illustrated in the following example.

Example 4.8 (Adjoint operator for a differential operator of order n). Consider the differential operator oforder n given by

Ax( · , t ) =n∑

j=0β j

d j x( · , t )

dz j, βn 6= 0 (4.55)

with x( · , t ) ∈ L2(0,1), with the domain

D(A) =

x ∈ L2(0,1)∣∣∣ d j x

dz jabs. continuous for j = 0, . . . ,n −1,

dn x

dzn ∈ L2(0,1),

n−1∑j=0

(γ0

j ,l

d j x

dz j(0)+γ1

j ,l

d j x

dz j(1)

)= 0, l = 1, . . . ,n

. (4.56)

• A linear operator T is said to be closed if for xn ∈D(T ), n ∈N with limn→∞ xn = x and limn→∞ T xn = y it follows that x ∈D(T )and T x = y (Curtain and Zwart, 1995, Definition A.3.43).

• A subset V of a normed linear space X is called dense in X , if for the closure V of V it holds true that V = X . The closureV of a normed linear space V is defined as the set V together with the set of all limits of (converging) sequences in V , i.e.,

V := x ∈ V |∃xn n∈N, xn ∈ V ∀n ∈N so that ‖xn − x‖→ 0 for n →∞. A set V in a normed linear space X is called closed if allconvergent sequences of elements of V have a limit in V .

3Recall that in a finite-dimensional vector space Rn the inner product is given by ⟨x ,ξ⟩ = xT ξ for x ,ξ ∈Rn . Now consider ⟨Ax , y⟩ withA :Rn →Rm and let x ∈Rn , y ∈Rm . Then it holds that ⟨Ax , y⟩ = (Ax)T y = (xT AT )y = xT (AT y) = ⟨x , AT y⟩ = ⟨x , y∗⟩ with y∗ = AT y ∈Rn .Here n and m are not necessarily identical.


According to the definition 4.4 the adjoint operator A∗ can be obtained from

⟨Ax, y⟩L2 = ⟨n∑

j=0β j

d j x

dz j, y⟩

L2

=∫ 1

0

n∑j=0

β jd j x

dz jydz = ⟨x, A∗y⟩L2 . (4.57)

By n–times partiel integration one obtains∫ 1

0

n∑j=0

β jd j x

dz jydz =

n−1∑j=0

[β j+1

j∑k=0

(−1)k d j−k x

dz j−k

dk y

dzk

]z=1

z=0+

∫ 1

0x

n∑j=0

(−1) jβ jd j y

dz j.dz. (4.58)

A comparison of (4.58) with (4.57) immediately leads to the equation for the adjoint operator

A∗x =n∑

j=0(−1) jβ j

d j x

dz j. (4.59)

The determination of the domain D(A∗) follows from the condition

n−1∑j=0

[β j+1

j∑k=0

(−1)k d j−k x

dz j−k

dk y

dzk

]z=1

z=0= 0

for all x ∈D(A). Obvously there exist constants γ0,∗j ,l and γ1,∗

j ,l , j = 0, . . . ,n −1, l = 1, . . . ,n so that the preceding

equation is fulfilled. Thus

D(A∗) =

x ∈ L2(0,1)∣∣∣ d j x

dz jabs. continuous for j = 0, . . . ,n −1,

dn x

dzn ∈ L2(0,1),

n−1∑j=0

(γ0,∗

j ,l

d j x

dz j(0)+γ1,∗

j ,l

d j x

dz j(1)

)= 0, l = 1, . . . ,n

. (4.60)

Some important properties of adjoint operatos are summarized below (see also (Heuser, 1992; Curtain andZwart, 1995)).

Theorem 4.4

The adjoint operator A∗ : D(A∗) ⊂ X → X is a continous linear mappinga, i.e., A∗ ∈ L (D(A∗),X ) with‖A∗‖ = ‖A‖. Let B ∈L (X ,X ) and α ∈C, then the following rules apply:

(A+B)∗ = A∗+B∗, D((A+B)∗) =D(A∗) (4.61)

(αA)∗ =αA∗, D((αA)∗) =

D(A∗), α 6= 0

X , α= 0(4.62)

(AB)∗ = B∗A∗ (4.63)

I∗ = I , 0∗ = 0. (4.64)

If A is bijective then also A∗ is bijective and (A∗)−1 = (A−1)∗. Further, it holds that A∗∗ = (A∗)∗ = A.

aThe space L (X ,Y ) indicates the space of linear operator from X to Y .

The concept of adjoint operator can be further used to introduce the following notions.

Definition 4.5: Normal and unitary operators

Let A ∈L (X ,X ). Then A is called

• normal, if A A∗ = A∗A

• unitary, if A A∗ = A∗A = I .

In addition, one can introduce the important notions of self–adjoint operator.


Definition 4.6: Symmetric and self–adjoint operators

A linear operator A in the Hilbert space X is called symmetric if

⟨Ax , y⟩ = ⟨x , Ay⟩ ∀x , y ∈D(A).

A symmetric operator is called self–adjoint if in addition D(A) =D(A∗).

4.1.3.2 Compact operators

For a certain class of operators, the so called compact operators, one can make statement about eigenvaluesand -functions (or -vectors) in analogy to the finite-dimensional case (Curtain and Zwart, 1995; Heuser, 1992;Naylor and Sell, 1982).

Definition 4.7: Compact operator

Let X and Y be normed linear spaces. An operator A ∈L (X ,Y ) is called compact if the image Axnn ofevery bounded sequence xnn ∈ X contains a convergent subsequence.

For the analysis of compactness of a given operator the following lemma is usefull (Curtain and Zwart, 1995;Heuser, 1992; Naylor and Sell, 1982).

Lemma 4.1

Let X and Y be normed linear spaces and let A : X → Y be a linear operator. Then it holds true that:

a) A is a compact operator if A is bounded and dim(A(X )) <∞.

b) If the sequence Ann of compact operators An : X → Y , n ∈ N converges uniformly to A then A iscompact.

c) Let A ∈ L (X ,Y ) be compact and let its range be a closed subspace of Y . Then the range of A isfinite–dimensional.

The application of this lemma can be illustrated through the following example.

Example 4.9 (Proof of compactness of an operator). Let X = L2(0,1) and A : X →X be given by

Ax =∞∑

j=1

1

λ jx jψ j , λ j =−( jπ)2, x j = ⟨x,ψ j ⟩L2 , ⟨ψk ,ψ j ⟩L2 = δk, j (4.65)

with spann≥1ψn = X . Obviously A is linear. Let

An x =n∑

j=1

1

λ jx jψ j .

It directly follows that also An is linear and for all finite n has a finite-dimensional range. According to Lemma4.1(i) An is compact.

To show that An converges uniformly to A consider

‖Ax − An x‖2X =

∫ 1

0

( ∞∑j=n+1

1

λ jx jψ j

)2

dz ≤ 1

λ2n+1

∞∑j=n+1

x2j ≤

1

λ2n+1

∞∑j=1

x2j =

1

λ2n+1

‖x‖2.

It follows that

‖A− An‖ ≤ 1

|λn+1|= 1

((n +1)π)2 → 0 for n →∞.


Thus A is compact according to Lemma 4.1(ii).

Further, integral operators represent an important class of compact operators (Curtain and Zwart, 1995).

Theorem 4.5: Compatness of integral operators

Let k(z, s) ∈ L2([a,b]× [a,b]). The the integral operator K : L2(a,b) → Lp (a,b) defined by

(K x)( · ) =∫ b

ak( · , s)x(s)ds (4.66)

is a compact operator.

4.1.3.3 Eigenvalue problems

The analysis of the eigenvalue problem

(λI − A)x = 0, x ∈D(A), (4.67)

requires first to clarify what should be understood by the concept of an eigenvalue of an operator A. This pointis addressed in the following definition, which additionally introduces the notions of resolvent and its domain,the resolvent set, which will play an important role in the following considerations.

Definition 4.8: Resolvent set and spectrum

Let A be a linear operator on a normed space X . The set

ρ(A) :=λ ∈C : (λI − A) is injective, D((λI − A)−1) =X , (λI − A)−1 is closed

is called the resolvent set of A. The mapping R( · , A) : λ 7→ (λI − A)−1 is the resolvent of A. The set

σ(A) :=C\ρ(A)

is called the spectrum of A. The set of eigenvalues of A

σp (A) := λ ∈C : λI − A is not injective

is called the point spectrum of A.

Remark 4.4

The resolvent R( · , A) of the infinitesimal generator A of a C0–semigroup T (t ) corresponds to the Laplace–transform of the semigroup, given that

dx

dt(t ) = Ax(t )+bu(t ), t > 0, x(0) = x0 ∈D(A)

implies that the Laplace–transform of the solution reads

x( · , s) = (sI − A)−1(bu(s)+x0) = R(s, A)(bu(s)+x0).

The element 0 6= x ∈ D(A) for which (λI − A)x = 0 for a given λ ∈ σp (A) is called eigenvector or eigenfunc-tion. Besides this one can define, e.g., also the algebraic and geometric multiplicty of eigenvalues as well asgeneralized eigenfunctions.

For general linear operators the spectral analysis is considerably more involved then for finite-dimensionaloperators. Nevertheless, given the similarities to finite-dimensional operators for the class of compact operatorsone can deduce additional spectral properties. For a general treatment refer to (showalter:94; Curtain andZwart, 1995; Heuser, 1992; Naylor and Sell, 1982). In the following only those results are summarized which arenecessary for the subsequent discussion.


Lemma 4.2

Let A be a closed linear operator in X whose resolvent R(λ, A) exists and is compact for a given λ ∈ ρ(A).Then the spectrum of A consists only of isolated eigenvalues with finite (algebraic) multiplicity and R(λ, A)is compact for all λ ∈ ρ(A).

This result from (Kato, 1980, S. 187) is particularly important for applications. Compact, normal operators on aHilbert space have no generalized eigenfunctions and one can perform a spectral decomposition in analogy tonormal matrices.

Theorem 4.6

Let A ∈L (X ,X ) be a compact, normal operator in the Hilbert space X . Then there exist an orthonormalbasis of eigenfunctions φn∞n=1 and associated eigenvalues λn∞n=1 so that

Ax =∞∑

n=1λn ⟨x ,φ j ⟩φ j ∀x ∈X . (4.68)

The application of this theorem is illustrated by the following example.

Example 4.10. Let X = L2(0,1) and

(K x)(z) =−∫ 1

0k(z, s)x(s)ds (4.69)

with

k(z, s) =

(1− s)z, 0 ≤ z ≤ s ≤ 1

(1− z)s, 0 ≤ s ≤ z ≤ 1.(4.70)

Given that k(z, s) = k(s, z) it is easily shown that K is self–adjoint. According to Theorem 4.5, K is compact.Thus, the eigenvalues and eigenfunctions of K can be determined using the condition Kφ=λφ, i.e.,

−∫ z

0(1− z)sφ(s)ds −

∫ 1

z(1− s)zφ(s)ds =λφ(z).

The left-hand side of the equation is twice continuously differentiability with respect to z, implying

φ(z) =λd2φ

dz2 (z). (4.71)

In addition it holds true that (Kφ)(0) = 0 and (Kφ)(1) = 0, or

λφ(0) = (Kφ)(0) = 0, λφ(1) = (Kφ)(1) = 0. (4.72)

For λ= 0 thus only the trivial solution exists, so that λ= 0 is no eigenvalue of K . Now, let λ 6= 0 so that from(4.72) the BCS φ(0) =φ(1) = 0 follow. The solution of the BVP (4.71) with these BCs leads to the eigenvaluesand associated eigenfunctions

λn =− 1

(nπ)2 , φn(z) =p2sin(nπz), n ≥ 1. (4.73)

According to Theorem 4.6, it follows that p

2sin(nπz)∞n=1 is an orthonormal basis of X = L2(0,1).

In addition, one can show (Curtain and Zwart, 1995) that every compact – but not neccesarily normal – linearoperator can be decomposed into a spectral representation similar to (4.68).


4.1.3.4 Riesz operators

The following theorem from (Curtain and Zwart, 1995) builds the basis for the definition of the so-called Rieszoperators.

Theorem 4.7

Let A be a linear operator in the Hilbert space X with domain D(A) and let 0 ∈ ρ(A). Let further A−1 benormal and compact. According to Theorem 4.6 it follows that for all x ∈ X the operator A−1 has thespectral representation

A−1x =∞∑

n=1λ−1

n ⟨x ,φn⟩φn , (4.74)

where λ−1n andφn are the eigenvalue and associated eigenfunction of A−1, and φn∞n=1 form an orthonor-

mal basis of X . For x ∈D(A) one has the spectral representation

Ax =∞∑

n=1λn ⟨x ,φn⟩φn (4.75)

with

D(A) =

x ∈X∣∣∣ ∞∑

n=1|λn |2| ⟨x ,φn⟩ |2 <∞

. (4.76)

Further, A is a closed linear operator.

The application of this theorem is illustrated in the following for the spatial operator of the heat equa-tion.

Example 4.11. Let X = L2(0,1) and

Ax = d2x

dz2

D(A) :=

x ∈X∣∣∣x,

dx

dzabs. continuous,

d2x

dz2 ∈ L2(0,1), x(0) = x(1) = 0

.

(4.77)

The inverse A−1 of A can be determined in form of the integral equation

(A−1x)(z) =−∫ 1

0k(z, s)x(s)ds

with k(z, s) from (4.70)a. Like it is shown in Example 4.10, A−1 is a compact, self–adjoint linear operator witheigenvalues −1/(nπ)2∞n=1 and associated eigenfunctions

p2sin(nπz)∞n=1. According to Theorem 4.7, A is a

closed linear operator with the spectral decomposition

Ax =∞∑

n=1−(nπ)2 ⟨x,

p2sin(nπz)⟩L2

p2sin(nπz)

and the domain of definition

D(A) =

x ∈X∣∣∣ ∞∑

n=1(nπ)4| ⟨x,

p2sin(nπz)⟩L2 |2 <∞

.

aHere k(z, s) is the so-called Green’s function.

With these preliminaries one can introduce the class of Riesz operators, that is very important from an applica-tion point of view. These operators include all those for which the eigenfunctions form a so-called Riesz basis(Curtain and Zwart, 1995).


Definition 4.9: Riesz–basis

A sequence of vectors φn , n ≥ 1 in a Hilbert space X forms a Riesz–basis for X if:

a) spann≥1φn =X

b) there exist positive constants m, M so that for arbitrary N ∈N and arbitrary scalars αn , n = 1, ..., N itholds that

mN∑

n=1|αn |2 ≤ ‖

N∑n=1

αnφn‖2 ≤ MN∑

n=1|αn |2.

It is important to notice here, that eventhough every orthonormal basis of a function space is a Riesz–basisthe orthogonality is not a necessary condition. Thus, even more general and in particular also non self–adjointlinear operators (cp. Example 4.8) can be analyzed using this notion.

Lemma 4.3

Let A be a closed linear operator in the Hilbert space X with simple eigenvalues λn∞n=1 and associatedeigenfunctions φn∞n=1 that forma Riesz–basis. Then the following holds true:

a) The eigenfunctions ψn∞n=1 of the adjoint operator A∗ for the associated eigenvalues λn ∞n=1 can bescaled in such a way that ⟨φn ,ψm⟩ = δn,m .

b) Every x ∈X can be uniquely represented in the form of a Fourier series

x =∞∑

n=1⟨x ,ψn⟩φn , (4.78)

so that there exist constants m, M > 0 for which

m∞∑

n=1| ⟨x ,ψn⟩ |2 ≤ ‖x‖2

X ≤ M∞∑

n=1| ⟨x ,ψn⟩ |2

holds true.

c) The eigenfunctions ψn∞n=1 of the adjoint operator A∗ form also a Riesz–basis for X . Every x ∈X

thus has a unique representation in form of a Fourier series

x =∞∑

n=1⟨x ,φn⟩ψn , (4.79)

so that there exist constants m, M > 0 for which

m∞∑

n=1| ⟨x ,φn⟩ |2 ≤ ‖x‖2

X ≤ M∞∑

n=1| ⟨x ,φn⟩ |2

holds true.

This enables the definition of a Riesz operator (Curtain and Zwart, 1995; Tucsnak and Weiss, 2009).

Definition 4.10: Riesz operator

Let A be a closed, linear operator in the Hilbert space X with simple eigenvalues λn∞n=1 and associatedeigenfunctions that form a Riesz–basis φn∞n=1for X . The operator A is called Riesz operator, if theclosure of the set λn∞n=1 is (totally) disconnected4.

Important properties of Riesz operators are summarized in the following theorem (Curtain and Zwart, 1995).

4Totally disconnected means here that no elements µ, η ∈ λn ∞n=1 exist which are connected through a line or a segment which

completely lies in λn ∞n=1.


Theorem 4.8: Properties of Riesz operators

Let A be a Riesz operator with simple eigenvalues λn∞n=1 and associated eigenfunctions φn∞n=1. Letfurther ψn∞n=1 denote the set of eigenfunctions of the adjoint operator A∗ with ⟨φn ,ψm⟩ = δn,m . Thenthe following holds true:

(i) A has a spectral representation

Ax =∞∑

n=1λn ⟨x ,ψn⟩φn (4.80)

for x ∈D(A) with

D(A) =

x ∈X∣∣∣ ∞∑

n=1|λn |2| ⟨x ,ψn⟩ |2 <∞

. (4.81)

(ii) A is the infinitesimal generator of a C0–semigroup T (t ) if and only if supn≥1ℜλn <∞. In this caseT (t ) has the spectral representation

T (t ) =∞∑

n=1eλn t ⟨ · ,ψn⟩φn . (4.82)

(iii) The growth bound of the semigroup is given by

ω0 = inft>0

(1

tlog‖T (t )‖

)= sup

n≥1ℜλn. (4.83)

(iv) The resolvent set and the spectrum are given by ρ(A) = λ ∈C | infn≥1 |λ−λn | > 0 andσ(A) = λn∞n=1.For λ ∈ ρ(A) the resolvent is given by

R(λ, A) = (λI − A)−1 =∞∑

n=1

1

λ−λn⟨ · ,ψn⟩φn . (4.84)

An application of this theorem is illustrated in the following for an extension of Example 4.11 for the linear heatconduction problem.

Example 4.12 (Linear heat conduction as Riesz operator). Consider the abstract Cauchy–problem

dx

dt(t ) = Ax(t ), t > 0 (4.85)

x(0) = x0 ∈D(A) (4.86)

with A and D(A) as in (4.77). According to the above, the eigenvalues and associated eigenfunctions of theoperator A are given by −(nπ)2∞n=1 and

p2sin(nπz)∞n=1. The later constitute an orthonormal basis for the

space X = L2(0,1). With Theorem 4.8(ii), the solution of (4.85), (4.86) can be written as

x(t ) = T (t )x0 =∞∑

n=1e−(nπ)2t ⟨x0,

p2sin(nπz)⟩L2

p2sin(nπz).

Furthermore Theorem 4.8(iii) implies that ‖T (t)‖ ≤ M exp(−π2t) with M > 0. Obviously, the properties ofRiesz operators enable important conclusions about the dynamic behavior of the solution.

Riesz operators include important linear examples from, e.g., mechanics (wave equation, Euler-Bernoullibeam, Timoshenko beam) and heat- and mass transfer (diffusion equation, diffusion-convection-reactionsystems). This includes also certain spatial variations of the system parameters.


4.2 Controllability and observability

In analogy to the considerations in (Curtain and Zwart, 1995, chapter 4) the structural properties of controllabil-ity and observability are introduced on the basis of the abstract representation of infinite-dimensional systemsΣ(A,B ,C ,D) with inputs u and outputs y , i.e.,

x(t ) = Ax(t )+Bu(t ), t > 0 (4.87)

x(0) = x0 ∈D(A) (4.88)

y(t ) =C x(t )+Du(t ), t ≥ 0. (4.89)

In the following the subsequent assumptions are assumed to hold true:

a) A is the infinitesimal generator of a C0–semigroup T (t ) on the Hilbert space X

b) B is a bounded linear operator mapping from a Hilbert space U to X

c) C is a bounded linear operator from X to a Hilbert space Y

d) D is a bounded linear operator from U to Y

e) u ∈ L2([0,τ];U ) ⇒ ∫ τ0 |ui (t )|2d t <∞, ui ∈U , i = 1, ...,m.

Under these assumptions

x(t ) = T (t )x0 +∫ t

0T (t − s)Bu(s)d s, 0 ≤ t ≤ τ (4.90)

represents a so-called mild solution of (4.87), (4.88) (Curtain and Zwart, 1995, Definition 3.1.4).

4.2.1 Controllability of linear DPS

The notion of controllability refers to the possibility to steer a system from an arbitrary initial state x0 within afinite time interval to a state x1, i.e.,

Definition 4.11

A system Σ(A,B ,−,−)5 is called (state–) controllable, if for a given x0 ∈ X and a given x1 ∈ X there exists atime 0 < τ<∞ and a function u ∈ L2([0,τ];U ) so that x(τ) = x1 in (4.90). If this holds true for all x0 ∈ X ,then Σ(A,B ,−,−) is called completely controllable.

Definition 4.12

The state x1 ∈ X is called reachable from x0 ∈ X , if there exists a time τ> 0 and a function u ∈ L2([0,τ];U )so that x(τ) = x1 in (4.90).

In contrast to finite-dimensional linear time-continuous systems, controllability and reachability for infinite-dimensional systems are not equivalent. Further there exist several different notions of controllability andreachability. The following considerations should thus be considered only as an introduction. For this thenotion of controllability map is necessary, whose definition results from the solution (4.90).

Definition 4.13

5In the following Σ(A,B ,−,−) denotes the system (4.87)–(4.89) with C = D = 0.


The bounded linear map Bτ : L2([0,τ];U ) → X

Bτu :=∫ τ

0T (τ− s)Bu(s)d s (4.91)

is called controllability map for Σ(A,B ,−,−) in [0,τ] (0 < τ<∞).

With this definition the notions of exact and approximate controllability can be discussed.

Definition 4.14

The system Σ(A,B ,−,−) is exactly controllable in [0,τ] (0 < τ<∞), if all points of the state space X arereachable from its origin6 x ′ with t = τ, i.e., if

ranBτ = X . (4.92)

This implies that the image ranBτ = v(t) ∈ X : v(t) =Bτu, u ∈ L2([0,τ];U ) of Bτ, coincides with the wholestate space X . Thus, for any x ′ ∈ X and any x1 ∈ X there exists an input u ∈ L2([0,τ];U ) so that X 3 v (t ) =Bτuresults to v(t) = x1 −T (τ)x ′, implying x(τ) = x1 with (4.90). Anyway, the following theorem shows that exactcontrollability is always lost, wenn finite-dimensional spatially distributed control inputs are analyzed..

Theorem 4.9

Let A be the infinitesimal generator of the C0–semigroup T (t ) in the Hilbert space X . If the input operatorB ∈L (Cm ,X ) for a finite m ∈N, then Σ(A,B ,−,−) is not exactly controllable for all t ∈ [0,τ], τ<∞.

Proof. In the following it is shown that under the assumptions of the theorem for all τ> 0 the controllabilitymap Bτ is a compact operator with closed range ranBτ, what implies, that ranBτ is finite-dimensional (cp.(Curtain and Zwart, 1995, Lemma A.3.22g)). For this is sufficient to consider the case m = 1 with Bu = bu.

Let ti = iτN , i = 0,1, ..., N , N ∈N and let FN : L2([0,τ];U ) →X be the operator defined by

FN u :=N∑

i=1T (ti )b

∫ ti

ti−1

u(s)d s, b ∈X .

Given that FN is bounded and the range ranFN is finite-dimensional, it follows from Lemma 4.1a, that FN iscompact. Let

F∞u =∫ τ

0T (s)bu(s)d s

and consider

‖FN u −F∞u‖X =∥∥∥ N∑

i=1

∫ ti

ti−1

[T (s)b −T (ti )b]u(s)d s∥∥∥

X

≤N∑

i=1

∫ ti

ti−1

‖T (s)b −T (ti )b‖X |u(s)|d s

≤N∑

i=1ε

∫ ti

ti−1

|u(s)|d s, N sufficiently large.

The preceding bound is a direct consequence of the equivalence of the strong continuity of the C0–semigroupT (t ) (for all b ∈X it holds that ‖T (s)b −T (ti )b‖X → 0 for s → ti ) and the boundedness of ‖T (t )‖ over all finitesubinterval of [0,∞) — see (Curtain and Zwart, 1995, Theorem 2.1.6). With Hölders inequality∫ τ

0|u(s)v(s)|d s ≤ ‖u‖Lp ([0,τ])‖v‖Lp [q]([0,τ]),

1

p+ 1

q= 1, (4.93)

6Given the linearity the initial state x0 can be transformed in the x ′ of the state space X without any loss of generality.

4.2 Controllability and observability 45

it follows, that

‖FN u −F∞u‖X ≤ ε‖u‖L2pτ,

where ε→ 0 for N →∞. Due to the uniform convergence F∞ is a compact operator, directly implying thecompactness of Bτ. For compact operators it holds, that the range is closed if the dimension of the rangeis finite. With regard to definition 4.14, it follows that the necessary boundedness of ranBτ implies its finitedimension, and thus the impossibility of having exact controllability with ranBτ =X , dimX =∞.

Obviously the controllability notion has to be weakened in order to account for systems that are of relevance inpractice. This yields the notion of approximative controllability, which represents the generic controllabilityconcept for infinite–dimensional systems.

Definition 4.15

The system Σ(A,B ,−,−) is approximatively controllable in [0,τ] (0 < τ<∞), if for an arbitrary ε> 0 it ispossible to reach any point in the state space from the origin x ′ at a time τ up to a distance ε, i.e.,

ranBτ = X , (4.94)

with ranBτ being the closure of ranBτ.

To further explain this definition (see also (Russell, 1978)), we consider the mild solution (4.90) at time t = τ(0 < τ<∞) (x0 is used redundant to x ′)

S(x0, x(τ)) := x(τ)−T (τ)x0 =Bτu.

If (4.94) holds true there exists for all ε> 0 a uε so that

‖Bτuε−S(x0, x(τ))‖ < ε.

This means that there exists a xε with ‖xε‖ < ε and the property

T (τ)x0 +Bτuε = x(τ)+xε

so that uε steers the initial state x0 to the final state x(τ)+xε with distance ε to x(τ).

Remark 4.5

For linear finite –dimensional systems the definitions 4.14 and 4.15 are equivalent, given that in this caseranBτ = ranBτ.

A further, frequently encountered notion of controllability is the zero–controllability.

Definition 4.16

The system Σ(A,B ,−,−) is exactly zero–controllable in [0,τ] (0 < τ<∞), if it is possible to reach the originx of X in the time τ from all x0 ∈ X , i.e.,

ranBτ ⊃ ranT (τ). (4.95)

4.2.2 Observability notions for linear DPS

The dual structural property to the controllability is the observability (Russell, 1978; Curtain and Zwart, 1995).Here also exist several notions, from which in the following the most important are discussed. In analogy tothe controllability map, the observability map is essential. For this purpose the mild solution (4.90) is put inrelation with the output equation (4.89), under the assumption that B = 0, D = 0 (autonomous system without


feedthrough), i.e.,

y(t ) =C T (t )x0. (4.96)

The observability map is then introduced by the following definition.

Definition 4.17

The bounded linear mapping C τ : X → L2([0,τ];Y )

C τx :=C T (τ)x

for Σ(A,−,C ,−) in [0,τ] (0 < τ<∞) is called observability map.

This definition allows to introduce the notions of exact and approximate observability (Curtain and Zwart,1995) in analogy to the corresponding notions on controllability discussed above. .

Definition 4.18

The system Σ(A,−,C ,−)7is exactly observable in [0,τ] (0 < τ<∞), if the initial state can be determineduniquely and continuously from the knowledge of the output in L2([0,τ];Y ), i.e., if C τ is injective withbounded inverse in ranC τ.

To explain this notion some functional analytic notions have to be introduced.

Definition 4.19

An operator A : D(A) ⊂ X → Y between linear spaces X and Y is invertible, if there exists a mappingS : D(S) := ranA ⊂ Y → X so that

S Ax = x , x ∈D(A)

AS y = y , y ∈ ranA.

S = A−1 is called the algebraic inverse of A.

Lemma 4.4

Let A be a linear operator from X to Y . Then it holds that A is invertible if and only if A is injective, i.e.,Ax = 0 implies x = 0.

Exact observability thus means, that from C τx = 0 it follows that x = 0, or equivalently from C τx1 = C τx0

it follows that x1 = x0 (states are univocally distinguishable). Boundedness of a linear operator implies itscontinuity (Curtain and Zwart, 1995, Theorem A.3.10).

Definition 4.20

The system Σ(A,−,C ,−) is approximatively observable in [0,τ] (0 < τ < ∞), if the initial state can beuniquely determined from the knowledge of the output in L2([0,τ];Y ), i.e.,

kerC τ = 0 = x ∈ X : C τx = 0. (4.97)

This definition means, that only the origin of the state space is mapped to 0, i.e., there does not exists anyx ∈ X \ 0 so that C τx = 0.

The duality of controllability and observability is illustrated through the following lemma (Curtain and Zwart,1995, Theorem 4.1.13)

7The system Σ(A,−,C ,−) will denote in the following the system (4.87)–(4.89) with B = D = 0.


Lemma 4.5

Let Σ(A,−,C ,−) be a linear system. Then it holds true that:

a) System Σ(A,−,C ,−) is approximately observable in [0,τ] (0 < τ <∞) if and only if the dual systemΣ(A∗,C∗,−,−) is approximately controllable in the same time interval.

b) SystemΣ(A,−,C ,−) is exactly observable in [0,τ] (0 < τ<∞) if and only if the dual systemΣ(A∗,C∗,−,−)is exactly controllable in the same time interval.

Remark 4.6

Often it is simpler to show the non–observability. Let x( · , t) = x(t) = T (t)x0 with T (t) being the C0–semigroup generated by A. Then it holds for the output vector that y(t) =C T (t)x0. The system is notobservable if there exist states x (1)

0 , x (2)0 ∈ X with x (1)

0 6= x (2)0 so that C T (t)x (1)

0 = C T (t)x (2)0 for t ∈ [t0, t1].

The states x (1)0 , x (2)

0 are indistinguishable in this case, contradicting the observability.

4.2.3 Analysis of approximative controllability and observability

The proof of exact controllability and observability is typically complex and often even intractable. Anyway, theanalysis of approximate controllability and observability is accessible. According to (Curtain and Zwart, 1995)these approximate properties can be considered the generic extension of the well–established notions fromfinite–dimensional systems theory.

The following theorem provides a direct criterion for approximate controllability and observability for Rieszspectral operators (Curtain and Zwart, 1995, S. 164).

Theorem 4.10

Let Σ(A,B ,C ,−) be a linear system with Riesz spectral operator A in a Hilbert space X , i.e., there ex-ist simple eigenvalues λn , eigenfunctions φn(z), adjoint eigenfunctions ψn(z), n ≥ 1, such that Ax =∑∞

n=1λn⟨x ,ψn⟩φn . Additionally, let B and C be operators of the form

Bu =m∑

i=1bi ui bi ∈ X (4.98)

C x = [⟨x ,c 1⟩, . . . ,⟨x ,c p⟩]T c i , x ∈ X . (4.99)

The system Σ(A,B ,−,−) is approximately controllable if and only if for all n ∈N

rank(⟨b1,ψn⟩, . . . ,⟨bm ,ψn⟩) = 1. (4.100)

The system Σ(A,−,C ,−) is approximately observable if and only if for all n ∈N

rank(⟨c 1,φn⟩, . . . ,⟨c p ,φn⟩) = 1. (4.101)

An illustrative explanation of the results of Theorem 4.10 is presented in continuation.

• Proof of approximate controllability of a Riesz system: According to the assumptions of Theorem 4.10and the considerations in Section 4.1.3.4 it follows that for the system

dx

dt(t ) = Ax(t )+Bu(t ), t > 0

x(0) = x0 ∈D(A),


it holds that

⟨ ∞∑n=1

⟨x ,ψn⟩︸︷︷︸= xn(t )

−λn ⟨x ,ψn⟩︸︷︷︸= xn(t )

φn ,ψm

⟩=

⟨Bu,ψm

⟩.

Given the bi-orthonormality, i.e., ⟨φn ,ψm⟩ = δn,m it follows that

xn(t )−λn xn(t ) = ⟨Bu,ψn⟩ (4.98)=m∑

i=1⟨bi ,ψn⟩ui (t ), n ∈N.

In accordance, approximate controllability means that every Fourier–coefficient, or modal state xn(t ), n ∈N is influenced by at least one ui , i.e.,

rank(⟨b1,ψn⟩, . . . ,⟨bm ,ψn⟩) = 1, ∀n ∈N.

For the above analysis one has to ensure that the eigenvalues λnn∈N are disjoint. In the case of highermultiplicity of an eigenvalue one has to make further analysis, and the controllability property dependson the algebraic and geometric multiplicity of the eigenvalues (see also Example 4.15).

• Proof of approximate observability for Riesz systems: According to the assumptions of Theorem 4.10and the considerations in Section 4.1.3.4 it follows that for the system

dx

dt= Ax +Bu, t > 0

x(0) = x0 ∈D(A),

mit

y =C x , t ≥ 0,

it holds true that

y =C x =∞∑

n=1⟨x ,ψn⟩Cφn

(4.99)=∞∑

n=1⟨x ,ψn⟩

⟨φn ,c 1⟩

...

⟨φn ,c p⟩

︸︷︷︸

=Cn

.

Accordingly, approximate observability implies that the measurement signal y(t ) contains information ofall spectral (or modal) states ⟨x ,ψn⟩, i.e.,

rank(⟨c 1,φn⟩, . . . ,⟨c p ,φn⟩) = 1.

Systems that satisfy the conditions of Theorem (4.100) are also often called modally controllable, i.e., eacheigenmode can be controlled. In general this is not equivalent to the approximate controllability, but thisholds true for Riesz operators. Anyway one has to take into account the restriction that modal controllability isnot necessarily equivalent to the approximate controllability in [0,τ] for arbitrary 0 < τ<∞. An example forthis difference is provided by the linear wave equation (Example 4.14), which requires a minimum time forcontrollability given the finite wave propagation speed. In addition, for the controllability and observabilityanalysis of non-Riesz operators different methods have to be employed.

Example 4.13 (Controllability– and observability analysis for the linear heat equation). For the configu-ration shown in Figure 4.1 one obtains the equations for the temperature evolution in the one-sided ideallyisolated rod as

∂t x(z, t ) = a∂2z x(z, t )−αx(z, t )+b(z)u(t ) z ∈ (0,L), t > 0 (4.102)

x(0, t ) = v(t ), ∂z x(L, t ) = 0 t > 0 (4.103)


x(z, t )

-

z0 L

-

? ? ? ? ? @@@@@@@@@@@@@@

v(t )

u(z, t ) = b(z)u(t )

Figure 4.1: Konfiguration des Wärmeleiters.

x(z,0) = x0(z) z ∈ [0,L]. (4.104)

Following the approach for homogenization in Example 4.7 the above system can be rewritten in state spaceform. In the following (4.102)–(4.104) a different approach based on Green’s theorem is used to write the systemin the modal or spectral form. This basically corresponds to the abstract view in the sense of C0–semigroupsand their infinitesimal generators.

With the properties of the self–adjoint Riesz operator

Ax = ad2x

dz2 −αx

according to Section 4.1.3.4 and the introduction of the spectral, or modal state xk = ⟨x( · , t ),ψk⟩ in the statespace X = L2(0,L) one obtains the spectral representation of (4.102)–(4.104) as

dxk

dt= ⟨∂t x,ψk⟩ =

∫ 1

0(a∂2

z x −αx +bu)ψk dz

= a

[∂z xψk −x

dψk

dz

]1

0+

∫ 1

0x

(a

d2ψk

dz2 −αψk

)dz +

∫ 1

0bψk dz u

and after substituting the inhomogenous boundary conditions

dxk

dt=λk xk +

∫ L

0b(z)ψk (z)dz︸︷︷︸

= bk

u(t )+adψk

dz(0)︸︷︷︸

= bk

v(t ), t > 0, k ∈N (4.105)

with the eigenvaluesλk =−[a( 2k−12

πL )2+α] and the associated eigenfunctionsψk (z) =φk (z) =p

2/L sin( 2k−12 π z

L )where ⟨ψk (z),φm(z)⟩ = δk,m . In the following first the modal, or approximative controllability of the system isanalyzed for the case of the boundary input or spatially distributed input.

• Let u(t ) = 0 and v(t ) 6= 0 (boundary input): Obviously (4.105) simplifies to

dxk

dt=λk xk + bk v(t ) mit bk =

√2

La

(2k −1

2

π

L

).

Given that λk 6= λm for k 6= m and bk 6= 0 for all k ∈ N the system is modally, or approximatelycontrollable through the boundary.

• Let u(t ) 6= 0 and v(t ) = 0 (spatially–distributed input): In this case it follows from (4.105) that

dxk

dt=λk xk +bk u(t ) mit bk =

∫ L

0b(z)ψk (z)d z,

showing that the modal controllability directly depends on the appropriate choice of adequate spatialcharacteristics of the input. This design problem is further analyzed in the sequel for the case of apointwise input, i.e., b(z) = δ(z −ζ), ζ ∈ (0,L). Given that λk 6= λm for k 6= m the controllability of the


system is ensured if and only if bk = ∫ L0 δ(z −ζ)ψk (z)d z =ψk (ζ) =p

2/L sin( 2k−12 π

ζL ) 6= 0 for all k ∈N.

The shape of the eigenfunctions ψk (z) is shown in Figure 4.2. It is easily seen that the above conditionrequires that ζ does not correspond to any zero of the eigenfunctions ψk . This means that

2k −1

2πζ

L6= mπ ⇒ ζ

L6= 2m

2k −1= even

odd, k,m ∈N

or that ζ/L has to be an irrational number.

• Reachability of stationary profiles: It should be highlighted at this point that the proof of controllabil-ity and the reachability of desired stationary profiles or operation points are not equivalent for DPS. Forexample, for the considered example of the heat conduction, taking u(t ) = 0 and v = vs , the stationary

solution is given by a xs (z) = cosh(√

αa [L− z])/cosh(

√αa L)vs . The variation of vs ∈R provides the set of

the stationary profiles that can be reached from the boundary input v. Exemplarily Figure 4.3 showssome of the reachable stationary profiles.

As shown in the general consideration above, the analysis of modal or approximate observability for Rieszsystems directly follows the lines of the controllability analysis. For example, in the case of a pointwisemeasurement with spatial characteristics c(z) = δ(z − ζ), ζ ∈ (0,L) it follows that

y(t ) =∞∑

k=1xk ⟨ψk (z),c(z)⟩︸︷︷︸

=ψk (ζ)

=∞∑

k=1xk

√2

Lsin

(2k −1

2πζ

L

)︸︷︷︸= ck

.

Obvously the system is modally controllable if and only if ck 6= 0 for all k ∈N. This design–problem for theappropriate choice of the measurement location ζ is solvable if the sensor location does not coincide with azero of the eigenfunctions ψk (z). This is the case if and only if

ζ

L6= 2p

2k −1= even

odd, k, p ∈N,

or equivalently, if ζ/L is a irrational number. A comparison with the above considerations shows the knownduality between controllability and observability.

aThis solution can be obtained using a Laplace transform of the pde (4.102)-4.104.

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

z/L

ψk

k=1

k=2

k=3

Figure 4.2: Shape of the eigenfunctions ψk (z) =p2/L sin( 2k−1

2 π zL ) for k = 1,2,3 and L = 1.


0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

z

xs

vs=−1

vs=1

vs=2

Figure 4.3: Stationary profiles xs reachable by a boundary input vs forpα/a = 1 and L = 1.

Example 4.14 (Controllability and observability analysis for the linear wave equation). In the precedingexample it was possible to show the approximate controllability and observability for the linear heat equation,with the particularity that no restrictions on the control time τ in Definition 4.15 were necessary. According tothe considerations in Chapter 3 on the characterization of DPS for hyperbolic systems, like the linear waveequation different results have to be expected. This is illustrated in the following on the basis of the PDE

∂2t x(z, t ) = c2∂2

z x(z, t ), z ∈ (0,L), t > 0 (4.106)

with BCs and ICs

x(0, t ) = v(t ), ∂z x(L, t ) = 0, t > 0, (4.107)

x(z,0) = x0(z), ∂t x(z,0) = x1(z), z ∈ [0,L], (4.108)

using the method of so-called non-harmonic Fourier analysis (see, e.g., (Russell, 1967)). Given the Rieszproperty of the system operator one can directly (e.g. using Green’s theorem) obtain the spectral representationof (4.106)–(4.108) in the form

d2xk

dt 2 = c2λk︸︷︷︸=−µ2

k

xk + c2 dψk

dz(0)︸︷︷︸

= bk

v(t ), k ∈N (4.109)

xk (0) = xk,0 = ⟨x0(z),ψk (z)⟩ (4.110)

dxk

dt(0) = xk,1 = ⟨x1(z),ψk (z)⟩. (4.111)

The associated eigenvalue problem is self–adjoint with orthonormal eigenfunctions

ψk (z) =φk (z) =p

2/L sin(2k −1

2π

z

L)

and discrete eigenvalues λk = ±i 2k−12

πL and eigenfrequencies µk = 2k−1

2πL , k ∈ N. The solution can thus be

obtained using Fourier–synthesis

x(z, t ) =∞∑

k=1

[xk,0 cos(µk t )+ xk,1

µksin(µk t )

]ψk (z)+

∞∑k=1

ψk (z)bk

µk

∫ t

0sin(µk (t −τ))v(τ)dτ. (4.112)

The subsequent analysis corresponds to the analysis of the zero–controllability. It concerns the question,whether there exist a function v and a time interval t = [0,T ], T > 0 such that for x0(z) 6= 0 and x1(z) 6= 0 oneobtains x(z,T ) = 0 and ∂t x(z,T ) = 0.

From (4.112) it follows that

0 = x(z,T )


=∞∑

k=1ψk (z)

[xk,0 cos(µk T )+ xk,1

µksin(µk T )+ bk

µk

=:αk︷︸︸︷∫ T

0sin(µk (T −τ))v(τ)dτ

]︸︷︷︸

= 0

0 = ∂t x(z,T )

=∞∑

k=1ψk (z)

[−µk xk,0 sin(µk T )+xk,1 cos(µk T )+bk

=:βk︷︸︸︷∫ T

0cos(µk (T −τ))v(τ)dτ

]︸︷︷︸

= 0

.

Obviously αk and βk are the Fourier–coefficients of the input function v. In addition one obtains equationsfor the explicit determination of αk and βk for k ∈N. For this certain existence requirements must be fulfilledthat are derived in the following. With

αk =∫ T

0sin(µk (T −τ))v(τ)dτ=

∫ T

0sin(µkτ) v(T −τ)︸︷︷︸

= v(τ)

dτ

and

βk =∫ T

0cos(µk (T −τ))v(τ)dτ=

∫ T

0cos(µkτ)v(τ)dτ

and µk = (2k −1)πc/(2L) it follows that

αk =∫ T

0sin(kωτ)v(τ)dτ, βk =

∫ T

0cos(kωτ)v(τ)dτ

where kω= (2k −1)πc/(2L). A comparison with the conditions for the coefficients of the Fourier–series yieldsthe condition

2πk

T= kω= (2k −1)πc

2Lor equivalently T = 4kL

(2k −1)c→ 2L

cfor k →∞.

This shows that at least the time Ts = 2L/c is required to uniquely determine a Fourier–coefficient, implying thatTs establishes the mininum control time. For the analysis of the approximate observability the measurement

y(t ) =C x( · , t ) mit C x( · , t ) =∫ L

0c(z)x(z, t )d z (4.113)

is considered. In analogy to the considerations in Example 4.13, given the Riesz system properties, modal,or approximate controllability are ensured, if ck = Cψk ( ·) = ∫ L

0 c(z)ψk (z)d z 6= 0 for all k ∈ N. In the caseof a pointwise measurement with c(z) = δ(z − ζ), ζ ∈ (0,L] it follows that ζ/L must be an irrational num-ber. Additionally, applying the method of Fourier–analysis shows that observability requires a minimummeasurement time.

For this purpose, in analogy to the proof of the existence of a minimum control time Fourier–analysis isemployed. With v = 0 it follows from (4.112) that

y(t ) = x(ζ, t ) =∞∑

k=1

[xk,0 cos(µk t )+ xk,1

µksin(µk t )

]ψk (ζ). (4.114)

Given that ζ/L is an irrational number it follows directly from the approximate observability of the system,that ψk (ζ) 6= 0 for all k. In particular y(t ) is a periodic function due to (4.114). A comparison with the generalformula for a Fourier–series

y(t ) = A0 +∞∑

k=1[Ak cos(kωt )+Bk sin(kωt )] ,


0 ζ L z

Lc

2Lc

Figure 4.4: Graphical interpretation of the existence of a minimum control and measurement time using characteristics for(4.106)–(4.108).

yields the conditions

A0 = 0

Ak =ψk (ζ)∫ L

0x0(z)ψk (z)dz

Bk = ψk (ζ)

µk

∫ L

0x1(z)ψk (z)dz

with kω= 2kπ/T = (2k −1)πc/(2L). In consequence

T = 4kL

(2k −1)c→ 2L

cfor k →∞,

implying the minimum measurement time Tm = 2L/c.

A graphical interpretation of this condition is provided in Figure 4.4. As shown in Chapter 3.2 the propaga-tion of x(z, t) takes place along charakteristic curves of the systems. Obviously the structural properties ofcontrollability and observability require that in the the (z, t )–plane the location of influence of v(t ) and themeasruement location for y(t ) all characteristics need to intersect at least once.

Remark 4.7

Obviously the property of a minimum measurement, or control time is characteristic for hyperbolicDPS due to the finite propagation velocity of signals. This property also holds in a similar way for certainclasses of of nonlinear hyperbolic DPS, with a considerably more envolved analysis.

Remark 4.8

The considered example of the linear wave equation is exactly controllable without a minimum controltime if instead of a boundary input at z = 0 one considers a spatially distributed infinite–dimensionalinput u( · , t ) in the PDE (4.106).


Temperature x(z1, z2, t )

Ω= (0,L1)× (0,L2)

(0,0)

(L1,L2)

z1

z2

Input

v(z1, t )

Figure 4.5: Configuration of a 2–dimensional heat conductor.

Example 4.15 (Controllability of a spatially 2–dimensional heat conduction problem). In the followingthe controllability of the heating of the plate shown in in Figure 4.5 is considered.

For this example the energy balance yields the 2–dimensional heat equation

∂t x = ∂2z1

x +∂2z2

x =∆x (z1, z2) ∈Ω, t > 0 (4.115)

x(0, z2, t ) = 0 = x(L1, z2, t ) z2 ∈ [0,L2], t > 0 (4.116)

x(z1,0, t ) = v(z1, t ), x(z1,L2, t ) = 0 z1 ∈ [0,L1], t > 0 (4.117)

x(z1, z2,0) = x0(z1, z2) (z1, z2) ∈Ω. (4.118)

The associated eigenvalue problem is self–adjoint

∆ψ=λψ (z1, z2) ∈Ωψ(0, z2) = 0 =ψ(L1, z2) z2 ∈ [0,L2]

ψ(z1,0) = 0 =ψ(z1,L2) = 0 z1 ∈ [0,L1]

with eigenvalues λk,l and eigenfunctions φk,l (z1, z2) =ψk,l (z1, z2) according to

λk,l =−π2[(

k

L1

)+

(l

L2

)](4.119)

ψk,l (z1, z2) = 2pL1L2

sin

(kπ

L1z1

)sin

(lπ

L2z2

)(4.120)

with k, l ∈ N. In consequence one can transform (4.115)–(4.118) with Green’s theorem into the spectralrepresentation with xk,l (t ) := ∫ L1

0

∫ L20 x(z1, z2, t )ψk,l (z1, z2)d z2d z1

dxk,l (t )

dt=λk,l xk,l (t )+

∫ L1

0∂z2ψk,l (z1,0)v(z1, t )d z1 t > 0 (4.121)

xk,l (0) = x0k,l k, l ∈N. (4.122)

Alternatively one can represent the equations (4.115)–(4.118) in an extended state-space following the homoge-nization approach from Example 4.7.

In the following the modal, or approximate controllability is analzyed in dependency of the type of input:

• spatially continuous input in z1, i.e., v( · , t ) 6= b( · )v(t )

• spatially discrete input in z1, i.e., v( · , t ) = b( · )v(t )

For this purpose we first need to analyze the point spectrum, i.e., the set of all eigenvalues of the system


operator. Obviously it holds that for multiple eigenvalues

λk,l =λm,n ⇔(

k

L1

)2

+(

l

L2

)2

=(

m

L1

)2

+(

n

L2

)2

⇔(

k2 −m2

n2 − l 2

)2

=(

L1

L2

)2

.

It can be easily seen that for a rational relation, L1/L2 ∈Q, multiple eigenvalues can appear. As an exampleconsider L1 = L2 = L, so that λk,l =−(πL )2(k2 + l 2) =λl ,k , k, l ∈N. In addition one has to take into account thatfrom λk,l =λm,n it not necessarily results that ψk,l =ψm,n . In the case that L1 = 2L2 one has, e.g., λ4,1 =λ2,2

but ψ4,1 ∼ sin( 2πz1L2

)sin(πz2L2

) while ψ2,2 ∼ sin(πz1L2

)sin( 2πz2L2

).

Having these considerations for the point spectrum and the related eigenfunctions as point of departure theanalysis of modal or approximate controllability is based on the modal or spectral representation (4.121), sothat with (4.120) the following system of ODEs is obtained

dxk,l (t )

dt=λk,l xk,l (t )+ 2p

L1L2

(lπ

L2

)︸︷︷︸=αl

∫ L1

0sin

(kπ

z1

L1

)v(z1, t )d z1 t > 0, k, l ∈N. (4.123)

• Let v(z1, t) be a spatially continuous input: with vk (t) := ∫ L10 sin(kπ z1

L1)v(z1, t)d z1 one can order

(4.123) with respect to the indices k into subsystems Σk , k ∈N

Σ1 :dx1,l (t )

dt=λ1,l x1,l +αl v1(t ), l ∈N

Σ2 :dx2,l (t )

dt=λ2,l x2,l +αl v2(t ), l ∈N

...

Σk :dxk,l (t )

dt=λk,l xk,l +αl vk (t ), l ∈N

...

In particular each subsystem Σk , k ∈N is controlled by a separate modal component vk (t ) of the inputv( · , t ). Further, the eigenvalues λk,l in each subsystem Σk are disjoint (in general, anyway, not for allsubsystems). Given that αl 6= 0 holds true for all l ∈N every subsystem Σk is modally, or approximatelycontrollable, implying that the complete system is approximately controllable.

• Let v(z1, t) = b(z1)v(t) be a spatially discrete input: With βk := ∫ L10 sin(kπ z1

L1)b(z1)d z1 one has that

from (4.123) it follows that

dxk,l (t )

dt=λk,l xk,l (t )+αlβk v(t ), k, l ∈N. (4.124)

According to the above analysis of the point spectrum for the analysis of the controllability the subsequentcases have to be differentiated.

– Let L1/L2 ∈ Q: There exist k, l ,m,n ∈ N, k 6= m, l 6= n with λk,l = λm,n . This implies that, given(4.124) one has

dxk,l (t )

dt=λk,l xk,l (t )+αlβk v(t )

dxm,n(t )

dt=λk,l xm,n(t )+αnβm v(t ),

so that a non–controllable subsystem is identified. In consequence the system is not modally, orapproximately controllable for L1/L2 ∈Q.

– Let L1/L2 6∈ Q: In this case the point spectrum is disjoint. Given αl = αn ⇔ l = k the system ismodally, or approximately controllable if and only if βk 6= 0 for all k ∈N.

Summarizing this example:


• Spatially higher–dimensional problems can present eigenvalue multiplicity leading to a loss of controlla-bility.

• A more detailed controllability analysis for the heated plate shows that not all final profiles x( · , · ,T ) isreachable (additional conditions are anyway only sufficient and not necessary).

Given the duality between controllability and observability, similar problems appear in the analysis of observ-ability. This is exemplarily shown in the following example provides an example of a non Riesz operator thatalso implies a minimum control time. For non Riesz systems this analysis required an alternative approach,given that Theorem 4.10 cannot be applied.

x(z, t )

-

z0 L

-

u(t )

---------

u?

y(t )

Figure 4.6: Configuration of the transport system considered in example 4.16.

Example 4.16 (Observability of a transport system). In the following, the following is explained using theexample shown in illustration 4.6 is shown Example of a flow through an incompressible medium tube isshown, how to select the appropriate measuring point to to ensure observability.

The system is governed by

∂t x(z, t )+ v∂z x(z, t ) = 0 z ∈ (0,L), t > 0 (4.125)

x(0, t ) = u(t ) t > 0 (4.126)

x(z,0) = x0(z) z ∈ [0,L] (4.127)

y(t ) = x(ζ, t ) ζ ∈ [0,L], t ≥ 0 (4.128)

described. The axial Flow velocity v is assumed to be constant. The measurement is to occur at the positionζ ∈ [0,L].

Using the Laplace-transformation or the characteristics-method the solution of the PDgl. can easily bedetermined. The latter method (derived from the classification of PDgln. in section ??) uses the informationthat the propagation takes place along the characteristics, which are to

dt

dz= 1

vbzw. t − t0 = 1

v(z − z0)

surrender. These are shown below.

z

t

L

Lv

In case u(t ) = 0 the solution is

x(z, t ) = x0(z − v t )σ(z − v t ).


For the observability analysis the output or measured variable according to y(t ) = x(ζ, t ) = x0(ζ− v t )σ(ζ− v t )selected with ζ ∈ [0,L]. As introduced above, the property of observability, that is derived from the knowledgeoutput variable y(t), t ∈ (0,Tm] the initial condition x0(z), z ∈ [0,L] continuously and unambiguouslyreconstruct lets. To analyze this, the substitution

z = ζ− v t ⇒ t = ζ− z

v

is inserted into the equation of the output variable, i.e.

y

(ζ− z

v

)= x0(z)σ(z) = x0(z),

where the last equation is derived from the property that z ≥ 0 results. The reconstruction of the initialprofile from the measured variable requires in particular the determination of x0(L). For this purpose first themeasuring point to ζ= L−ε with ε≥ 0 selected. Thus the above equation results in

x0(L) = y(− ε

v

).

Since −ε/v < 0 for ε> 0 requires in this case the determination of x0(L) the knowledge of y(t ) for t < 0 which isa a–causality. Obviously causality can only be can be produced when ε= 0 or ζ= L. In in this case there are

x0(0) = y

(L

v

), x0(L) = y(0),

where the first equation again describes the existence or necessity of a minimum measurement time Tm = L/vis illustrated.

References

Curtain, R. F. and H. Zwart (1995). An introduction to infinite-dimensional linear systems theory. Springer(cit. on pp. 25, 29, 32, 34–42, 44–48).

Dettman, J. W. (1988). Mathematical Methods in Physics and Engineering. Dover Publications, McGraw-Hill(cit. on p. 27).

Engel, K.J. and R.Nagel (2006). A short course on operator semigroups. Springer-Verlag New York (cit. on pp. 29,34, 35).

Evans, L. (2002). Partial Differential Equations. Vol. 19. Graduate studies in mathematics. American Mathemati-cal Society, Providence, Rhode Island (cit. on p. 29).

Fattorini, H. (1968). „Boundary control systems“. In: 6(3), pp. 349–388 (cit. on p. 34).Heuser, H. (1992). Funktionalanalysis. 3rd. B. G. Teubner, Stuttgart (cit. on pp. 37–39).Ho, L. and D. Russell (1983). „Admissible input elements for systems in Hilbert space and a Carleson measure

criterion“. In: SIAM J Control Optim 21(4), pp. 614–640 (cit. on p. 34).Jüngel, A. (2001). Eine Einführung in die Halbgruppentheorie. Vorlesungsskript, Univ. Konstanz (cit. on p. 29).Kato, T. (1980). Perturbation Theory for Linear Operators. Springer, Berlin (cit. on p. 40).Naylor, A. and G. Sell (1982). Linear Operator Theory in Engineering and Science. Vol. 40. Applied Mathematical

Sciences. Springer–Verlag, New York (cit. on pp. 38, 39).Russell, D. (1967). „Nonharmonic Fourier Series in the Control Theory of Distributed Parameter Systems“. In: J

Math Anal Appl 18(3), pp. 542–560 (cit. on p. 52).— (1978). „Controllability and stabilizability theory for linear partial differential equations: Recent progress

and open questions“. In: SIAM Review 20(4), pp. 639–739 (cit. on p. 46).Tucsnak, M. and G. Weiss (2009). Observation and Control of Operator Semigroups. Birkhäuser-Verlag, Basel

(cit. on pp. 34, 42).


Chapter 5

Stability theory

5.1 Introduction to the stability theory of SVPn

The investigation of the stability of a dynamical system presents one of the most important questions in systemstheory. From the point of view of control engineering, this also includes the question of the stabilizability, i.e.the possibility of stabilizing a system by means of control action. In the following an introduction to the basicconcepts is given using the abstract representation of an autonomous system

dx

dt(t ) = Ax(t ), t > 0, x(0) = x0 ∈D(A) (5.1)

with operator A ∈D(A) ⊂X in a Hilbert space X . Here A is an infinitesimal generator of a C0–semigroup T (t ),so that x(t ) = T (t )x0.

To motivate and illustrate the theory of stability for DPS, first some basic results for finite-dimensional linearsystems of the form

dx(t )

dt= Ax(t ), t > 0

x(0) = x0

with A ∈ Rn×n with solution given by x(t) = T (t)x0 = e At x0 are recalled. In particular, it is known that forfinite–dimensional systems the following statements are equivalent:

a) x(t ) is asymptotically stable

b) x(t ) is exponentially stable

c) all eigenvalues of A lie in the open left half complex plane

d) for all p ∈N, p ≥ 1 and t0 ≥ 0 it holds true that∫ ∞

t0‖x(t )‖p dt <∞

e) the state transition matrix T (t) = e At is exponential stable, i.e. ∃M , ω> 0 so that ‖T (t)‖ ≤ Me−ωt where‖T (t )‖ is the operator norm of T (t ).

It should be emphasized that for DPS the statements (i)–(v) are generally not equivalent. This will be ex-plained in more detail in the following sections.

Starting from the functional-analytical basics, it is easy to see in the infinite–dimensional case that the charac-terization of the stability of a DPS can be done directly via the solution, i.e. on the basis of the C0–semigroupT (t ) generated by the system operator A (5.1). However, since T (t ) can often not be analytically determined, itis often preferrable to analyze the stability of the system directly on the basis of the infinitesimal generator A.

59

5.1.1 Stability of C0–semigroups

There are different notions of stability for semigroups (see e.g. Luo et al., 1999, chapter 3), and the mostimportant ones, related in particular to asymptotic and exponential stability are defined next.

Definition 5.1

The C0–semigroup T (t ) is called

• „weakly stable", if ∀x1, x2 ∈X , ⟨x1,T (t )x2⟩→ 0 for t →∞• „strongly stable", if ∀x ∈X , T (t )x → 0 for t →∞• asymptotically stable, if for all x ∈X , ‖T (t )x‖X → 0 for t →∞• exponentially stable, if M , α> 0 exist so that

‖T (t )‖ ≤ Me−αt for t ≥ 0. (5.2)

The constant α is known as decay rate and the supremum over all possible α is called the stabilityreserve.

Note that ‖T (t)‖ denotes the operator norm. Obviously the term stability for infinite-dimensional systemsdepends on the norm ‖ ·‖X for the selected state space X . This means that, different from the results for CPSs,in the DPS case the stability with respect to one norm does not imply the stability with respect to anotherone1.

The problems appearing the stability analysis of infinite-dimensional systems will be discussed in the followingon the basis of two examples.

Example 5.1 (Linear operator in l 2 (Luo et al., 1999)). Let X = l 2 be the Hilbert space of quadratic summablesequences and

T (t )x = (e−t x1,e−t/2x2, ...,e−t/n xn , ...), t ≥ 0

for all x = (x1, x2, ..., xn , ...) ∈ X . The operator T (t) : X → X obviously meets the Definition ?? and thusrepresents a C0-semigroup on X . For all x ∈X it holds that

‖T (t )x‖2X =

∞∑n=1

e−2t/n x2n → 0 for t →∞.

According to ?? T (t ) is thus asymptotically stable. On the other hand, for every t ∈ [0,∞), it holds that

‖T (t )‖ = supx∈X ,‖x‖X =1

‖T (t )x‖X = supx∈X ,‖x‖X =1

( ∞∑n=1

e−2t/n x2n

) 12

= limn→∞e−t/n = 1.

This follows directly from the minimum decay rate of the expressions exp(−2t/n) for t ≥ 0, which is equal to 1for n →∞.

Hence, the semigroup T (t ) is not exponentially stable, showing that for infinite-dimensional systems asymp-totic stability does not imply exponential stability.

Example 5.2 (Shift–semigroup (Curtain and Zwart, 1995; Luo et al., 1999)). As an example of a non-Rieszoperator consider the so called Shift–semigroup

T (t )x(z) = x(t + z), ∀t ≥ 0. (5.3)

In the following it is shown that this semigroup is a C0–semigroup of contractions with ‖T (t)‖ = 1 in X =1This is a direct analogy to the results in chapter ??, where it was shown that for DPS the choice of a suitable state space (with

corresponding norm) or suitable states, respectively, is linked with the well-posedness property in the sense of Hadamard.

60 Chapter 5 Stability theory

X1 ∩Lp ([0,∞)) with X1 = x(z) ∈ Lp [1]([0,∞)) | |x|1 = ∫ ∞0 ez |x(z)|d z <∞, where the spectrum σ(A) of the

infinitesimal generator of T (t ), i.e. of

Ax = dx

dz, D(A) =

x ∈ Lp (0,∞) : x abs. continuous,

dx

dz∈ Lp (0,∞)

, (5.4)

is completely located in the (open) left half complex plane. With the norm

‖x‖X = ‖x‖Lp ([0,∞)) +|x|1X becomes a Banach-space.

• T (t ) : X →X is contracting: It holds that

‖T (t )x‖X =(∫ ∞

0|x(t + z)|p d z

) 1p +

∫ ∞

0ez |x(t + z)|d z

η = t+z=(∫ ∞

t|x(η)|p dη

) 1p +e−t

∫ ∞

teη|x(η)|dη

≤ ‖x‖Lp ([0,∞)) +|x|1 = ‖x‖X ,

implying

‖T (t )‖ = supx∈X ,x 6=0

‖T (t )x‖X

‖x‖X≤ 1.

According to Definition 4.2 T (t ) is thus a C0–semigroup of contractions.

• ‖T (t )‖ = 1: Consider the function xε ∈X with

xε(z) =

1, z ∈ [t , t +εp ],

0, non,

and ε> 0. It holds that ‖xε‖X = ε+e t (eεp −1), and ‖T (t )xε‖X = ε+eε

p −1, so that

‖T (t )xε‖X = ε+eεp −1

ε+e t (eεp −1)‖xε‖X .

For ε ↓ 0 the application of the rule of l’Hospital to ‖T (t )xε‖X /‖xε‖X = 1 implies that supx∈X ,x 6=0 ‖T (t )x‖X /‖x‖X ≥1. Therefore it holds that ‖T (t )‖ = 1.

• Spectrum σ(A) of the infinitesimal Generator: As shown in Definition 4.8, the spectrum σ(A) of thegenerator A can be directly determined on the basis of the resolvent set ρ(A), according toσ(A) =C\ρ(A).With (5.3) and (5.4) for all λ ∈ ρ(A), z ∈ [0,∞) the resolvent is given bya

R(λ, A)x(z) = (λI − A)−1x =∫ ∞

0e−λt T (t )x(z)dt =

∫ ∞

0e−λt x(t + z)dt

η = t+z=∫ ∞

ze−λ(η−z)x(η)dη=: gλ(z).

In particular it holds that x = (λI − A)gλ and thus λ ∈ ρ(A) if and only if gλ ∈X , i.e., ‖gλ‖X <∞. Inthe following, it is shown that all λ ∈Cwith ℜλ >−1 are contained in the resolvent set ρ(A).

Let ℜλ >−1. For all z ∈ [0,∞) it holds that

|gλ(z)| =∣∣∣∣∫ ∞

ze−λ(η−z)x(η)dη

∣∣∣∣≤ |eλz |∫ ∞

z|e−ληx(η)|dη

≤ eℜλz∫ ∞

ze−ℜλη︸︷︷︸

= e−(ℜλ+1)ηeη

|x(η)|dη (?)

5.1 Introduction to the stability theory of SVPn 61

In consequence, for |gλ|1 it follows that

|gλ|1 =∫ ∞

0ez |gλ(z)|dz

(?)≤∫ ∞

0ez eℜλz

∫ ∞

ze−(ℜλ+1)ηeη|x(η)|dηdz

=∫ ∞

0

∫ ∞

ze(ℜλ+1)(z−η)eη|x(η)|dηdz.

In virtue of Fubinis theorem (interchange of integration order), implying that (for sufficiently well-behaved functions g : [a,b]× [a,b] →R)∫ b

a

∫ b

zg (z, s)dsdz =

∫ b

a

∫ s

ag (z, s)dzds,

it holds true that∫ ∞

0

∫ ∞

ze(ℜλ+1)(z−η)eη|x(η)|dηdz =

∫ ∞

0

∫ η

0e(ℜλ+1)(z−η)dzeη|x(η)|dη.

Given that∫ η

0e(ℜλ+1)(z−η)dz = e−(ℜλ+1)η 1

ℜλ+1

(e(ℜλ+1)η−1

)≤ 1

ℜλ+1,

it finally follows that

|gλ|1 ≤1

ℜλ+1|x|1 <∞.

Furthermore, it holds that

‖gλ‖Lp =(∫ ∞

0|gλ(z)|p dz

) 1p ≤

(∫ ∞

0|gλ|p1 dz

) 1p ≤

(∫ ∞

0

[1

ℜλ+1|x|1

]p

dz

) 1p

= 1

ℜλ+1|x|1 <∞.

Thus, it holds true that ‖gλ‖ <∞ and in consequence gλ ∈X implying λ ∈ ρ(A). Thus, for the resolventset it follows that

λ ∈C| ℜλ >−1 ⊂ ρ(A).

In consequence it follows for the spectrum of A, that

σ(A) ⊂ λ ∈C| ℜλ ≤−1.

In summary, it was shown that the spectrum of the generator A is completely contained in the open lefthalf complex plane while ‖T (t )‖ = 1. In conclusion, for infinite-dimensional systems it is in general notpossible to conclude the stability properties from the spectrum of the system operator in contrast to thefinite–dimensional case.

aAs shown before, R(λ, A) corresponds to the Laplace transform of the C0–semigroup T , i.e. R(λ, A) = (λI − A)−1 = ∫ ∞0 T (t )e−λt dt

with ℜλ >ω as ‖T (t )‖ ≤ Meωt .

5.1.2 Exponential stability and the „spectrum determined growth assumption”

As shown in the previous section, for C0–semigroups T in Hilbert spaces with ‖T (t )‖ ≤ Meωt and generator Ait is not possible to determine the growth bound ω from the spectrum σ(A). In consequence it is in generalnot possible to conclude the exponential stability of the semigroup by analyzing the spectrum σ(A). This is instrong contrast to the finite–dimensional case.

It should be noted anyway, that for each C0–semigroup T the so called growth bound of the semigroup can be


determined according to

ω0 = inft>0

log‖T (t )‖t

= limt→∞

log‖T (t )‖t

⇒ ω0 ∈ [−∞,∞)

In particular, for each ω>ω0 there exists a constant Mω = M(ω) so that ‖T (t )‖ ≤ Mωeωt . This is a consequenceof the Hille–Yosida Theorem (see Appendix A). Furthermore it can be shown that the inequality

supλ∈σ(A)

ℜλ ≤ω0 (5.5)

is valid in general, where for the particular case of finite-dimensional linear systems it holds with equality. Asshown in Example 5.2 equality does in general not apply in the case of infinite-dimensional systems.

The so called „spectrum determined growth assumption”, or respectively „spectrum determined growthcondition” is fulfilled, if (5.5) holds with equality, i.e., if

supλ∈σ(A)

ℜλ =ω0. (5.6)

Using the shift-semigroup as an example, the spectrum σ(A) of the operator A is not determined directly, butby the determination of the resolvent set ρ(A) together with the fact that σ(A) =C\ρ(A). This underlines thesignificance of the resolvent operator R(λ, A), whose importance becomes clear especially in the Hille–Yosidatheorem for the characterization of C0–semigroups of contractions (see Appendix A).

Since the resolvent is given by the Laplace transform of the semigroup T , if follows with the known existencecriteria of the Laplace-transformation, that for T (t ) with ‖T (t )‖ ≤ Meωt and ω< 0, the entire right half complex

plane C+

0 = λ ∈C |ℜλ ≥ 0 is included in the resolvent set. This leads to the following theorem.

Theorem 5.1

Let A be the infinitesimal generator of a C0–semigroup T ( · ) in the Hilbert space X . T is exponentiallystable if and only if it holds true that R(s, A) ∈ H∞(L (X )) with

H∞(L (X )) :=

G :C+0 →L (X )| G holomorphic2and sup

ℜs>0‖G(s)‖ <∞

.

The proof of this necessary and sufficient condition is provided in (Curtain and Zwart, 1995, p. 222f), exploitingthe following time domain condition for exponential stability (Curtain and Zwart, 1995; Jüngel, 2001).

Lemma 5.1

Let A be the infinitesimal generator of a C0 semigroup T in the Hilbert space X . T is exponentially stableif and only if for all x ∈X there exists a positive constant γx <∞ so that∫ ∞

0‖T (t )x‖2

X d t ≤ γx . (5.7)

Thus the question arises, for which infinitesimal generators A or semigroups T (t) one can directly conclude(5.6). Curtain and Zwart (Curtain and Zwart, 1995) provide the following characterization.

Theorem 5.2

Let A be the infinitesimal generator of a C0–semigroup T (t ) in a Hilbert space X and let

ωσ := supλ∈σ(A)

ℜλ. (5.8)

2Recall that a function f : U ⊆C→C is called holomorphic in U , if it is complex differentiable at every point z ∈U .


The semigroup T satisfies the „spectrum determined growth assumption” if and only if for all ω>ωσ itholds true that ((s +ω)I − A)−1 ∈ H∞(L (X )).

Proof. The proof of this theorem can be carried out as follows:

• Sufficiency: It is assumed that T satisfies the „spectrum determined growth assumption”, i.e., ω0 =ωσ.Given ‖T (t)‖ ≤ M0eω0t it holds true that T (t) = e−ωt T (t) is exponentially stable for all ω > ωσ. Thegenerator A of T (t) is obtained according to (4.50) as A = −ωI + A. With Theorem 5.1 it follows that(sI − A)−1 = ((s +ω)I − A)−1 ∈ H∞(L (X )).

• Necessity: Letω≥ωσ. With Theorem 5.1 it follows that ((s+ω)I−A)−1 = (sI−(−ωI+A))−1 ∈ H∞(L (X )) sothat −ωI+A is the infinitesimal generator of an exponentially stable C0–semigroup T with T (t ) = e−ωt T (t )for all t ≥ 0. The growth bound ω0 of T ( · ) is smaller as ω for any ω ≥ ωσ. In consequence ω0 ≤ ωσ.According to (5.5) it holds that ωσ ≤ ω0, so that ω0 = ωσ. Thus, T satisfies the „spectrum determinedgrowth assumption”.

In particular, it can be shown immediately that for all Riesz systems the conditions of Theorem 5.2 are satisfied.For Riez operators A =∑∞

n=1λn⟨ · ,ψn⟩Xφn the resolvent can be (4.84) can be specified directly as

R(s, A) = (sI − A)−1 =∞∑

n=1

1

s −λn⟨ · ,ψn⟩Xφn , ∀s ∈ ρ(A).

Furthermore A has a pure point spectrum, i.e. σ(A) =σp (A), so that ωσ = supn∈Nλn follows. For all ω>ωσ itthus holds true that R(s +ω, A) is defined on the entire right half plane C+

0 and is holomorphic.

These results can be further generalized. In (Luo et al., 1999) it is shown that all compact C0- semigroups or allC0–semigroups with pure point spectrum the „spectrum determined growth assumption” holds true. Fromthis, possibilities for the analysis of exponential stability result focussing on the resolvent of the generator A.These are subject of the two following statements.

Theorem 5.3

Let T be a C0–semigroup in a Hilbert space X with infinitesimal generator A. T is exponentially stable ifand only if λ ∈C| ℜλ ≥ 0 ⊂ ρ(A) and

‖R(λ, A)‖ ≤ M

for all λ with ℜλ ≥ 0 and constant M > 0.

This leads to the so-called frequency range test for exponential stability analysis of C0–semigroups in Hilbertspaces (Luo et al., 1999, S. 139/140).

Corollary 5.1. Let T be an uniformly bounded C0–semigroup in a Hilbert space X with infinitesimal generatorA. T is exponentially stable if and only if iR⊂ ρ(A) and

M0 := supτ∈R

‖R(iτ, A)‖ <∞.

This approach offers an efficient method for stability analysis and is applied to the class of Riesz operators inthe following. First a linear heat conduction system is considered.

Example 5.3 (Determination of the resolvent for a linear heat conduction system). The temperature dis-tribution x(z, t ) in a rod obeys the equations

∂t x = ∂2z x, z ∈ (0,1), t > 0

x(0, t ) = x(1, t ) = 0, t > 0

x(z,0) = x0(z), z ∈ [0,1].


The resolvent, i.e. the Laplace transform of the corresponding C0 semigroup, can be can be determined in twoways:

• Application of the Laplace-transformation and Method of the Green function: Let L2(z, s) denote theLaplace-transform of x. With this, the associated boundary value problem is derived in the from

d2L2(z, s)

dz2 − sL2(z, s) =−x0(z) z ∈ (0,1)

L2(0, s) = L2(1, s) = 0.

In the following g (z,ζ, s) denotes the Green function, which allows an inversion of the differentialoperator

−∫ 1

0g (z,ζ, s)x0(ζ)dζ=

∫ 1

0g (z,ζ, s)

(d2L2(ζ, s)

dζ2 − sL2(ζ, s))dζ

=[

g (z,ζ, s)dL2(ζ, s)

dζ−∂ζ g (z,ζ, s)L2(ζ, s)

]ζ=1

ζ=0︸︷︷︸=g (z,1,s)︸︷︷︸

!=0

dL2dζ (1,s)−g (z,0,s)︸︷︷︸

!=0

dL2dζ (0,s)

+∫ 1

0

(∂2ζ g (z,ζ, s)− sg (z,ζ, s)

)︸︷︷︸

!=δ(ζ−z)

L2(ζ, s)dζ,

or equivalently

L2(z, s) =−∫ 1

0g (z,ζ, s)x0(ζ)dζ= R(s, A)x0.

Obviously the resolvent can be used directly as an integral operator in form of the Greens function. Thecorresponding equations for the determinatino of the Greens function and thus the resolvent are given by

∂2ζ g (z,ζ, s)− sg (z,ζ, s) = δ(ζ− z), ζ ∈ (0,1)

g (z,0, s) = g (z,1, s) = 0.

The solution of this boundary value problem for g (z,ζ, s) can be obtained, e.g., using a sectoral approachand the formulation of jump and continuity conditions (butkovskiy:82; gilles:73). This yields

g (z,ζ, s) =

−sinh(

psζ)sinh(

ps(1− z))

sinh(p

s), 0 ≤ ζ≤ z

−sinh(p

sz)sinh(p

s(1−ζ))

sinh(p

s), z ≤ ζ≤ 1.

• Spectral (or modal) system representation: By solving the, in this case self–adjoined eigenvalue prob-lem for the differential operator one can calculate the spectral representation of the PDE with boundaryand initial conditions. In accordance with the above comments on Riesz operators this can be done byby considering the infinite number of ordinary differential equations for the spectral (or modal) statesxn(t ) = ⟨x,ψn(z)⟩X , X = L2(0,1), according to

dxn(t )

dt=λn xn(t ), t > 0, n ∈N

mit

xn(0) = x0n = ⟨x0(z),ψn(z)⟩X .

with the adjoint eigenfunctions ψn(z) and eigenvalues λn =−(nπ)2 for n ∈N. The application of theLaplace-transformation leads to

L2n(s) = 1

s −λnx0

n , n ∈N,

from which the spectral representation of L2(z, s) or equivalently, of the resolvent R(s, A) is obtained by


Fourier–synthesis, cf. (??), as

L2(z, s) = ∑n∈N

1

s −λn⟨x0(z),ψn(z)⟩Xψn(z) = R(s, A)x0.

The solution in the time domain can thus be easily obtained by the inverse Laplace transform, yielding

x(z, t ) = ∑n∈N

xn(t )ψn(z) = ∑n∈N

eλn t ⟨x0(z),ψn(z)⟩Xψn(z),

from which the exponential stability of the system is directly verified, given thatω= supn∈Nλn =−π2 < 0.

In the following, it is outlined how the resolvent of a Riez system can be used to conclude the exponentialstability of the corresponding C0–semigroup.

Example 5.4 (Exponential stability analysis for Riesz systems). Let λk be the eigenvalues of A with eigen-functions φk and adjoint eigenfunctions ψk , k ∈N. In virtue of the Riesz basis properties, there exists M > 0 sothat

‖R(λ, A)x‖2X = ‖

∞∑k=1

1

λ−λn⟨x,ψk⟩φk‖2

X ≤ M∞∑

n=1

∣∣∣∣ 1

λ−λn

∣∣∣∣2

⟨x,ψk⟩2 .

For λ= a + ib and λn = an + ibn it holds that

1

|λ−λn |= 1

(a −an)2 + (b −bn)2 ≤ 1

(a −an)2 ≤ 1

(a − supn∈N

an)2 = 1

(ℜλ− supn∈N

ℜλn)2 .

In consequence, there exists an m > 0 so that

‖R(λ, A)x‖2X ≤ M

(ℜλ− supn∈N

ℜλn)2

∞∑n=1

⟨x,ψk⟩2 ≤ M

m(ℜλ− supn∈N

ℜλn)2 ‖x‖2

This implies that

‖R(λ, A)‖ = sup‖x‖6=0

‖R(λ, A)x‖‖x‖ ≤

√M

m

1

|ℜλ− supn∈N

ℜλn| .

According to the Hille-Yosida theorem (see Appendix A) it follows that

‖T (t )‖ ≤√

M

meωt , ω= sup

n∈Nℜλn.

Thus, the exponential stability is ensured if ℜλn < 0 for all n ∈N.

In summary, the following results for stability of Riesz systems hold true (Curtain and Zwart, 1995).

Corollary 5.2. Let A be a Riesz operator with simple eigenvalues λn , n ≥ 1, corresponding eigenfunctionsφn , n ≥ 1 and adjoint eigenfunctions ψn , n ≥ 1. Then the spectrum determined growth assumption is satisfiedand the exponential stability can be concluded directly from the spectrum of A.

5.1.3 Lyapunov stability

Besides conditions on the spectrum, for finite–dimensional systems the stability of an equilibrium point isoften analyzed using Lyapunov–equations, or Lyapunov’s direct method. In the following, it is discussed to whatextend these approaches can be extended to infinite–dimensional function spaces.


5.1.3.1 The Lyapunov operator equation

Parting from Lemma 5.1, which characterizes the exponential stability of a C0–semigroup one can directly showthe following Lyapunov–like result.

Theorem 5.4

Let A be the infinitesimal generator of a C0–semigroup T ( · ) in the Hilbert space X . T is exponentiallystable if and only if there exists a positive operator P ∈L (X ), such that

⟨Ax ,P x⟩X +⟨P x , Ax⟩X =−‖x‖2X ∀x ∈D(A). (5.9)

Equation (5.9) is known as Lyapunov operator equation.

Note that a self–adjoint operator is called positive, if

∀x ∈X : ⟨x ,P x⟩X ≥ 0,

with the identity applying only for x = 0. The proof of the necessity in Theorem 5.4 is more of technical natureand employs the notion of Gramian maps, which is not further develped in these notes. The proof of sufficiency,anyway is quite straight forward, showing to a good extend a direct analogy to the results for finite–dimensionalsystems.

Proof. (Sufficiency) Let P be a bounded, positive operator and consider

V (t , x) = ⟨PT (t )x ,T (t )x⟩X .

Given that P is positive and bounded it holds true that V (t , x) ≥ 0 ∀t ≥ 0. For all x ∈D(A), the differentiation ofV (t , x) along solutions of ϕ= Aϕ,ϕ(0) = x ∈D(A) shows that

dV (t , x)

dt= ⟨PAT (t )x ,T (t )x⟩X +⟨PT (t )x , AT (t )x⟩X(5.9)= −‖T (t )x‖2

X .

Integration with respect to t yields

0 ≤V (t , x) =V (0, x)−∫ t

0‖T (τ)x‖2

X dτ.

Given that∫ t

0‖T (τ)x‖2

X dτ≤V (0, x) = ⟨P x , x⟩X ∀t ≥ 0, x ∈D(A)

and D(A) is dense in X (see also Corollary 4.1), there exists a γx = ⟨P x , x⟩X such that∫ t

0‖T (τ)x‖2

X dτ≤ γx

implying the expopnential stability of T in virtue of Lemma 5.1.

For the proof of necessity see, e.g., (Curtain and Zwart, 1995, S. 217).

Eventhough one can employ this theorem to show the exponential stability of a semigroup T , the applicabilityof the Lyapunov operator equation (5.9) is limited. In particular there exist no general analytic or numericmethods for the determination of the operator P , in contrast to the finite–dimensional case. Further results,as well as some applications of this approach are shown, e.g., in (Pritchard and Zabczyk, 1981). Furthermore,dissipativity-based results yielding operator inequalities have been studied in recent years (see, e.g., (Pandolfi,1998; Brogliato et al., 2007; Schaum, Moreno, et al., 2013; Schaum, Meurer, and Moreno, 2018)).


5.1.3.2 Lyapunov’s direct method

The well-established notion of Lyapunov stability can be extended to the distributed parameter case. Anequilibrium point xR of (5.1) determined by

AxR = 0, xR ∈D(A) (5.10)

Without loss of generality one can assume3 in the following that xR ≡ 0.

Definition 5.2: Stability in the sense of Lyapunov

The equilibrium xR = 0 of (5.1) is called stable in the sense of Lyapunov with respect to the norm ‖ ·‖X ,if for all ε> 0 there exists a δ(ε) > 0, so that under the assumption that ‖x0‖X ≤ δ(ε) for all t ≥ 0 it holdstrue that ‖x(t )‖X ≤ ε. If in addition ‖x(t )‖X → 0 for t →∞ the equilbirium xR = 0 is called asymptoticallystable in the sense of Lyapunov.

For the proof of stability in the sense of Lyapunov for finite–dimensional (nonlinear) systems one typicallyemploys Lyapunov’s direct method (see, e.g., (Khalil, 1996)). This basically states that, if there exists a positivedefinite functional V : X →Rwith negative (semi-) definite derivative dV

dt along solutions, one can conclude the(asymptotic) stability of the equilibrium point xR = 0. The proof of this statement is based on the compactnessof level sets of the Lyapunov function V (x), which for finite–dimensional state spaces is always ensured.

Definition 5.3: Compact set

A set M in a norm space X with norm ‖ ·‖ is called compact, or respectively relatively compact, if allsequences in M have a convergent subsequence with limit in M , or respectively M .

For subsets of finite–dimensional spaces the closedness implies the compactness. At this place it needs to bementioned, that infinite–dimensional function spaces are in general not compact. In consequence, no generalextension of the direct method of Lyapunov exists for distributed parameter systems and particular additionalconsiderations needs to be employed for the different cases (see, e.g., (Zubov, 1964)). The following exampleillustrated the non-compactness of subsets in L2(0,1).

Example 5.5 (Non–compactness of the unit sphere in L2(0,1)). Let X = L2(0,1) with standard inner product⟨x, y⟩X = ∫ 1

0 x(z)y(z)dz for all x, y ∈ L2(0,1) and induced norm ‖x‖2X

= ⟨x, x⟩. For the space X there existdifferent orthonormal basis, e.g.,

EB = 1,p

2sin(2πnz),p

2cos(2πnz), n ≥ 1.

As can be easily calculated, for each sequence xn∞n=1 of orthonormal elements xn ∈ EB , n ≥ 1 (i.e., ‖xn‖X = 1,n ≥ 1) it holds that ‖xn −xm‖2

X= 2 for all n 6= m. Thus, there does not exist any convergent subsequence xni i

with limit contained in the unit sphere, given that all elements have the distancep

2 > 1.

This argument can easily be extended to arbitrary Hilbert spaces with orthonormal basis.

For a special case, which is analyzed in the sequel, one can obtain an analogue to the finite–dimensional case inspite the lack of compactness. To introduce this result, first the concept of Lyapunov functional is introduced(Luo et al., 1999). Recall that a functional is function mapping from a vector space X to R.

Definition 5.4: Lyapunov functional

Let V : D(V ) ⊆X →R be continuous in D(V ). For x ∈D(V ) define

dV

dt(x) := lim

t↓0

1

t[V (T (t )x)−V (x)]. (5.11)

If dVdt (x) ≤ 0 for all x ∈D(V ), then V is called a Lyapunov functional.

3This can be achieved by a state transformation ∆x(t ) = x(t )−xR with xR from (5.10).


The following theorem is then a straight–forward extension of the finite–dimensional case (cp., e.g., (Rahn,2001)).

Theorem 5.5

The C0–semigroup T (t) is exponentially stable, if there exists a Lyapunov functional V : D(V ) ⊆ X → R

such that there exist α,β,γ> 0 for which

(i) α‖x‖2X

≤V (x) ≤β‖x‖2X

, and

(ii)dV

dt(x) ≤−γ‖x‖2

X .

Proof. Proof:[Outline] From (ii) it follows that

dV

dt(x) ≤−γ‖x‖2

X ≤−γβ

V (x) =−µV (x),

with µ= γ/β, implying that V (x(t )) ≤V (x0)e−µt along solutions x : [0,∞) →X with x(0) = x0. Thus

‖x(t )‖X ≤√β

α‖x0‖X e−

µ2 t .

Taking x(t ) = T (t )x0 it holds for the operator norm of the associated C0–semigroup T : [0,∞) →L (X ,X ) that

‖T (t )‖ = supx0∈D(T ),x0 6=0

‖T (t )x0‖X

‖x0‖X≤

√β

αe−

µ2 t ,

and thus, according to Definition 5.1 the exponential stability of the C0–semigroup T follows given thatα,β,µ> 0(and thus the exponential stability of the system (5.1).

The application of this theorem is illustrated in the following on the basis of examples.

Example 5.6 (Stability analysis of a heat conduction problem). The heating of a conductor with tempera-ture distribution x is described by the PDE

∂t x = ∂2z x, z ∈ (0,1), t > 0 (5.12)

with boundary and initial conditions

∂z x(0, t ) = K x(0, t ), x(1, t ) = 0, t > 0 (5.13)

x(z,0) = x0(z), z ∈ [0,1]. (5.14)

The stability of the equilibrium state xR (z) = 0 is analyzed using Lyapunov’s direct method. As state space

X = L2(0,1) is chosen with the system operator A = d2

dz2 with domain

D(A) = x ∈ L2(0,1) | d2x

dz2 ∈ L2(0,1), x,dx

dzabs. continuous,

dx

dz(0)−K x(0) = 0, x(1) = 0.

As a candidate for a Lyapunov functional consider

V (x) = 1

2

∫ 1

0x2(z, t )dz = 1

2‖x‖2

X .

Obviously it holds that α‖x‖2X

≤V (x) ≤β‖x‖2X

for all α ∈ [0, 12 ] and β≥ 1

2 . In addition one has that a

dV

dt(x) =

∫ 1

0x∂t xdz =

∫ 1

0x∂2

z xdz = [x∂z x]10 −

∫ 1

0(∂z x)2 dz =−K x2(0)−‖∂z x‖2

X .

For the purpose of employing Theorem 5.5 the question appears of how to bound ‖∂z x‖2X

by ‖x‖2X

. For this


purpose consider∫ 1

01 · x2dz = [zx2]1

0 −2∫ 1

0zx∂z xdz. (5.15)

Given that

0 ≤∫ 1

0

(xp2+p

2z∂z x

)2

dz = 1

2

∫ 1

0x2dz +2

∫ 1

0z2 (∂z x)2 dz +2

∫ 1

0zx∂z xdz,

or equivalently

−2∫ 1

0zx∂z xdz ≤ 1

2

∫ 1

0x2dz +2

∫ 1

0z2 (∂z x)2 dz ≤ 1

2

∫ 1

0x2dz +2

∫ 1

0(∂z x)2 dz

it follows from (5.15), that∫ 1

0x2dz ≤ 2x2(1)+4

∫ 1

0(∂z x)2 dz

or equivalently

‖x‖2X ≤ 2x2(1)+4‖∂z x‖2

X . (5.16)

This is also known as Poincaré–(Wirtinger) inequality. Given x(1) = 0 it thus holds true that

dV

dt(x) ≤−K x2(0)− 1

4‖x‖2

X ≤−1

4‖x‖2

X =−1

2V (x)

for all K ≥ 0. The exponential stability of the equilibrium xR = 0 in the L2–norm follows from Theorem 5.5. Itshould be noticed at this place that, in case that y(t ) = x(0, t ) is the measured output, the exponential stabilityhas been proven for a proportional boundary feedback control. Furthermore it should be mentioned that theexponential stability in the L2–norm does not imply the pointwise exponential stability of the solution, i.e.,|x(z, t )| ≤ Me−µt , M ,µ> 0, for all z ∈ [0,1].

aIn analogy to the abstract representation of DPS one would have to use the notation ∂z x = A12 x, or ∂2

z x = Ax. For the sake ofreadability the more intuitive notation is used here.

As shown in the above example the application of Lyapunov’s direct method typically requires the employmentof adequate inequalities. A selection of important inequalities in this context is given below (see, e.g., (Hardyet al., 1952; Krstic, 1999; Franke, 1987)).

• Cauchy–Schwarz inequality: Let x(z), y(z) ∈ L2(0,1), then it holds that[∫ 1

0x(z)y(z)dz

]2≤

∫ 1

0x2(z)dz ·

∫ 1

0y2(z)dz. (5.17)

• Poincaré inequality: Let x(z) ∈ H 1(0,1), then it holds that∫ 1

0x2(z)dz ≤ 2x2(0)+4

∫ 1

0(∂z x(z))2 dz (5.18)∫ 1

0x2(z)dz ≤ 2x2(1)+4

∫ 1

0(∂z x(z))2 dz (5.19)

• Ungleichung von Agmon: Let x(z) ∈ H 1(0,1), then it holds true that

maxz∈[0,1]

|x(z)|2 ≤ x2(0)+2(∫ 1

0x2(z)dz

) 12(∫ 1

0(∂z x(z))2 dz

) 12

(5.20)

maxz∈[0,1]

|x(z)|2 ≤ x2(1)+2(∫ 1

0x2(z)dz

) 12(∫ 1

0(∂z x(z))2 dz

) 12

(5.21)


Obviously the Agmon inequality can be used for the proof of pointwise stability as shown next for the example5.6.

Example 5.7 (Proof of pointwise exponential stability for a heat conduction problem). Having the aboveresults as point of departure consider

V (x) = 1

2

∫ 1

0x2dz + 1

2

∫ 1

0(∂z x)2 dz = 1

2‖x‖2

H 1(0,1)

as Lyapunov functional cancidate. In addition, let K = 0, so that

dV

dt(x) =

∫ 1

0x∂t xdz +

∫ 1

0∂z x∂t (∂z x)dz =−

∫ 1

0(∂z x)2 dz −

∫ 1

0

(∂2

z x)2

dz.

The Poincaré inequality yields after substituting x(z) by ∂z x(z) that

dV

dt(x) ≤−1

4

∫ 1

0x2dz − 1

4

∫ 1

0(∂z x)2 dz =−1

2V (x).

Thus, by the definition of V (x)

‖x‖2X +‖∂z x‖2

X ≤ e−t2

(‖x0‖2

X +‖∂z x0‖2X

),

so that the application of Agmon’s inequality yields

maxz∈[0,1]

|x(z)|2 ≤ 2‖x‖X ‖∂z x‖X ≤ ‖x‖2X +‖∂z x‖2

X ≤ e−t2

(‖x0‖2

X +‖∂z x0‖2X

).

Obviously, this implies the pointwise exponential stability of the equilibrium xR = 0.

These simple examples show first possibilities of the application of Theorem 5.5. Nevertheless, it should bementioned that already simple modifications of the heat equation, e.g., in the boundary conditions, can implyconsiderable higher effort in the mathematical analysis.

Example 5.8 (Exponential stability of a heat equation with spatially discrete state feedback). Having theexample 5.6 as point of departure, the exponential stability of the equilibrium xR = 0 is analyzed for the case ofa feedback of x(ζ, t ) at the bounary, with ζ ∈ [0,1) being an arbitrary location. This implies that instead of theBCs (5.13) one has to consider the conditions

∂z x(0, t ) = K x(ζ, t ), x(1, t ) = 0 t > 0.

As Lyapunov functional candidate consider

V (x) = 1

2‖x‖2

X .

It holds that

dV

dt(x) =

∫ 1

0x∂t xdz =−K x(0)x(ζ)−‖∂z x‖2

X .

For the further analysis of dVdt (x) the following property is exploited which is ensured due to x(1) = 0

x(0)x(ζ) =∫ 1

0∂z xdz

∫ 1

ζ∂z xdz =

(∫ 1

ζ∂z xdz

)2+

∫ ζ

0∂z xdz

∫ 1

ζ∂z xdz = x(ζ)2 − (x(ζ)−x(0))x(ζ).

In the sequel the cases K ≥ 0 and K < 0 are considered separately.

• K ≥ 0: In this case it holds that

dV

dt(x) ≤ K

∫ ζ

0|∂z x|dz

∫ 1

ζ|∂z x|dz −‖∂z x‖2

X


By virtue of the Cauchy–Schwarz inequality∫ ζ

0|∂z x|dz ≤

√ζ(∫ ζ

0(∂z x)2 dz

) 12

∫ 1

ζ|∂z x|dz ≤

√1−ζ

(∫ 1

ζ(∂z x)2 dz

) 12

implying

dV

dt(x) ≤ K

√ζ(1−ζ)

(∫ ζ

0(∂z x)2 dz

) 12(∫ 1

ζ(∂z x)2 dz

) 12 −‖∂z x‖2

X

≤−(1−K√ζ(1−ζ))‖∂z x‖2

X ,

so that for 1−K√ζ(1−ζ) > 0 or equivalenlty, K < 1/

√ζ(1−ζ) and by virtue of the Poincaré inequality

one has that

dV

dt(x) ≤−1

2(1−K

√ζ(1−ζ))V (x).

• K < 0: In this case it holds that

dV

dt(x) ≤−K

∫ 1

0|∂z x|dz

∫ 1

ζ|∂z x|dz −‖∂z x‖2

X

and the analogous application of the Cauchy–Schwarz as well as the Poincaré inequalities yields that

dV

dt(x) ≤−1

2(1+K

√1−ζ)V (x)

for K >−1/√

1−ζ.

Summarizing, the exponential stability of xR = 0 in the L2–norm can be shown for the heat equation withproportional feedback of x(ζ, t ) for an arbitrary location ζ ∈ [0,1) as long as

− 1√1−ζ

< K < 1√ζ(1−ζ)

.

Note that, in spite of the successful application of Lyapunov’s direct method according to Theorem 5.5 to simplelinear DPSs, one cannot deduce a general method at this place. This is due to the fact, that there are almmostno systematic approaches for the construction of a Lyapunov functional V . For certain mechanical systems,choosing the total energy as Lyapunov functional sometimes further results can be derived. For this purposefrequently the so–called energy multipliyer method (Komornik, 1994; Luo et al., 1999) is employed, which willbe illustrated in the next section. Besides this approach, e.g., passivity and dissipativity–based approaches havebeen studied in recent years (Pandoli; Brogliato et al., 2007; Schaum, Meurer, and Moreno, 2018; Schaum andMeurer, 2019).

5.1.3.3 The energy multiplier method

The subyacent idea in the energy multiplier method ist illustrated in the following on the basis of an Euler–Bernoulli–beam that is clamped on one side. For further discussions and results the reader is referred to theliterature, in particular to (Komornik, 1994; Luo et al., 1999).

Example 5.9 (Exponential stability of an Euler–Bernoulli beam). According to Euler-Bernoulli beam theory,the equations of motion in normed, dimensionless form for a one-sided clamped beam are given by

∂2t v +∂4

z v = 0, z ∈ (0,1), t > 0. (5.22)


Here, v(z, t ) denotes the beam deflection. The associated BCs at the clamped end are given by

v(0, t ) = 0, ∂z v(0, t ) = 0, t > 0 (5.23)

and at the unconstrained end

∂2z v(1, t ) = u1(t ), ∂3

z v(1, t ) = u2(t ), t > 0. (5.24)

Hereby it is assumed that the applied moment u1(t ) and the applied lateral force u2(t ) are linear combinationsof the form

u1(t ) = kT1

v(1, t )

∂t v(1, t )

+kT2

∂z v(1, t )

∂2v∂z∂t (1, t )

(5.25)

u2(t ) = kT3

v(1, t )

∂t v(1, t )

+kT4

∂z v(1, t )

∂2v∂z∂t (1, t )

(5.26)

with k j = [k j ,1,k j ,2]T , j = 1,2,3,4. The ICs are given by

v(z,0) = v0(z), ∂t v(z,0) = v1(z). (5.27)

It can be shown that (see, e.g., (Luo et al., 1999, S. 200f)) (5.22)–(5.27) using the state vector x(z, t ) = [x1(z, t ), x2(z, t )]T

with x1(z, t ) = v(z, t ), x2(z, t ) = ∂t v in the state space

X =

x | x1 ∈ H 2(0,1), x2 ∈ L2(0,1), x1(0) = 0 = ∂z x1(0)

with inner product and induced norm

⟨x , y⟩X =∫ 1

0

(∂2

z x1∂2z y1 +x2 y2

)dz, ‖x‖X = ⟨x , x⟩

12X

, x , y ∈X

the system corresponds to a well–defined abstract Cauchy problem of the form (5.1) with system operator

Ax = x2

−∂4z x1

and domain of definition

D(A) =

x ∈X |x1 ∈ H 4(0,1), x2 ∈ H 2(0,1), x1(0) = dx1

dz(0) = 0, x2(0) = 0,

dx2

dz(0) = 0,

d2x1

dz2 (1) = kT1 x(1)+kT

2dx

dz(1),

d3x1

dz3 (1) = kT3 x(1)+kT

4dx

dz(1)

. (5.28)

In a first step the total energy of the system

E(x) = 1

2‖x‖2

X (5.29)

is considered as candidate Lyapunov functionala. It holds true that

dE(x)

dt=−

∫ 1

0x2∂

4z x1dz +

∫ 1

0∂2

z x1∂2z x2dz

=[−x2∂

3z x1 +∂z x2∂

2z x1

]1

0

=−x2(1)∂3z x1(1)+∂z x2(1)∂2

z x1(1) (5.30)

=−x2(1)(kT

3 x(1)+kT4 ∂z x(1)

)+∂z x2(1)

(kT

1 x(1)+kT2 ∂z x(1)

). (5.31)


The equation (5.30) shows a connection between the applied moment at z = 1 and the angular velocity ∂z x2(1)as well as between the laterial force at z = 1 and the velocity x2(1). This relation is also called collocation inthis context, so that the moment and the velocity as well as the lateral force and the angular velocity formso–called collocated variables. The property of collocation results useful in particular for the further stabilityanalysis. This is illustrated next for the case that kT

1 = kT2 = kT

4 = 0, k3,1 = 0, k3,2 = K , in which with (5.31) itfollows that

dE(x)

dt=−K x2

2(1). (5.32)

The total energy thus reduces along the solution trajectories of the system, i.e., the considered system isdissipativeb. Nevertheless, it is not possible to obtain a bound like E ≤ −γE for some γ > 0 from the aboverelation, so that it is not possible here to conclude the exponential stability of the equilibrium xR = 0. Toachieve this, in the following the Lyapunov functional is chosen as

V (x) = E(x)+ερ(x)

with E(x) given in (5.29), ε ≥ 0 and ρ > 0. The additive term ερ(x) is called enery multiplier, where ε is adegree of freedom which in the following has to be determined adequately, and ρ(x) has the form

ρ(x) =∫ 1

0zx2∂z x1dz. (5.33)

To proof that there exists a constant γ> 0 so that dVdt (x) ≤−γV (x) some further estimations are needed. From

dρ(x)

dt=

∫ 1

0z(x2∂z x2 −∂4

z x1∂z x1

)dz

and ∫ 1

0z∂4

z x1∂z x1dz =[

z∂z x1∂3z x1 −∂2

z x1∂z x1

]1

0+

∫ 1

0

(∂2

z x1

)2dz −

∫ 1

0z∂2

z x1∂3z x1dz

with ∫ 1

0z∂2

z x1∂3z x1dz =

[z(∂2

z x1

)2]1

0−

∫ 1

0

(∂2

z x1

)2dz −

∫ 1

0z∂2

z x1∂3z x1dz

⇒∫ 1

0z∂2

z x1∂3z x1dz = 1

2

[z(∂2

z x1

)2]1

0− 1

2

∫ 1

0

(∂2

z x1

)2dz

as well as∫ 1

0zx2∂z x2dz = [

zx22

]10 −

∫ 1

0x2

2dz −∫ 1

0zx2∂z x2dz

⇒∫ 1

0zx2∂z x2dz = 1

2

[zx2

2

]10 −

1

2

∫ 1

0x2

2dz

and the boundary conditions according the definition of D(A) in (5.28), it follows that

dρ(x)

dt=−K x2(1)∂z x1(1)+ 1

2x2

2(1)− 3

2

∫ 1

0

(∂2

z x1

)2dz − 1

2

∫ 1

0x2

2dz.

To further bound this expression the following result can be used.

Proposition 5.1. Let a, b ∈R, than it holds for all ε> 0 that

2ab ≤ εa2 + b2

εbzw. −2ab ≤ εa2 + b2

ε.

Proof. Given that (εa ∓b)2 ≥ 0 one has that for all a, b ∈R it holds that ε2a2 ∓2εab +b2 ≥ 0 or equivalentlyε2a2 +b2 ≥±2εab. With ε> 0 the statement is obtained.


With this it can be easily seen that for K ≥ 0

−K x2(1)∂z x1(1) ≤ K

2

[ε(∂z x1(1)

)2+x2

2(1)

ε

]≤ K

2

[ε

∫ 1

0

(∂2

z x1

)2dz + x2

2(1)

ε

],

where the last estimation follows from the Cauchy–Schwarz inequality and ∂z x1 =∫ 1

0 ∂2z x1dz. In summary it

holds true that

dρ(x)

dt≤ 1

2

(1+ K

ε

)x2

2(1)− 1

2(3−K ε)

∫ 1

0

(∂2

z x1

)2dz − 1

2

∫ 1

0x2

2dz.

For 3−K ε> 1 or equivalently ε< 2/K it follows that

dρ(x)

dt≤ 1

2

(1+ K

ε

)x2

2(1)− 1

2‖x‖2

X =−E(x)+ 1

2

(1+ K

ε

)︸︷︷︸

=C

x22(1).

In the sequel, let C ≥ 0, i.e. ε≥−K what can be met for K ≥ 0 or equivalently ε> 0.

Besides the bound for ρ(x) by means of E (x) one can also bound |ρ(x)| by the total energy. For this the followingresult can be used.

Proposition 5.2. For x ∈X it holds true that

∓∫ 1

0x2∂z x1dz ≤ 1

2

∫ 1

0

[x2

2 +(∂z x1

)2]dz.

Proof. It holds that

0 ≤∫ 1

0

( x2p2± 1p

2∂z x1

)2dz = 1

2

∫ 1

0

[x2

2 +(∂z x1

)2]dz ±

∫ 1

0x2∂z x1dz

implying the statement.

Obviously it holds that

|ρ(x)| ≤∫ 1

0|z||x2|

∣∣∣∂z x1

∣∣∣dz ≤∫ 1

0|x2|

∣∣∣∂z x1

∣∣∣dz

≤ 1

2

∫ 1

0

[x2

2 +(∂z x1

)2]dz

≤(∂z x1(0)

)2+ 1

2

∫ 1

0

[x2

2 +4(∂2

z x1

)2]dz

≤ 2∫ 1

0

[x2

2 +(∂2

z x1

)2]dz = 4E(x)

where for the term 4(∂2z x1)2 the Poincaré inequality is used. With this result it follows inmediately that for

ρ(x) ≥ 0 or for ρ(x) < 0 the Lyapunov functional V (x) ≤ (1+4ε)E (x) or V (x) ≥ (1−4ε)E (x) for − 14 < ε< 1

4 . The

determined bounds can now be used to bound dVdt (x) according to

dV (x)

dt= dE(x)

dt+εdρ(x)

dt≤−εE(x)− (K −εC )x2

2(1)

≤−εE(x) for K −εC ≥ 0 or ε≤ K

≤− ε

1+4εV (x) given that E ≥V /(1+4ε). (5.34)

Using the preceding bounds for ε it follows that

ε> 0 ∧ ε< min

K ,2

K,

1

4

. (5.35)


In particular an exponential decay of the Lyapunov functional V (x), or respectively the total energy E(x)follows from (5.34) according to

V (x) ≤ exp(− εt

1+4ε

)V (x(0)) ≤ (1+4ε)exp

(− εt

1+4ε

)E(x(0)) (5.36)

or

E(x) ≤ 1+4ε

1−4εexp

(− εt

1+4ε

)E(x(0)), (5.37)

so that with E (x) = 12‖x‖2

Xthe exponential stability of the equilibrium xR = 0 follows under velocity feedback

u2(t ) = K x2(1, t ) for K > 0.

aNote that the relation of the norm with the total energy motivates the notion of energy norm.bFor a more detailed discussion of the wide field of dissipativity and its applications the reader is referred to more specific literature

(Brogliato et al., 2007; Schaum, Meurer, and Moreno, 2018).

This, basically simple example illustrates the mathematical effort required for the application of Lyapunov’sdirect method for the proof of exponential stability.

5.1.3.4 Generalizations

There exist generalizations of the direct method of Lyapunov which enable the analysis of asymptotic (but notexponential) stability when the conditions 5.5 can not be established. These generalizations are summarized inthe sequel for the sake of completeness. Hereby the focus is put on the fact that infinite–dimensional functionspaces are in general not compact, which implies that from dV

dt (x) ≤ 0 or dVdt (x) < 0 one can not conclude the

stability.

In the sequel T (t ) is assumed to be a continuous, contracting (nonlinear) semigroup on a closed subset D(T ) ofa real Banach space X . Let x ∈D(T ). The following notions are important:

• the orbit γ(x) =⋃t≥0 T (t )x passing through x

• theω–limit set ω(x) = y ∈D(T ) | y = limn→∞ T (tn)x with , tn < tn+1, and tn →∞ for n →∞.

The central question is under which conditions ω(x) is a non–empty set. A sufficient condition is provided bythe following theorem.

Theorem 5.6

Let γ(x) be relatively compact for x ∈ D(T ), i.e., γ(x) is compact. Then ω(x) is non–empty, compact,connected4and it holds that

limt→∞d(T (t )x ,ω(x)) = 0.

Here d(y ,Ω) denotes the distance between y andΩ for y ∈ Y andΩ⊂ Y , i.e.,

d(y ,Ω) = infw∈Ω

‖y −w‖Y .

The proof of the Theorem can be found, e.g., in (Luo et al., 1999). Considering that the relative compactnessof the orbit γ(x) has been established and the ω–limit set of x can be determined, the characterization of theasymptotic behavior of the solution ϕt (x) = T (t )x is possible. This yields the invariance principle of LaSalle.

Theorem 5.7: Invariance principle of LaSalle

Let V (x) be a Lyapunov functional in D(T ) and let E be the largest positively invariant subset5of x ∈

4This implies that ω(x) can not be separated in disjoint non–empty open sets.


D(T ) | dVdt (x) = 0. If x ∈D(T ) and γ(x) is relatively compact, then it holds true that

limt→∞d(T (t )x ,E ) = 0.

For a proof of the theorem the reader is referred to (Luo et al., 1999). The theorems 5.6 and 5.7 thus provide acriterion which enables to show that for all x ∈D(T ) the distance of the trajectory T (t )x from the ω–limit set, orrespectively from E tends to zero for t →∞. This implies asymptotic stability in case that E = 0.

From a mathematical point of view the establishment of the relative compactness of the orbit γ(x) resultsextremely restrictive. A possibility to establish this property is provided by the following theorem.

Theorem 5.8

Let A be a dissipative operator in a real Banach space X with

D(A) ⊂ ranI −λA

for all sufficiently small λ and let T (t ) be the contracting semigroup defined by

T (t )x = limn→∞

(I − t

nA

)−nx .

Let 0 ∈ ranA and there exists a λ> 0 so that (I −λA)−1 is compact. Then γ(x) is relatively compact for allx ∈D(A).

Applications of the methods that are summarized in this section can be found, e.g., in (Shifman, 1993; Luo et al.,1999; Kugi and Thull, 2005; Meurer and Kugi, 2011).

References

Brogliato, B., R. Lozano, B. Maschke, and O. Egeland (2007). Dissipative Systems Analysis and Control: Theoryand Applications. 2nd. Springer-Verlag, London (cit. on pp. 67, 72, 76).

Curtain, R. F. and H. Zwart (1995). An introduction to infinite-dimensional linear systems theory. Springer(cit. on pp. 60, 63, 66, 67).

Franke, D. (1987). Systeme mit örtlich verteilten Parametern, Eine Einführung in die Modellbildung, Analyseund Regelung. Springer-Verlag, Berlin Heidelberg (cit. on p. 70).

Hardy, G., J. E. Littlewood, and G. Pólya (1952). Inequalities. Second. Cambridge University Press (cit. on p. 70).Jüngel, A. (2001). Eine Einführung in die Halbgruppentheorie. Vorlesungsskript, Univ. Konstanz (cit. on p. 63).Khalil, H. (1996). Nonlinear Systems. 2nd. Prentice-Hall, Upper Saddle River, New Jersey (cit. on p. 68).Komornik, V. (1994). Exact Controllability and Stabilization – The Multiplier Method. John Wiley & Sons,

Chichester (cit. on p. 72).Krstic, M. (1999). „On global stabilization of Burgers’ equation by boundary control“. In: Systems & Control

Letters 37, pp. 123–141 (cit. on p. 70).Kugi, A. and D. Thull (2005). „Infinite–dimensional decoupling control of the tip position and the tip angle of a

composite piezoelectric beam with tip mass“. In: Control and Observer Design for Nonlinear Finite andInfinite Dimensional Systems. Ed. by T. Meurer, K. Graichen, and E. D. Gilles. Vol. 322. Lecture Notes inControl and Information Sciences. Springer, pp. 351–368 (cit. on p. 77).

Luo, Z.-H., B.-Z. Buo, and O. Morgul (1999). Stability and Stabilization of Infinite Dimensional Systems withApplications. Springer–Verlag, London (cit. on pp. 60, 64, 68, 72, 73, 76, 77).

Meurer, T. and A. Kugi (2011). „Tracking control design for a wave equation with dynamic boundary conditionsmodeling a piezoelectric stack actuator“. In: Int. J. Rob. Nonlin. Cont. 21 (5), pp. 542–562 (cit. on p. 77).

Pandolfi, L. (1998). „Dissipativity and the Lur’e problem for parabolic boundary control systems“. In: SIAM J.Control Optim 36 (6), pp. 2061–2081 (cit. on p. 67).

Pritchard, A. and J. Zabczyk (1981). „Stability and Stabilizability of Infinite Dimensional Systems“. In: SIAMReview 23 (1), pp. 25–52 (cit. on p. 67).

5A set E is called positively invariant for a given system if for all x ∈ E it holds that T (t )x ∈ E for all t ≥ 0.

REFERENCES 77

Rahn, C. (2001). Mechatronic Control of Distributed Noise and Vibration – A Lyapunov Approach. Springer–Verlag,Berlin (cit. on p. 69).

Schaum, A. and T. Meurer (2019). „Dissipativity-based backstepping boundary control for a class of semilinearpartial integro-differential equations“. In: Prodeedings of the IFAC Workshop on control of partial differentialequations (CPDE) (Oaxaca, Mexico) (cit. on p. 72).

Schaum, A., T. Meurer, and J. A. Moreno (2018). „Dissipative observers for coupled diffusion-convection-reactionsystems“. In: Automatica 94, pp. 307–314 (cit. on pp. 67, 72, 76).

Schaum, A., J. A. Moreno, E. Fridman, and J. Alvarez (2013). „Matrix Inequality-Based observer design for a classof distributed transport-reaction systems“. In: Int. J. Rob. Nonlin. Cont. 24(16), doi:10.1002/rnc.2981 (cit. onp. 67).

Shifman, J. (1993). „Lyapunov functions and the control of the Euler–Bernoulli beam“. In: Internat J Control57(4), pp. 971–990 (cit. on p. 77).

Zubov, V. I. (1964). Methods of A. M. Lyapunov and their application. P. Noordhoff, Ltd. Groningen (cit. on p. 68).


Chapter 6

Feedback control for PDE systems

The preceding functional-analytic and system-theoretic fundamentals for the analysis of distributed–parametermodels in infinite–dimensional function spaces are used in the following for the design of stabilizing state–feedback controllers. In the abstract operator theoretic representation according to (4.87)–(4.89), i.e.,

dx(t )

dt= Ax(t )+Bu(t ), t > 0 (6.1a)

x(0) = x0 ∈D(A) (6.1b)

y(t )(t ) =C x(t ), t ≥ 0. (6.1c)

this corresponds to the question, whether there exists a realizable state feedback of the form

u(t ) = K x(t ) bzw. u(t ) = K x(t ), x ∈XN ⊂X (6.2)

with a bounded linear operator K : X →U , where U is the space in which the control variable u(t ) is defined attime t . In practically meaningful situations typically U =Rp for some p ∈N.

In the following the backstepping–based control design and the spectral, or modal approach. While thefirst makes use of appropriate state transformations into a target system with desired properties (Krstic andSmyshlyaev, 2008; Meurer, 2013), in the second one amplitudes and eigenvalues are explicitely assigned, as faras possible (Curtain and Zwart, 1995; Gilles, 1973; Franke, 1987).

6.1 Backstepping–based control design

This section summarizes some basic results for the design of backstepping controllers for PDE systems. Forfinite–dimensional nonlinear control systems backstepping is a well–established Lyapunov–based methodto determine stabilizing feedback control (Krstic, Kanellakopoulos, et al., 1995; Sepulchre et al., 1997). Theextension of the backstepping approach to certain classes of infinite–dimensional systems has been developedin (Krstic and Smyshlyaev, 2008) and is exemplarily presented for a diffusion-reaction system.

6.1.1 Feedback control for a linear diffusion–reaction system using backstepping

In the following an introduction to backstepping–based stabilization of PDEs is given by considering a lineardiffusion–reaction system.

79

6.1.1.1 Stabilization problem

Consider the stabilization of the diffusion–reaction system

∂t x = ∂2z x +αx, z ∈ (0,1), t > 0 (6.3a)

x(0, t ) = 0, x(1, t ) = u, t > 0 (6.3b)

x(z,0) = x0(z), z ∈ [0,1] (6.3c)

which is unstable1 for α>π2, by means of properly designing a state-feedback controller using the boundaryinput u. The main idea of the backstepping design approach is to use the Volterra integral transformation

w(z, t ) = x(z, t )−∫ z

0k(z,ζ)x(ζ, t )dζ (6.4)

to transfer (6.3) into the target system

∂t w = ∂2z w −βw, z ∈ (0,1), t > 0 (6.5a)

w(0, t ) = 0, w(1, t ) = 0, t > 0 (6.5b)

w(z,0) = w0(z), z ∈ [0,1]. (6.5c)

Using Lyapunov’s direct method, as discussed in the preceding chapter, it is straigt–forward to show theexponential stability for the target system in the L2 norm for the case that β+ 1

4 > 0. For this purpose considerthe Lyapunov functional candidate

V (w) = 1

2

∫ 1

0w2dz

whose rate of change over time along solutions of (6.5) given by

d

dtV (w) =

∫ 1

0w∂t wdz =

∫ 1

0w

[∂2

z w −βw]

dz

= w∂z w∣∣∣1

0−

∫ 1

0(∂z w)2 +βw2dz.

Given the homogeneous Dirichlet boundary conditions and recalling the Poincaré-Wirtinger inequality (5.16) itholds that

−∫ 1

0(∂z w)2 dz ≤−1

4

∫ 1

0w2dz,

implying

d

dtV (w) ≤−

(1

4+β

)∫ 1

0w2dz =−

(1

2+2β

)V.

Given that V (w) = 12‖w‖L2 the exponential stability in the L2 norm follows under the established condition2 on

β.

To achieve the transformation of the system (6.3) into the target system (6.5) it is at first necessary to determinethe integral kernel k.

6.1.1.2 Kernel computation

To derive of the equations governing the kernel k substitute the target system (6.5) into the transformation(6.4) and take into account the equations of the original system (6.3) for evaluation. For this, successive

1This can be easily seen by calculating the dominant eigenvalue λ1 =α−π2 of the homogeneous problem.2Actually, this result is slightly conservative, given that the dominant eigenvalue is given by λ1 = −β−π2 and thus the exponential

stability is ensured even for β>−π2.

80 Chapter 6 Feedback control for PDE systems

0

z

1

ζ1

0

η

1

2

χ1

Figure 6.1: Domain of kernel PDE: (left) in the (z,ζ)–plane with ζ ∈ (0,1), z ∈ (ζ,1) and (right) in the (χ,η)–plane withχ ∈ (0,1),η ∈ (χ,2−χ).

differentiation of (6.4) is needed which yields

∂z w = ∂z x −k(z, z)x −∫ z

0∂z k(z,ζ)dζ

∂2z w = ∂2

z x − dk

dz(z, z)x −∂z k(z, z)∂z x −∂z k(z, z)x −

∫ z

0∂2

z k(z,ζ)dζ

with the total differential dkdz (z, z) = ∂z k(z, z)+∂ζk(z, z) and

∂t w = ∂t x −∫ z

0k(z,ζ)∂t xdζ

= ∂2z x +αx −

∫ z

0k(z,ζ)

(∂2ζx +αx

)dζ

= ∂2z x +αx − [

k(z,ζ)∂ζx −∂ζk(z,ζ)x]ζ=zζ=0 −

∫ z

0

(∂2ζk +αk

)xdζ

= ∂2z x +αx −k(z, z)∂z x(z, t )−∂ζk(z, z)x(z, t )+k(z,0)∂z x(0, t )−∂ζk(z,0)x(0, t )−

∫ z

0

(∂2ζk +αk

)xdζ.

Substituting these expressions into (6.5a) yields

0 = ∂t w −∂2z w +βw

= x

(α+β+2

dk

dz(z, z)

)−x(0, t )∂ζk(z,0)+∂z x(0, t )k(z,0)+

∫ z

0

(∂2

z k −∂2ζk − (α+β)k

)xdζ.

In view of the BCs (6.3b) this provides the so–called kernel PDE

∂2z k −∂2

ζk = (α+β)k, ζ ∈ (0,1), z ∈ (ζ,1) (6.6a)

2dk

dz(z, z)+ (α+β) = 0 (6.6b)

k(z,0) = 0. (6.6c)

The triangular domain ζ ∈ (0,1), z ∈ (ζ,1) is shown in Figure 6.1 (left). The BCs (6.6b), (6.6c) are equivalent to

k(z, z) =−α+β2

z. (6.6d)

6.1 Backstepping–based control design 81

6.1.1.3 Solution of the kernel PDE

The solution of the kernel PDE (6.6) makes use of the fact, that the PDE (6.6a) is similar to the wave equation(see the considerations in Example 3.2). Hence, solution techniques developed for the wave equation can bedirectly applied to determine k. For this, the so–called method of characteristics is applied which exploits asuitable change of the independent coordinates according to

η= z +ζ, χ= z −ζ. (6.7)

Introducing k(η,χ) = k(z,ζ) allows to transfer (6.6) into the normal form (cp. Example 3.2 for more details onthe derivation)

4∂η∂χk = (α+β)k, χ ∈ (0,1), η ∈ (χ,2−χ) (6.8a)

k(η,0) =− (α+β)

4η (6.8b)

k(η,η) = k(χ,χ) = 0. (6.8c)

The transformed (still triangular) domain in the (χ,η)–plane is depicted in Figure 6.1 (right). With this, thesolution of (6.8) can be obtained by formally integrating the PDE with respect to χ (form 0 to χ), i.e.,

∂ηk(η,χ) = ∂ηk(η,0)+ (α+β)

4

∫ χ

0k(η, q)dq

(6.8b)= − (α+β)

4

(1−

∫ χ

0k(η, q)dq

)followed by an integration with respect to η from χ to η ( see Figure 6.1 (right)), yielding

k(η,χ) =− (α+β)

4

(η−χ−

∫ η

χ

∫ χ

0k(p, q)dqdp

). (6.9)

Note that in the last step the integration constant is zero due to the boundary condition (6.8c). The resultingimplicit solution in terms of an integral equation can be made explicit by considering the so–called method ofsuccessive approximation. For this purpose consider the series representation

k(η,χ) =∞∑

n=0kn(η,χ). (6.10)

Substituting this into (6.9) yields

k0(η,χ)+ k1(η,χ)+∞∑

n=2kn(ηχ) =−α+β

4(η−χ)+

∫ η

χ

∫ χ

0k0(p, q)+

∞∑n=1

(kn(p, q)

)dqdp

motivating a recursive determination of the coefficient functions kn according to

k0(η,χ) =− (α+β)

4(η−χ)

kn(η,χ) = (α+β)

4

∫ η

χ

∫ χ

0kn−1(p, q)dqdp, n ≥ 1.

In the next step it is shown using complete induction, that the series coefficients fulfill

kn =−(

(α+β)

4

)n+1 (η−χ)ηnχn

n!(n +1)!. (6.11)

For this purpose note that for n = 0 the formula fits the above identified coefficient functions k0. For n > 0 itholds that

kn(η,χ) = (α+β)

4

∫ η

χ

∫ χ

0kn−1(p, q)dqdp

= (α+β)

4

∫ η

χ

∫ χ

0−

((α+β)

4

)n (p −q)pn−1qn−1

(n −1)!n!


=−(

(α+β)

4

)n+1 ∫ η

χ

∫ χ

0

pn qn−1 −pn−1qn

(n −1)!n!dqdp

=−(

(α+β)

4

)n+1 ∫ η

χ

[pn qn

n(n −1)!n!− pn−1qn+1

(n +1)(n −1)!n!

]χ0

dp

=−(

(α+β)

4

)n+1 ∫ η

χ

pnχn

n(n −1)!n!− pn−1χn+1

(n +1)(n −1)!n!dp

=−(

(α+β)

4

)n+1 [pn+1χn

(n +1)n(n −1)!n!− pnχn+1

n(n +1)(n −1)!n!

]ηχ

=−(

(α+β)

4

)n+1 1

(n +1)!n!

(ηn+1χn −ηnχn+1 −χn+1χn +χnχn+1)

=−(

(α+β)

4

)n+1 (η−χ)ηnχn

n!(n +1)!,

showing that the formula (6.11) holds true for all n ∈N.

Evaluation of the series ansatz (6.10) with series coefficients (6.11) yields

k(η,χ) =−(η−χ)∞∑

n=0

((α+β)

4

)n+1 ηnχn

n!(n +1)!. (6.12)

Recalling the definition of the modified Bessel functions of first kind

I1(ξ) = ξ

2

∞∑n=0

(ξ2

)2n

n!(n +1)!

and setting ξ=√(α+β)ηχ, so that the right-hand side of (6.12) can be rewritten as

−(η−χ)∞∑

n=0

((α+β)

4

)n+1 ηnχn

n!(n +1)!=−(η−χ)

(α+β)

4

∞∑n=0

(p(α+β)ηχ

2

)2n

n!(n +1)!=−(η−χ)

(α+β)

4

2

ξI1(ξ).

Accordingly, it holds that

k(η,χ) =− (α+β)

2(η−χ)

I1(√

(α+β)ηχ)√(α+β)ηχ

.

Reverting the coordinate transformation (6.7) thus results in the kernel in original coordinates

k(z,ζ) =−(α+β)ζI1(

√(α+β)(z2 −ζ2))√

(α+β)(z2 −ζ2). (6.13)

Note that on the boundary where z = ζ the relation

limξ→0

I1(ξ)

ξ= 1

2

holds true (Kstic; Arfken, 1985), so that the expression (6.13) remains finite on the boundary.

6.1.1.4 Inverse transformation

To establish a one–to–one correspondence between the original and the target system it is necessary to verifyinvertibility of the Volterra transformation (6.4). Consider

x(z, t ) = w(z, t )+∫ z

0g (z,ζ)w(ζ, t )dζ (6.14)


and proceed as before, i.e. differentiate (6.14) with respect to t , twice with respect to z, and substitute the ob-tained expressions into the equations (6.3) for the diffusion–reaction system in x(z, t ). After some intermediatebut standard computations it can be shown, that the equations governing the inverse transformation g (z,ζ)are identical to those of the original kernel except for the coefficient (α+β) in the kernel PDE, who has to bereplace by −(α+β), i.e.,

∂2z g (z,ζ)−∂2

ζg (z,ζ) =−(α+β)g (z,ζ), ζ ∈ (0,1), z ∈ (ζ,1) (6.15a)

2dz g (z, z)+ (α+β) = 0 (6.15b)

g (z,0) = 0. (6.15c)

Thus the same solution procedure can be applied to explicitly determine g (z,ζ). However, as is shown subse-quently, the desired stabilization assertion is obtained immediately without computing g (z,ζ) but by verifying,that g (z,ζ) is bounded in z and ζ.

6.1.1.5 State–feedback controller and closed–loop stability

Having computed the integral kernel k allows to determine the state-feedback controller, that is required torealize the (invertible) transformation from the original diffusion–reaction system (6.3) into the exponentiallystable target system (6.5). Evaluating the BC (6.3b) at z = 1 with (6.4) provides in view of (6.5b) at z = 1 thestate-feedback controller

u(t ) =∫ 1

0k(1,ζ)x(ζ, t )dζ=−(α+β)

∫ 1

0ζ

I1(√

(α+β)(1−ζ2))√(α+β)(1−ζ2)

x(ζ, t )dζ. (6.16)

The notion state-feedback controller becomes immediately apparent since the complete spatial–temporalevolution of the state variable x is required for evaluation. Hence, the implementation of (6.16) relies on anadequate state-observer for the reconstruction of x( · , t ) which can be similarly achieved using backstepping(see the Exercise 8).

For the verification, that the state-feedback controller (6.16) does indeed stabilize the original diffusion–reactionsystem (6.3) recall, that the target system is exponentially stable so that its solution satisfies

‖w( · , t )‖L2 ≤ e−λt‖w0‖L2 (6.17)

for λ=β+ 14 > 0. In addition, (6.4) implies the bound

‖w( · , t )‖L2 ≤ ‖x( · , t )‖L2 +∥∥∥∥∫ z

0k(z,ζ)x(z, t )dζ

∥∥∥∥L2

.

The last term can be further estimated as follows∥∥∥∫ z

0k(z,ζ)x(z, t )dζ

∥∥∥2

L2=

∫ 1

0

∣∣∣∫ z

0k(z,ζ)x(z, t )dζ

∣∣∣2dz ≤

∫ 1

0

(∫ 1

0|k||x(z, t )|dζ

)2dz

≤∫ 1

0‖k(z, · )‖2

L2‖x( · , t )‖2L2 dz (Cauchy–Schwarz inequality)

≤C 2‖x( · , t )‖2L2 , (boundedness of k)

for a sufficiently large constant C > 0. As a result, we have

‖w( · , t )‖L2 ≤ (1+C )‖x( · , t )‖L2 . (6.18)

Applying a similar sequence of estimates on the inverse kernel (6.14) provides

‖x( · , t )‖L2 ≤ (1+D)‖w( · , t )‖L2


for a sufficiently large constant D > 0. Combining both bounds finally yields

‖x( · , t )‖L2(6.17)≤ (1+D)e−λt‖w0‖L2

(6.18)≤ (1+C )(1+D)︸︷︷︸=M

e−λt‖x0‖L2 , (6.19)

and thus the exponential stability of the closed–loop control system consisting of (6.3) with state-feedbackcontroller (6.16).

Example 6.1. Consider the system (6.3) with α = 12 > π2, implying that the open–loop system is unstable.Setting β=−0.2 >− 1

4 the stability of the closed–loop system is evaluated using a finite–difference solution withN = 40 discretization points. The corresponding open–loop and closed-loop behaviors are shown in Figure 6.2.On the left side the unstable open–loop solution evolution can be seen, while on the right side the closed–loopexponentially stable counterpart is shown.

t

0.00.1

0.20.3

0.4

z

0.00.2

0.40.6

0.81.0

x(z, t)

024

6

8

t

0.00.1

0.20.3

0.4

z

0.00.2

0.40.6

0.81.0

x(z, t)

−15

−10

−5

0

Figure 6.2: Numerical evaluation of the open–loop (left) and closed–loop (right) behavior for (6.3) with α= 12,β=−0.2using the backstepping–base state feedback controller (6.16).

As already commented above, the controller can not be implemented in its present form in a real set-up, giventhat it requires the knowledge of the state profile at any point z ∈ [0,1]. As only pointwise information isavailable from measurements a state observer is required for the purpose of implementation. Assume that anexponentially converging observer exists (see, e.g., Exercise 8 for a backstepping observer design of the presentsystem) with observed/ estimated state x( · , t ) ∈ L2(0,1), i.e., there exist Mo ,γo > 0 so that

‖x( · , t )−x( · , t )‖L2 ≤ Moe−γo t‖x0 −x0‖L2

where x( · ,0) = x0. Writing x = x + x with the observation error x the controller, based on the observed state xcan be written as

u(t ) = K x = K (x + x) = K x +K x.

Accordingly, given the linearity of the considered PDE system, the closed-loop solution will satisfy

x = xs f +xe

where xs f denotes the exponentially convergent solution with the state-feedback controller and xe the onedue to the observation error. As the observation error converges exponentially to zero, so does the solution3

xe , and it results that the observer and controller can be independently designed but ensure the exponentialstability of the closed-loop system. This is a direct extension of the well-known separation principle for linearfinite–dimensional systems.

3Actually this is most easily seen in the transformed state w , where the only effect of the observation error is accumulated in the rightboundary condition, while the unperturbed system dynamics is exponentially stable.


6.2 Spectral, or modal control design

6.2.1 Modal (or spectral) control design

The following considerations are based mainly on the results presented in (Curtain and Zwart, 1995; M. Balas,1998). The central concern hereby is on how to design finite–dimensional feedback controllers whose realizationrequires only a small amount of actuators and sensors. This of course involves the question about the stabilityof the closed–loop dynamics. The point of departure for the analysis is the abstract formulation

dx

dt(t ) = Ax(t )+Bu, t > 0 (6.20a)

x(0) = x0 ∈D(A) (6.20b)

y(t ) =C x(t ), t ≥ 0. (6.20c)

with

a) an unbounded closed linear operator A, which is the infinitessimal generator of a C0–semigroup T (t ) on aHilber space X ,

b) a bounded linear operator B mapping from the Hilbert space U ⊆Rm to X

c) a bounded linear operator C mapping from the Hilbert space X to a Hilbert space Y ⊆Rp

d) the domain of definition D(A) of the operator A, which is dense in X .

6.2.1.1 Decomposition of the state space through projections

The design of a finite–dimensional state feedback is based in the following on the decomposition of the statespace X into a finite–dimensional subspace XN ⊆D(A) and a residual (infinite–dimensional) subspace XR , sothat

X =XN ⊕XR .

The corresponding maps from X to XN , or from X to XR , respectively, represent so–called projectionsand aredenoted by PN , or PR , respectively. A projection is defined as follows (curtain_zwart:95).

Definition 6.1: Projection

A bounded operator P : X →X is called a projection, if P 2 := PP = P . The operator is called an orthogonalprojection, if in addition P∗ = P , i.e., P is self–adjoint.

Example 6.2 (Riesz system). Let A : D(A) →X be a Riesz operator according to Definition 4.10 with (simple)eigenvalues λnn∈N and associated eigenfunctions φnn∈N, which form a Riesz basis for X . Let furtherψnn∈N be the set of eigenfunctions of the adjoint operator A∗, which also form a Riesz basis for X , so that⟨φn ,ψm⟩ = δn,m . According to the preceding considerations A allows a representation in the form

Ax = ∑n∈N

λn ⟨x ,φn⟩ψn .

Let the operator PN : X →XN be given by

PN =N∑

n=1⟨ · ,ψn⟩φn . (6.21)

Obviously, PN defines a projection, as it holds true that

PN PN =N∑

n=1⟨

N∑j=1

⟨ · ,ψ j ⟩Xφ j ,ψn⟩

X

φn =N∑

n=1

N∑j=1

⟨ · ,ψ j ⟩X⟨φ j ,ψn⟩X

φn = PN .


Further PN is an orthogonal projection if A is self–adjoint, as in this caseφn =ψn holds true. The projectionPN is used often in the context of the so–called modal decomposition of the state space, in which, by means ofPN and the projection

PR =∞∑

n=N+1⟨ · ,ψn⟩φn , (6.22)

a decomposition of the spectral representation into an N –dimensional subsystem (through PN ) as well as aresidual infinite–dimensional subsystem (through PR ) is achieved.

In correspondence to the decomposition of the state space, a decomposition of the state vector x(t ) is associatedaccording to

x(t ) = x N (t )+xR (t ) = PN x(t )+PR x(t ) = PN x N (t )+PR xR (t )

The last equation is a direct consequence of P 2N = PN and P 2

R = PR . This allows in particular to decompose thesystem (6.20a)–(6.20c) into the subsystems

dx N (t )

dt= AN x N (t )+ AN R xR (t )+BN u, t > 0 (6.23)

dxR (t )

dt= ARN x N (t )+ AR xR (t )+BR u, t > 0 (6.24)

y(t ) =CN x N (t )+CR xR (t ), t ≥ 0, (6.25)

with AN = PN A = PN APN , AN R = PN A = PN APR , BN = PN B , ARN = PR A = PR APN , AR = PR A = PR APR ,BR = PR B , CN =C PN and CR =C PR .

Example 6.3 (Riesz system (Example 6.2 contd.)). Let the input and output operators are given by

Bu =m∑

i=1bi ui , bi : U →X (6.26)

C x = [⟨x ,c 1⟩X , . . . ,⟨x ,c p⟩X ]T , c k : X → Y (6.27)

With x N (t ) = PN x(t ) and xR (t ) = PR x(t ), where PN and PR are given by (6.21) and (6.22), the system

dx(t )

dt= Ax(t )+Bu, x(0) = x0 ∈D(A)

y(t ) =C x(t ),

is decomposed in the two subsystems according to

dx N (t )

dt=

N∑n=1

⟨Ax(t )+Bu,ψn⟩φn =N∑

n=1λn ⟨x(t ),ψn⟩φn +

m∑i=1

ui (t )N∑

n=1⟨bi ,ψn⟩φn

=ΦN

λ1

. . .

λN

⟨x ,ψ1⟩

...

⟨x ,ψN ⟩

+ΦN

⟨b1,ψ1⟩ ⟨b2,ψ1⟩ . . . ⟨bm ,ψ1⟩

......

⟨b1,ψN ⟩ ⟨b2,ψN ⟩ . . . ⟨bm ,ψN ⟩

u (6.28)

withΦN = [φ1,φ2, . . . ,φN ] and

dxR (t )

dt=

∞∑n=N+1

⟨Ax(t )+Bu,ψn⟩φn =∞∑

n=N+1λn ⟨x(t ),ψn⟩φn +

m∑i=1

ui (t )∞∑

n=N+1⟨bi ,ψn⟩φn

=ΦR

λN

λN+1

. . .

⟨x ,ψN+1⟩⟨x ,ψN+2⟩

...

+ΦR

⟨b1,ψN+1⟩ . . . ⟨bm ,ψN+1⟩⟨b1,ψN+2⟩ . . . ⟨bm ,ψN+2⟩

......

u, (6.29)

where ΦR = [φN+1,φN+2, . . .]. It should be highlighted that both x N (t) and xR (t) are functions of z and t,

6.2 Spectral, or modal control design 87

according to the construction of the projections PN and PR . A comparison with (6.23) and (6.24) directly shows,that due to the Riesz property of the componentsφn andψn the coupling terms satisfy AN R = 0 and ARN = 0.

The known modal, or spectral system representation in the form of ordinary differential equations for the asso-ciated Fourier coefficients is obtained through (6.28) and (6.29), given that x N (t ) =∑N

n=1 ⟨x(t ),ψn⟩X φn andxR (t ) =∑∞

n=N+1 ⟨x(t ),ψn⟩X φn , through the suczessive projection on the basis ψnNn=1 of XN , or ψn∞n=N+1

of XR , respectively. With xn,N (t ) := ⟨x(t ),ψn⟩X , n = 1,2, . . . , N and xn,R (t ) := ⟨x(t ),ψn⟩X , n = N +1, N +2, . . .one can write (6.28) as

dxn,N (t )

dt=λn xn,N (t )+

[⟨b1,ψn⟩X . . . ⟨bm ,ψn⟩X

]u, n = 1,2, . . . , N , (6.30)

while (6.29) is reduced to the form

dxn,R (t )

dt=λn xn,R (t )+

[⟨b1,ψn⟩ . . . ⟨bm ,ψn⟩

]u, n = N +1, N +2. . . . (6.31)

This formal equivalence between (6.28), (6.29) and (6.30), (6.31), or in particular between XN and RN is alsocalled an isomorphism.

Alternative methods for the decomposition of the state space are provided, e.g., by the method of weightedresiduals, in particular the Galerkin method.

6.2.1.2 State feedback with state observer

The design of a realisable finite–dimensional state feedback based on (6.23) and (6.25) is carried out unter theneglection of the residual parts AN R xR (t ) and CR xR (t ), so that the design model is given by

dx N (t )

dt= AN x N (t )+BN u, t > 0 (6.32a)

y(t ) =CN x N (t ), t ≥ 0. (6.32b)

It is obvious that this approach requires the exponential stability of the residual dynamics, i.e., the dimension ofthe design model has to be chosen such that AR is the generator of an exponentially stable C0–semigroup TR (t )with ‖TR (t )‖ ≤ MR exp(−ωR t ), MR ≥ 1, ωR > 0. The feedback then be designed, e.g., according to

u = KN x N (t ) (6.33a)

with the state observer

dx N

dt= AN x N (t )+BN u +LN

[y(t )−CN x N (t )

], t ≥ 0 (6.33b)

x N (0) = x0. (6.33c)

This approach (typically) requires the system Σ(AN ,BN ,CN ) to be completely observable and controllable.

The block–diagramm of the closed–loop for (6.23)–(6.25) with (6.33a) as well as (6.33b) and (6.33c) is shown inFigure 6.3. In correspondence, the dynamics of the closed–loop system are given by

d

dt

x N (t )

xR (t )

x N (t )

=

AN AN R BN KN

ARN AR BR KN

LN CN LN CR AN +BN KN −LN CN

︸︷︷︸

= M

x N (t )

xR (t )

x N (t )

. (6.34)

In particular the so–called „Spillover”–effects are directly identified (balas:78c; balas:78a; balas:78b; guelich:84).The so–called „control spillover” causes an excitation of the residual dynamics xR (t ) through the terms BR KN .If there is no observer contained in the feedback loop, i.e., x N (t) = x N (t), or if the measurement signals donot include components of xR (t) (i.e., if CR = 0) then this only causes an undesired excitation of the residualdynamics, while the behavior remains mainly unchanged. In contrary, the so–called „observation spillover”


dxN (t)

dt= ANxN (t) +ANRxR(t) +BNu(t)

dxR(t)

dt= ARNxN (t) +ARxR(t) +BRu(t)

dxN

dt= AN xN (t) +BNu(t) + LN

[y(t)− CN xN (t)

]

CN

CR

BN

BR

KN

yN

yR

yu

“Control Spillover” ‘‘Observation Spillover”

Verteilt–parametrische Regelstrecke

Figure 6.3: Closed–loop distributed parameter system (6.32) with (6.33).

Distributed parameter system

produces a feedback of the residual part y(t)R (t) = CR xR (t) contained in the measurement, causing an un-desired influence on the residual dynamics. It can be easily seen that this has effects on the stability of theclosed–loop dynamics and in extreme cases may lead to a destabilization.

The preceding considerations can be inmediately applied to the example of a general Riesz system. Anyway, inorder to illustrate the resulting spillover effects the following two particular cases are considered.

Example 6.4 (Diffusion–reaction system). (Example 6.3 cont.)]

It is straightforward to show that the eigenvalues and eigenfunctions of the self–adjoint operator

Ax = d2x

dz2 +µx, D(A) = x ∈ H 2(0,1)|x(0) = x(1) = 0 ⊂X = Lp [2](0,1)

are given by λn = µ− (nπ)2 and φn(z) =p2sin(nπz), n ∈N. For µ > π2 there are thus positive eigenvalues,

causing the open–loop system to be unstablea.

The control input is considered as pointwiese at the location ζ ∈ (0,1), i.e., b(z) = δ(z −ζ), so that with theanalysis steps from Example 6.3 and in particular (6.30), (6.31) the following modal, or spectral representationsare obtained

d

dt

x1,N (t )

...

xN ,N (t )

=

λ1

. . .

λN

x1,N (t )...

xN ,N (t )

+

φ1(ζ)

...

φN (ζ)

u(t )

for the subsystem considered for control design, and

d

dt

xN+1,R (t )

xN+2,R (t )...

=

λN+1

λN+2

. . .

xN+1,R (t )

xN+2,R (t )...

+

φN+1(ζ)

φN+2(ζ)...

u(t )

for the residual system. In particular it should be noted that the dimension N of the design system has to bechosen such, that all eigenvalues with positive real part are represented by it.


As shown in Section 4.2.3, the approximative controllability requires that φn(ζ) 6= 0 for all n ∈ N, what isassumed as given in the following. Specifically, the following values are considered µ = π2 +2, N = 2 andζ = 1/π. As output y(t) = x(ζ, t) with ζ = 2/π is considered, leading to CN = [

p2sin(πζ),

p2sin(2πζ)] =

[p

2sin(2),p

2sin(4)].

The design of the state feedback and the state observer is based on an eigenvalue assignment for the subsystemwith the components x1(t) and x2(t). The eigenvalues for the state controller and observer are chosen asλ

regn = λn −5 and λbeo

n = λn −10 for n = 1,2. For the numerical determination of KN and LN the MATLAB

command place was usedb.

The resulting eigenvalues of the closed–loop dynamics are illustrated on the basis of the eigenvalues of thematrix M in (6.34) for an increasing number of components of the residual system in Figure 6.4. Thesenumerical results give at least an indicator that the closed–loop system with state controller and observeron the basis of the N = 2–dimensional design system is exponentially stable. Possibilities to confirm thisobservation theoretically are presented on an introductory level in the next section.

aIt should be noted that for the considered operator the „spectrum determined growth assumption” holds truebAlternatively, e.g., PYTHON with control.place can be used.

−12 −10 −8 −6 −4 −2 0

x 104

−1

−0.5

0

0.5

1

ℜλ

nR = 0nR = 1nR = 10nR = 100

−40 −35 −30 −25 −20 −15 −10 −5 0 5−1

−0.5

0

0.5

1

ℜλ

nR = 0nR = 1nR = 10nR = 100

Figure 6.4: Eigenvalues of the closed–loop system for the unstable diffusion–reaction system with variation of the dimensionnR of the residual system (with zoom in the right figure). The case nR = 0 corresponds to the analysis of thecontrolled (two–dimensional) design system.

Example 6.5 (Undamped wave equation). As shown in earlier sections, introducing the state vector x(t) =[x( · , t ),∂t x( · , t )]T , the linear wave equation can be written in the form (6.20a) with

Ax = x2

d2x1dz2

, D(A) = x ∈ H 2(0,1)⊕H 1(0,1)|x1(0) = x1(1) = 0 ⊂X =H 1(0,1)⊕Lp [2](0,1).

The further configuration and the input–ouput operators correspond with those of the linear diffusion–reactionsystem in Example 6.4. The resulting system can easily be written in the modal, or spectral form correspondingto (6.30), (6.31), where it should be noted that all eigenvalues are strictly imaginary with λn = inπ, n ∈Z. Thefurther design follows the steps of the preceding examplea.

The eigenvalues of the closed–loop system, based on the analysis of the matrix M given in (6.34) are shownin Figure 6.5 for an increasing number of components in the residual system. For this case, in contrary tothe diffusion–reaction system considered above, the effect of the „control spillover” and in particular of the„observation spillover” can be clearly seen, causing an unstable closed–loop for nR > 1. It should be notedthat the instability insuded by the state control with observer is a typical phenomeon in waekly or undampedmechanical structures and thus can be observed also experimentally.

aAll formulations can be easily written in terms of the index set Z. Alternatively, (6.20a) can also be written as an abstract differentialequation of second order.

To reduce the „spillover”–effects additional measure must be implemented. Inmediately, the requirementBR = 0 or CR = 0 comes into mind. None of these requirements can in general be exactly satisfied for real


−30 −25 −20 −15 −10 −5 0 5 10−400

−300

−200

−100

0

100

200

300

400

ℜλ

nR = 0nR = 1nR = 10nR = 100

−20 −15 −10 −5 0 5 10−30

−20

−10

0

10

20

30

ℜλ

nR = 0nR = 1nR = 10nR = 100

Figure 6.5: Eigenvalues of the closed–loop system for the wave equation under variation of the dimension nR of the residualsystem (with zoom in the right figure). The case nR = 0 corresponds to the analysis of the closed–loop designsystem.

systems, as this in turn requires the placement of actuators and sensors in such a way that no excitation of theresidual dynamics occurs. A reduction of the „spillover” effects can anyway be obtained, e.g., considering theapplication of dynamical controllers or filters (M. J. Balas, 1978).

6.2.1.3 Stability of the closed–loop system

Given the considerations in the preceding section, the question arises under which conditions the applicationof the designed feedback on the distributed parameter system yields an exponential stable closed–loop system.To answer this question the, kind of intuitive concept of stabilising subspaces is considerd (M. Balas, 1998).

Definition 6.2: Stabilising subspaces

The system Σ(A,B ,−) given in (6.20a) has the pair of stabilising subspaces XN and XR , if

(i) AR is the infinitesimal generator of an exponentially stable C0–semigroup, and

(ii) there exists K : X → Rm with K = K PN = KN such that A0 = A +BK is exponentially stable with a(desired) stability margin ω0.

Note that the requirement K = K PN = KN means that KR = K PR = 0, i.e., the feedback gain only affetcs the finite–dimensional subspace XN . Accordingly, it requires that all, but a finite number of eigenvalues lie sufficiently tothe left of the line −ω0, and that the remaining ones can be moved with a finite–dimensional control.

The associated closed–loop system is then given by

A0 = AN +BN KN AN R

ARN +BR KN AR

so that an obvious condition for the stabilizability by an observer–based feedback control consists in thecontrollability and observability of the system Σ(AN ,BB ,CN ). The main condition for the pair (XN ,XR ) to bestabilizing subspaces thus consists in that ‖AN R‖ is small enough, so that the achieved stability properties arenot destroyed by the coupling.

Based on these considerations, in (M. Balas, 1998) the following result for the exponential stability assessmentof the closed–loop system (6.20) with (6.33).


Theorem 6.1

Let the following assumption be satisfied:

a) Σ(AN ,BN ,CN ) is completely controllable and observable,

b) the observer gain LN in (6.33b) is chosen such that the spectrum of AN −LN CN is located the left of−ωN , ωN > 0 in the complex plane, i.e., the matrix exponential TN (t ) = exp((AN −LN CN )t ) associatedto AN −LN CN satisfies ‖TN (t )‖ ≤ MN exp(−ωN t ), for some MN ≥ 1.

c) AR is dissipativea and generates an exponentially stable C0–semigroup TR with ‖TR (t )‖ ≤ exp(−ωR t )for some ωR > 0, and

d) the subspaces XN and XR are stabilizing subspaces for Σ(A,B ,−) and KN is such that A +BKN

generates an exponentially stable semigroup T0 satisfying ‖T0(t )‖ ≤ M0 exp(−ω0t ) with 0 <ω0 < ωN .

Then the feedback (6.33) yields the exponentially stable closed–loop system

dx(t )

dt= A0x(t )+BKN eN (t ), t > 0

deN (t )

dt= (AN −LN CN )eN (t )+ (LN CR − AN R )x(t ), t > 0

with eN (t ) = x N (t )−x N (t ), if

‖LN CR − AN R‖ ≤ ω0

M

with M = M0MN√

1+η+η2 ≤ K1K2(1+η) where η= ‖BKN‖/|ω0 − ωN |.aIn this context the operator A : D(A) ⊂ X → X is called dissipative if ⟨Ax, x⟩X ≤−σ⟨x, x⟩X for some σ> 0. In the theorem it is

assumed that AR is dissipative with σ=ωR .

Note that the last inequality for M allows for an easier bound and follows from the fact that 1+a +a2 ≤ (1+a)2

for all positive a.

Possible applications of this theorem and related results in (Curtain and Zwart, 1995) require a detailed analysisof the underlying distributed parameter systems, so that, similar to the considerations in 5.1.3 on the Lyapunovstability analysis, no general design criteria can be deduced.

Remark 6.1

In the case that the projections and the associated decomposition of the state space exploits the eigen-functions of the system operator more precise analysis steps and stability assessments are possible. In thisregard the reader is refered in particular to the discussion in (Curtain and Zwart, 1995), which in additiongive concrete criteria for the design and the choice of the dimensions in the design system.

Remark 6.2

The underyling idea of exploiting a decomposition into a finite–dimensional and an infinite–dimensionalsubsystem has been further exploited in the literature, including applications to semilinear systems. Theinterested reader is referred to, e.g., (Christofides and Daoutidis, 1997; Christofides, 2001; Schaum et al.,2018).

References

Arfken, G. (1985). Mathematical Methods for Physicists. 3rd. Orlando (FL), Academic Press, pp. 610–616 (cit. onp. 83).


Balas, M. (1998). „Stable Feedback Control of Linear Distributed Parameter Systems: Time and FrequencyDomain Conditions“. In: J Math Anal Appl 225(1), pp. 144–167 (cit. on pp. 86, 91).

Balas, M. J. (1978). „Active control of flexible systems“. In: J. Optimization theory and applications 25 (3),pp. 415–436 (cit. on p. 91).

Christofides, P. D. (2001). Nonlinear and Robust Control of PDE Systems - Methods and Applications toTransport-Reaction Processes. Systems & Control: Foundations & Applications - Birkhäuser (cit. on p. 92).

Christofides, P. D. and P. Daoutidis (1997). „Finite-Dimensional Control of Parabolic PDE Systems Using Approx-imate Inertial Manifolds“. In: J. Math. Anal. Appl. 216, pp. 398–420 (cit. on p. 92).

Curtain, R. F. and H. Zwart (1995). An introduction to infinite-dimensional linear systems theory. Springer(cit. on pp. 79, 86, 92).

Franke, D. (1987). Systeme mit örtlich verteilten Parametern, Eine Einführung in die Modellbildung, Analyseund Regelung. Springer-Verlag, Berlin Heidelberg (cit. on p. 79).

Gilles, E. (1973). Systeme mit verteilten Parametern. R. Oldenbourg Verlag München Wien (cit. on p. 79).Krstic, M., I. Kanellakopoulos, and P. V. Kokotovic (1995). Nonlinear and adaptive control design. Wiley, N.Y.

(cit. on p. 79).Krstic, M. and A. Smyshlyaev (2008). Boundary Control of PDEs: A Course on Backstepping Designs. SIAM

(cit. on p. 79).Meurer, T. (2013). Control of Higher Dimensional PDEs. Communication and Control Engineering. Springer

(cit. on p. 79).Schaum, A., T. Meurer, and J. A. Moreno (2018). „Dissipative observers for coupled diffusion-convection-reaction

systems“. In: Automatica 94, pp. 307–314 (cit. on p. 92).Sepulchre, R., M. Jankovic, and P. Kokotovic (1997). Constructive Nonlinear Control. Springer-Verlag, London

(cit. on p. 79).

REFERENCES 93


Chapter 7

Frequency-domain analysis and design

Frequency domain methods have established as very powerful analysis and design tools for finite–dimensionallinear, time–invariant systems. Given that the Laplace transform can be directly applied to functions dependingon additional variables (like, e.g., space), it is desirable to extend the theory for DPSs. This is the subject ofthis chapter, with the aim to give an introduction to the modern and generalized frequency–domain analysisapproach.

7.1 Introduction

The application of the (half–sided) Laplace transform

x(z, s) :=L x(z, t ) =∫ ∞

0e−st x(z, t )dt (7.1)

to the linear, time–invariant, inhomogeneous IBVP

nt∑j=1

α j ∂jt x(z, t )︸︷︷︸

= Dt x(z, t )

+nz∑j=0

β j ∂jz x(z, t )︸︷︷︸

= Dz x(z, t )

= u(z, t ), z ∈ (0,L), t > 0

∂jt x(z,0) = x0, j (z), z ∈ [0,L], j = 0,1, . . . ,nt −1(nz−1∑k=0

γ0k, j ∂

kz x(z, t )

)︸︷︷︸

= R0z, j x(z, t )

∣∣∣z=0

= u0j (t ), t > 0, j = 1,2, . . . ,n0

z

(nz−1∑k=0

γLk, j ∂

kz x(z, t )

)︸︷︷︸

= RLz, j x(z, t )

∣∣∣z=L

= uLj (t ), t > 0, j = 1,2, . . . ,nL

z

yields the equivalent frequency domain BVP

Dz x(z, s)+nt∑

j=1α j s j x(z, s) = u(z, s)+

nt∑j=1

α j

j−1∑i=0

s j−1−i xi (z), z ∈ (0,L)

R0z, j x(z, s)|z=0 = u0

j (s), j = 1,2, . . . ,n0z

RLz, j x(z, s)|z=L = uL

j (s), j = 1,2, . . . ,nLz .

For the solution of this BVP different approaches can be used:

• direct solution, e.g., using the Ansatz eλz .

95

• modal transformation

• Green’s function.

In the following the application of the Laplace transform for the analyiss of DPSs is illustrated for two examples: a heatconduction problem and the transport equation. For a summary of the most important properties and correspondence ofthe Laplace transform the reader is referred to Appendix B.

Example 7.1 (Heat equation). Consider a heated rod with distributed input u acting over the domain, and boundaryinputs u0,uL acting on the left and right boundary, respectively.

∂t x −a∂2z x = u, z ∈ (0,L), t > 0

x(z,0) ≡ 0, z ∈ [0,L]

x(0, t ) = u0(t )

x(L, t ) = uL(t )t > 0

so that γ00 = γL

0 = 1 and γ01 = γL

1 = 0.

x(z, t )

-z0 L

- ?????????

u0(t ) uL(t )

u(z, t )

The application of the Laplace transform yields

sx(z, s)−ad2x(z, s)

dz2= u(z, s), z ∈ (0,L)

L2(0, s) = u0(s), L2(L, s) = uL(s).

In the sequel it is assumed that u(z, s) ≡ 0, so that with

x(z, s) = A(s)sinh

(√s

az

)+B(s)cosh

(√s

az

)

the general solution reads

x(z, s) =sinh

(√sa (L− z)

)sinh

(√sa L

)︸︷︷︸

=: g0(z, s)

u0(s)+sinh

(√sa z

)sinh

(√sa L

)︸︷︷︸=: gL(z, s)

uL(s)

Using the convolution theorem it follows that

x(z, t ) =∫ t

0g0(z, t −τ)u0(τ)dτ+

∫ t

0gL(z, t −τ)uL(τ)dτ

= g0(z, t )?u0(t )+ gL(z, t )?uL(t ).

where the inverse transfer functions g0(z, t) =L−1g0(z, s) and gL(z, t) =L−1gL(z, s) can be determined, e.g., usingthe residue theorem.

In addition, some further properties can be analyzed:

Stationary solution: Using the final value theorem of the Laplace transform one has

xS (z) = x(z, t →∞)−• lims→0

sx(z, s) for lims→0

su0,L(s) = u0,L = const.

Thus one has

x(z, t →∞) = lims→0

sinh(√

sa (L− z)

)sinh

(√sa L

)︸︷︷︸

→ 0

0

su0(s)︸︷︷︸→ u0

+sinh

(√sa z

)sinh

(√sa L

)︸︷︷︸

→ 0

0

suL(s)︸︷︷︸→ uL

l’Hospital= u0 +z

L(uL − u0).

96 Chapter 7 Frequency-domain analysis and design

Poles sk of the transfer functions g0(z, s) und gL(z, s): From

sinh

(√s

aL

)= 0 = isin

(√s

a

L

i

)⇒

√s

a

L

i= kπ, k ∈N0

or equivalently

sk =−a

(kπ

L

)2, k = (0),1, . . .

∞–dimensional pole spectrum with ℜsk < 0, and thus exponential stability.

Back transform and solution properties: To determine the inverse x(z, t) = L−1x(z, s) using the residue theoremrequires the expansion of the transfer functions g0(z, s) and gL(z, s) at an ∞ number of poles sk , k = 1,2, .... This leads to

x(z, t ) = g0(z, t )?u0(t )+ gL(z, t )?uL(t )

where ? denotes the convolution operator and

g0(z, t ) =∞∑

k=1

2kπ(−1)k+1

L2e−a

(kπL

)2t sin

(kπ

L(L− z)

), gL(z, t ) =

∞∑k=1

2kπ(−1)k+1

L2e−a

(kπL

)2t sin

(kπz

L

).

Example 7.2 (Plug–flow system). Consider a plug–flow system with input u0 on the left boundary.

∂t x(z, t )+ v∂z x(z, t ) = u(z, t ), z ∈ (0,L), t > 0

x(0, t ) = u0(t ) t > 0

x(z,0) = x0(z) z ∈ [0,L].

x(z, t )

-z0 L

-

? ? ? ? ? ? ? ? ?u0(t )

u(z, t )

---------

v

Using the Laplace transform one obtains

sx(z, s)+ vdx(z, s)

dz= u(z, s)+x0(z), z ∈ (0,L)

L2(0, s) = u0(s).

implying that

x(z, s) = u0(s)e−sv z︸︷︷︸

boundary input

+ 1

v

∫ z

0u(ζ, s)e−

sv (z−ζ)dζ︸︷︷︸

distr. input

+ 1

v

∫ z

0x0(ζ)e−

sv (z−ζ)dζ︸︷︷︸

IC

Considering x0(z) ≡ 0 it follows that

x(z, s) = u0(s)e−sv z + 1

v

∫ z

0u(ζ, s)e−

sv (z−ζ)dζ

with

g0(z, s) = x(z,s)u0(s) = e−

zv s subject to a space–dependent delay Tt (z) = z

v

∫ z0 u(ζ, s) e−

sv (z−ζ)

v dζ= ∫ L0 g (z,ζ, s)u(ζ, s)dζ with Green’s function (GF)

g (z,ζ, s) =

1

ve−

(z−ζ)v s , ζ≤ z

0, ζ> z.

which determins in particular the region of influence of the distributed input u(z, s)on the solution x(z, s) at a fixedpoint z ∈ [0,L]:

-z0 L

?u(ζ, s), ζ< z

?x(z, s)

GF6= 0 GF= 0

---------

v

7.1 Introduction 97

Considering y(t ) = x(L, t ) and u(z, t ) = u(t ) it follows that

y(s) = e−sTt︸︷︷︸= g0(s)

u0(s)+ 1

s

(1−e−sTt

)︸︷︷︸

= g1(s)

u(s), Tt = L

v.

In the time domain this corresponds to

y(t ) = u0(t −Tt )σ(t −Tt )+∫ t

0(u(τ)−u(τ−Tt )σ(τ−Tt ))dτ.

The poles and zeros of the transfer functions can then determined setting s =σ+ iω according to

lims→sk

g0(s) =∞ ↔ σ=−∞ and lims→sk

g0(s) = 0 ↔ σ=∞

as well as (noting that according to the rule of lHopital lims→0 g1(s) = Tt <∞)

lims→sk

g1(s) =∞ ↔ σ=−∞ and

lims→sk

g1(s) = 0 ↔ lims→sk

esTt = 1 = eσk Tt(cos(ωk Tt )+ isin(ωk Tt )

) ↔ σk = 0, ωk = 2kπ

Tt.

7.2 Feedback–control design

To illustrate the main ideas, in the following the feedback control design for scalar, spatially discrete or pointwise inputs andoutputs is analyzed. The general setup of the feedback control loop is shown in Figure 7.1.

-y∗(t ) c - Controller

gr (s)

-u(t ) I/O–behavior

(guy (s) irrational,transcendental)

guy (s)

-y(t )r

6−

Figure 7.1: Control loop with output–feedback: controller gr (s) and transfer function guy (s).

The transfer function of the closed–loop system is given by

y(s) = g0(s)

1+ g0(s)y∗(s)

with the transfer function of the open–loop system g0(s) = guy (s)gr (s) and the desired output behavior y∗. As in finite–dimensional systems, for the design of the controller the Bounded Input Bounded Output (BIBO) stability, or equivalentlyL∞–stability concept are employed. Given the transcedental nature of the transfer functions for PDE systems someadditional considerations are necessary. For this purpose some mathematical fundamentals are discussed first.

7.2.1 Mathematical fundamentals

Besides the classical Lp spaces

Lp (Ω) = f :Ω→R| f is Lebesgue measurable, ‖ f ‖p <∞ with the norm ‖ f ‖p = (∫Ω | f (t )|p d t

) 1p für 1 ≤ p <∞,

L∞(Ω) = f :Ω→R| f is Lebesgue measurable, ‖ f ‖∞ <∞ with the norm ‖ f ‖∞ = esssupΩ | f |

one can introduce so–called extended spaces using the concept of truncated functions.

Definition 7.1

Let f :R+ →R. For each T ∈R+ the function fT :R+ →R is defined by


fT (t ) =

f (t ), 0 ≤ t ≤ T,

0, t > T

is called truncation of f (t ) to the interval [0,T ].

Lpe ([0,∞)) = f :R+ →R| fT ∈ Lp ∀T <∞,

so that obviously it holds true that Lp ⊂ Lpe .

Example 7.3. Let f (t ) = t , t ∈R+. Then fT (t ) =

t , t ∈ [0,T ]

0, t > T

‖ fT ‖pLp =

∫ ∞

0| fT (t )|p dt =

∫ T

0| f (t )|p dt = T p+1

p +1

⇒ ‖ fT ‖Lp = T1+ 1

p

(p +1)1p

→∞ for T →∞

so that f ∉ Lp and f ∈ Lpe for any 1 ≤ p ≤∞.

Lemma 7.1: (Desoer and Vidyasagar, 1975)

For any p ∈ [1,∞] the space Lpe is a vector space over R. It holds that for any fixed p and f ∈ L

pe , one has (i) ‖ fT ‖p is a

non–decreasing function in T and (ii) f ∈ Lp if and only if there exists an m with 0 < m <∞ so that ‖ fT ‖p ≤ m, ∀T ≥ 0.In this case it holds true that ‖ f ‖p = limT→∞ ‖ fT ‖p .

For the definition of Input–Output stability one has to relate input and output. For this purpose the concept of binaryrelation can be used.

Definition 7.2

A binary relation R is a subset of the cartesian product of two sets M1 and M2

R ⊆ M1 ×M2 := (a1, a2) | a1 ∈ M1 ∧a2 ∈ M2

In particular, having a mapping A : M1 → M2, it defines a binary relation RA of the form

RA := (a1, Aa1)|a1 ∈ M1.

This notion can be used to introduce the concept of Lp stability.

[Lp stability]

Definition 7.3

Let R be a binary relation on Lpe .

(i) The relation R is called Lp –stable if ( f , g ) ∈ R with f ∈ Lp implies that g ∈ Lp .

(ii) The relation R is called Lp –stable with finite gain and without offset if R is Lp –stable and there exists a constantγp <∞ so that ( f , g ) ∈ R with f ∈ Lp implies that ‖g‖p ≤ γp‖ f ‖p .

The application of this concept to the binary relation defined by the mapping A the concept of Lp stable mappings isobtained.

Definition 7.4

The mapping A : Lp [pe] → Lp [pe] is called Lp –stable (with finite gain and without offset) if the induced bindaryrelation RA on L

pe is Lp –stable (with finite gain and without offset).

An immediate application of this concept yields the next definition.

7.2 Feedback–control design 99

Definition 7.5

An input–output mapping that is L∞–stable (with finite gain and without offset) is called input–output stable.

Example 7.4. Let (A f )(t ) = ∫ t0 e−α(t−τ) f (τ)dτ with α 6= 0.

(I) Let f ∈ L∞e so that ‖ fT ‖L∞ = esssupt∈[0,T ] | f (t )| ≤ mT for all finite T > 0. Then it holds that

‖A f ‖L∞ =∥∥∥∥∫ t

0e−α(t−τ) f (τ)dτ

∥∥∥∥L∞

= esssupt∈[0,T ]

∣∣∣∣∫ t

0e−α(t−τ) f (τ)dτ

∣∣∣∣≤ esssup

t∈[0,T ]

∫ t

0e−α(t−τ) ∣∣ f (τ)

∣∣dτ≤ mT

α , α> 0mT|α|

(e−αt −1

), α< 0

so that ‖A f ‖L∞ <∞ for all 0 < T <∞ but A f ∉ L∞.

(II) Let f ∈ L∞ so that mT → m for T →∞ and thus A f ∈ L∞ only for α > 0, implying that for α > 0 the mapping A isinput–output stable with ‖A f ‖L∞ ≤ m

α .

7.2.2 Input–output stability of the open loop

-u(t ) I/O–behavior

(guy (s) irrational,transcendental)

guy (s)

-y(t )

In the case of rational transfer functions guy (s) the input–output stability is ensured if the impulse resposne guy (t) ∈L1([0,∞)), or equivalently, if

(i) guy (s) is proper and

(ii) all poles of guy (s) have a negative real part.

This conclusion is not longer valid in the case of irrational transfer functions. On the other hand, the analysis of input–outputbehavior (e.g., using the extended Nyquist criterion) in principle enables to address both kinds (rational and transcedental)of transfer function analysis within a common framework.

Before addressing particular specializations and extensions, some notes on the algebraic structure of the set of transferfunctions are in order.

Series connection (product)

-u(s)

g1(s) - g2(s) -y(s) ⇒ y(s) = g2(s)g1(s)u(s)

Parallel interconnection (addition)

-u(s)

g1(s)

?

-u(s)

g2(s)6c -y(s) ⇒ y(s) = [g1(s)+ g2(s)]u(s)

Amplification (scalar product)

-u(s)

K - g (s) -y(s) ⇒ y(s) = K g (s)u(s)


Feedback (algebraic inverse)

-y∗(t ) c - gr (s) -

u(t )guy (s) -

y(t )r6−

⇒ y(s) = [1+ gr (s)guy (s)]−1 gr (s)guy (s)y∗(s)

According to these considerations the class of transfer functions forms an algebra. As the associated mappings in the timedomain are of interest, there must be some corresponding operations defined in the time domain also. This leads to thefunction sets A (β) and A (β) introduced in the sequel, with A (0) (or A (0)) corresponding to the set of input–output stableimpulse responses (or their Laplace transforms).

Definition 7.6

Let β ∈R and let A (β) denote the set of all generalized functions (distributions) f (t ) of the form

f (t ) =

fa (t )+∑∞n=1 fnδ(t − tn ), t ≥ 0,

0, t < 0,

where tn ∈ [0,∞) with t1 = 0, tn < tn+1, fn ∈C, Delta–distribution δ(t − tn ) centerd at tn , e−βt fa (t) ∈ L1([0,∞)) and∑∞n=1 | fn |e−βtn <∞.

The norm ‖ f ‖A (β) of a distribution in A (β) is defined by

‖ f ( · )‖A (β) =∫ ∞

0e−βt | fa (t )|d t +

∞∑n=1

| fn |e−βtn

In addition, the convolution of two distributions f , g ∈A (β) is defined by

( f ? g )(t ) =∫ t

0f (t −τ)g (τ)dτ=

∫ t

0f (τ)g (t −τ)dτ

For β= 0 one denotes A instead of A (0).

The motivation to introduce the set A (β) can be easily seen from considering the following example.

Example 7.5. Consider the SISO system

x = Ax +bu, t > 0, x(0) = 0

y = cTx +du

so that

y(s) =[

cT (sI − A)−1 b +d]

u(s)

and

y(t ) = cT∫ t

0e A(t−τ)bu(τ)dτ+du(t ) =

[cTe At b +d

]︸︷︷︸

g (t ) (impulse response)

?u(t )

As it can be seen, the direct feed–through can be interpreted as a Dirac δ included in the input and amplified with the gaind. This motivates to consider the general class of systems introduced in Definition 7.6.

Lemma 7.2

A (β) equippped with the norm ‖ ·‖A (β) and the convolution operation ? has the following properties

(i) ‖ ·‖A (β) defines a norm on A (β) and A (β) is complete with respect to this norm.


(ii) The convolution is commutative, i.e., f ? g = g ? f , ∀ f , g ∈A .

(iii) The convolution is bilinear, i.e.,

f ? (αg ) =α( f ? g ), ∀α ∈R, ∀ f , g ∈A (β),

f ? (g +h) = f ? g + f ?h, ∀ f , g ,h ∈A (β).

(iv) For all f , g ∈A (β) it holds that f ? g ∈A (β) mit ‖ f ? g‖A (β) ≤ ‖ f ‖A (β)‖g‖A (β).

(v) The Delta–distribution δ( · ) is the identity element in A (β), i.e., δ? f = f ?δ= f , ∀ f ∈A (β).

(vi) A (β) is free of zero divisors, i.e., f ? g = 0 ⇒ f ≡ 0 or g ≡ 0.

Now, let f ∈A (β), then it holds for ℜs >β and the properties e−βt fa (t ) ∈ L1([0,∞)) as well as∑∞

n=1 | fn |e−βtn <∞, that∫ ∞

0f (t ) e−st dt =

∫ ∞

0fa (t ) e−st dt +

∫ ∞

0

∞∑n=1

fnδ(t − tn ) e−st dt = fa (s)+∞∑

n=1fn e−stn = f (s).

In other words, all elements of A (β) are Laplace–transformable with domain of the transform given by C+β

:= s ∈C|ℜs >β.

In the following A (β) = f | f ∈A (β) denote the set of Laplace–transforms of elements of f ∈A (β). For β= 0 this set issimply denoted by A instead of A (0).

Recall: A linear time–invariant finite–dimensional SISO system is exponentially stable if and only if the impulse response isan L1 function.

This result is extended in the sequel to the case of irrational transfer functions.

Definition 7.7

The extension Ae (β) of A (β) denotes the set of all generalized functions f (t) whose truncated function fT (t) ∈A (β), ∀T <∞. For β= 0 this set is denoted by Ae instead of Ae (0).

Theorem 7.1: Input–output stability and impulse response

Let the operator (H f )(t ) = h? f be given with h ∈Ae . Then the following statements are equivalent

(i) H is input–output stable.

(ii) h ∈A .

Proof. The proof is provided in two steps:

(ii)⇒(i): Let h ∈A and f ∈ L∞([0,∞)). Then it holds that

‖(H f )(t )‖∞ = ‖∫ t

0h(τ) f (t −τ)dτ‖∞

≤ ‖∫ t

0ha (τ) f (t −τ)dτ‖∞+‖

∫ t

0

∞∑n=1

hnδ(τ− tn ) f (t −τ)dτ‖∞

≤ ess supt

( ∞∑n=1

|hn || f (t − tn )|+∫ t

0|ha (τ)|| f (t −τ)|dτ

)

≤ ess supt | f (t )|( ∞∑

n=1|hn |+

∫ ∞

0|ha (τ)|dτ

)= ‖h‖A ‖ f ‖∞.

In consequence, it also holds true that ‖(H f )(t )‖∞ ≤ γ∞‖ f ‖∞ with γ∞ = ‖h‖A <∞, implying the input–output stability ofH .

(i)⇒(ii): The proof is given by contradiction, i.e., showing that h 6∈A ⇒ H is not input–output stable. As a simplification it isassumed that hn = 0, n ∈N, so that h(t ) = ha (t ). AS h(t ) 6∈A it follows that

∫ t0 |h(t −τ)|dτ is unbounded. It is shown that for

an f ∈ L∞([0,∞)) with ‖ f ‖∞ = 1, the mapping (H f )(t ) is unbounded. With f (τ) = sign(h(t −τ)) for τ ∈ [0, t ] it follows that

(H f )(t ) =∫ t

0h(t −τ)sign(h(t −τ))dτ=

∫ t

0|h(t −τ)|dτ.


In the limit t →∞ it follows that (H f )(t) is unbounded and thus not input–output stable. The general case with hn 6= 0follows similarly.

As a consequence of these results there is a direct analogy between rational transfer functions and irrational transferfunctions of class A with regard to the input–output stability, so that both can be in principle analyzed within the sameframework. Anyway, for this purpose the framework has to be extended in a form that is presented next, in the context of theanalysis of the closed–loop system.

7.2.3 Input–output stability of the closed–loop system

In the following the closed–loop system as shown in Figure 7.2 is analyzed. This system basically corresponds to a trajectorytracking problem, where u∗ denotes the feedforward input, associated with the desired output signal y∗. The deviation e2from this desired output leads to the control input ∆u, yielding the input offset e1.

y∗(s)e−

+gr (s)

e2(s)∆u(s)

6

-u∗(s) e+

+-

e1(s)guy (s) -ry(s)

?

Figure 7.2: Closed–loop system for trajectory tracking control.

The associated transfer matrix M(s) between [e1(s), e2(s)]T and [y∗(s), u∗(s)]T results ase1(s)

e2(s)

= 1

1+ gr (s)guy (s)

gr (s) 1

1 −guy (s)

︸︷︷︸

=: M(s)

y∗(s)

u∗(s)

.

As shown above, the input–output stability of the closed–loop system is ensured if M(s) ∈ A 2×2. The following Lemma givesa direct way to analyze this property.

Lemma 7.3

For guy (s), gr (s) ∈ A it is equivalent that

(i) M(s) ∈ A 2×2

(ii)1

1+ gr (s)guy (s)∈ A .

Proof. The proof follows in two steps:

(i)⇒(ii): Let M(s) ∈ A 2×2. It holds that Mi , j (s) ∈ A , i , j = 1,2 and from M2,1(s) it follows that 11+gr (s)guy (s) ∈ A .

(ii)⇒(i): Let 11+gr (s)guy (s) ∈ A . With M1,2(s) = 1

1+gr (s)guy (s) = M2,1(s), M1,1(s) = gr (s) 11+gr (s)guy (s) as well as M2,2(s) =

guy (s) 11+gr (s)guy (s) it directly follows that Mi , j (s) ∈ A , i , j = 1,2.

To check the condition [1+ gr (s)guy (s)]−1 ∈ A the following theorem can be employed.

Theorem 7.2

Let f (s) ∈ A (β). It holds true that f (s) is invertible in A (β), i.e., f −1(s) ∈ A (β), if and only if

infℜs≥β

| f (s)| > 0.


The proof can be found, e.g., in (Hille and Phillips, 1957, Theorem 4.18.6). In the literatue this condition is also referred to as

bounded away from zero over C+β := s ∈C|ℜs ≥β.

In consequence for guy (s) ∈ A and gr (s) ∈ A the input–output stability of the closed–loop is ensured if and only if

infℜs≥0

|1+ gr (s)guy (s)| > 0. (*)

To include unstable systems the following extension is considered (see, e.g., (Curtain and Zwart, 1995)).

Theorem 7.3

Let guy (s) be given by

guy (s) = g auy (s)+ g r

uy (s)

with g auy (s) ∈ A and rational, strict proper g r

uy (s). The resulting transfer matrix M(s) is element of A 2×2 if and only if(*) holds true.

While for rational transfer functions it is sufficient to evaluate these conditions along the imaginary axis, this is not truefor irrational ones, given that for ω→ ∞ not necessarily a limit is obtained. Thus, it is typically assumed that gr (s) =gr,a (s)+ gr,r (s) with gr,a (s) ∈ A and rational, strict proper component gr,r (s).

Theorem 7.4: Nyquist–stability test

Let guy (s) and gr (s) be given as in the preceding theorem. The transfer matrix

M(s) is element of A 2×2 if and only if

(i) 1+ gr (s)guy (s) 6= 0, ∀s ∈ N∞ and

(ii) (1 + gr (s)guy (s))|s∈N∞ encircles the origin p+ times counterclockwise,

with p+ being the number of (isolated) poles of the open–loop transferfunction gr (s)guy (s) in the open right–half complex plane, i.e.,

ind(1+ gr (s)guy (s)) = p+

with the so–called Nyquist index

ind(g (s)) = 1

2πlimω→∞

[arg(g (iω)|

ω∈N∞ )−arg(g (−iω)|ω∈N∞ )

].

Pol

N∞

σ

iω

For a proof of this theorem the reader is refered to (Curtain and Zwart, 1995, Theorem 9.1.8). Further interesting discussionsand the analysis of special cases can be found, e.g., in (Desoer and Vidyasagar, 1975, p.92 ff). The application of this theoremis shown in the sequel for the control of a plug–flow system and is also subject of Exercise 10.

Example 7.6 (Boundary control of a plug–flow system). Consider again the plug–flow system from Example 7.2.

x(z, t )

-z0 L

-

? ? ? ? ? ? ? ? ?u0(t )

u(z, t )

---------

With u(z, t ) = 0, u0(t ) = u(t ), and y(t ) = x(1, t ) it holds that

y(s)

u(s)= guy (L, s) = e−sTt , Tt = L

v.

According to the prior considerations it holds that p+ = 0.


P-controller: With gr (s) = K one has

f (iω) := 1+K guy (iω) = 1+K cos(ωTt )− i K sin(ωTt ).

The intersections with the ℜ–axis are obtained for the frequencies where sin(ωTt ) = 0, i.e., for ωk = kπTt

and it holds that

f (iωk ) = 1+K cos(ωk Tt ) = 1+K (−1)k , k ∈N0

Accordingly, for −1 < K < 1 the closed–loop system with the P–controller is exponentially stable according to Theorem 7.4,given that the graph of (1+ gr (s)guy (s))|s=iω, ω∈R+ does not encircle the origin.

A simulation result for

u(t ) = u −K (y(t )− y)

with u = 0.2, y = 0.2 and K = 0.5 using a finite–difference semidiscretization with N = 200 discretization points is shown inthe sequel

t 02468101214

z

0.00.20.40.6

0.81.0

x(z, t)

0.100.150.200.250.300.350.40

PI-controller: With gr (s) = KP + KIs one has

f (s) = 1+(KP + KI

s

)guy (s).

Again p+ = 0 so that there are no poles in the open right–half complex plane. Anyway there is a pole at s = 0, so that onehas to consider the modified Nyquist path

N∞ = (−i∞,−iε)∪Γ∪ (iε, i∞)

with Γ= εexp(iξ), ξ ∈ [−π/2,π/2] and ε¿ 1. For s ∈ (iε, i∞) one has

f (iω) =ℜ f (iω)+ iℑ f (iω)

= 1+KP cos(ωTt )− KI

ωsin(ωTt )− i

(KP sin(ωTt )+ KI

ωcos(ωTt )

). (7.2)

Obviously f (iω) is symmetric to the real axis, because f (−iω) =ℜ f (iω)− iℑ f (iω). In consequence the results for therange s ∈ (−i∞,−iε) can be directly extended by inversion along the real axis.

The analysis of the range s ∈ Γ+ = εexp(iξ), ξ ∈ [0,π/2] can be carried out by evaluating f (s) = f (a(ξ)+ i b(ξ)), given thatεexp(iξ) = ε(cos(ξ)+ i sin(ξ)) = a(ξ)+ i b(ξ), i.e.,

f (s) =1+e−a(ξ)Tt

(Kp cos(b(ξ)Tt )+ KI

ε2 [a(ξ)cos(b(ξ)Tt )−b(ξ)sin(b(ξ)Tt )]

)− i e−a(ξ)Tt

(KP sin(b(ξ)Tt )+ KI

ε2 [b(ξ)cos(b(ξ)Tt )+a(ξ)sin(b(ξ)Tt )]

). (7.3)

Again the symmetrie with respect to the real axis is obvious. In the following, first the input–output stability of the plug–flow


system with I –control alone is discussed and then the extension to the combined PI –control is considered.

• KP = 0: In this case the set of intersection of (7.2) with the real axis is determined by the condition KI cos(ωTt ) = 0,i.e., ωk = (2k −1)π/(2Tt ), k = 1,2, . . ., and thus

SRe =

1+ KI

ωk(−1)k

∣∣∣∣ k = 1,2, ...

.

Obivously, given the scaling with ωk , ω1 < ω2 < . . . < ωn < . . . there exist a minimum and maximum element.Requiring that

1− KI

ω1> 0 ∧ 1+ KI

ω2> 0

one can thus detemrine the interval −ω2 < KI <ω1 over which all intersection of f (iω), ω ∈ (ε,∞) with the real axisare located in the open right–half complex plan. Thus the question is if this interval is influenced by the contributionalong the remaining curve (7.3).

In particular, the valid interval for KI is influenced in the case that additional intersections with the real axis appear,which are outside the interval [1−KI /ω1,1+KI /ω2], or [1+KI /ω2,1−KI /ω1] for KI ≥ 0, or KI < 0, respectively.Given that a(ξ) ∈ [0,ε] with a(0) = ε, a(π/2) = 0 and b(ξ) ∈ [0,ε] with b(0) = 0, b(π/2) = ε it can be easily seen that

ℑ f (s) = 0 = b(ξ)cos(b(ξ)Tt )+a(ξ)sin(b(ξ)Tt )

only for b(ξ) = 0, i.e., ξ = 0. The curve f (s) thus has one additional intersection at the point 1+KI exp(−εTt )/ε.Requiring that this point is contained in the open right–half complex plane for all ε¿ 1 yields KI >−εeεTt , ε¿ 1,so that, given −εexp(εTt ) → 0 for ε→ 0 the modified interval is obtained as 0 ≤ KI <π/(2Tt ). It has to be taken intoaccount at this place that ε can be arbitrary small, but not identical to zero, so that εexp(εTt ) 6= 0.

According to the preceding analysis, all intersections with the real axis of the curve f (s), s ∈ N∞ are contained inthe open right–half complex plane as long as 0 ≤ KI <π/(2Tt ), so that the phase condition (ii) in Theorem 7.4 issatisfied, given that p+ = 0.

• KP 6= 0: In the general case with KP , KI 6= 0 the input–output statbility can no longer be analytically accessed.Nevertheless, one can use Theorem 7.4 in combination with the curves (7.2) and (7.3) in a numerical determinatinoof the allowed parameter set for (KI ,KP ). The result of such an analysis is shown in Figure ?? for different values ofTt ∈ 0.5,1,2. The combinations of KP ,KI for which the input–output stability is ensured are contained within thebounded curve. As expected, this sets become smaller for increasing delays Tt .

0 1 2 3 4−1.5

−1

−0.5

0

0.5

1

1.5

2

KP

KI

Tt = 0.5Tt = 1Tt = 2

Simulation results, considering v = 1 (i.e., Tt = 1), KP = 0.5,KI = 1.0 (included in the region determined above) anddesired output value y = 0.2 (as above), with the before described finite–difference scheme and

u(t ) = KP (y(t )− y)+KI

∫ t

0(y(τ)− y)dτ

are shown in the following figure.


t 0246810z

0.00.20.40.60.8

1.0

x(z, t)

−0.2−0.10.00.10.20.30.4

Note that there is on feedforward component included in the controller (the input corresponding to the desiredoutput is assumed unknown!), and the final value is obtained only by the integral part of the controller. Using thesame simulation setup it can be easily shown that, e.g., decreasing v from 1 to 0.5, i.e., increasing Tt from 1 to 2, thesame controller gains yield an unstable closed–loop behavior with an increasing oscillation.

References

Curtain, R. F. and H. Zwart (1995). An introduction to infinite-dimensional linear systems theory. Springer(cit. on p. 104).

Desoer, D. and M. Vidyasagar (1975). Feedback Systems: Input–Output Properties. Academic Press, New York(cit. on pp. 99, 104).

Hille, E. and R. Phillips (1957). Functional Analysis and Semi–Groups. American Mathematical Society, Provi-dence, Rhode Island (cit. on p. 104).

REFERENCES 107


Chapter 8

Flatness-based control

The notion of (differential) flatness, as introduced by M. Fliess and co–workers (Fliess, J. Lévine, et al., 1995; Fliess, P. Lévine,et al., 1999), has turned out to provide a useful approach for solving trajectory tracking problems (Rothfuß et al., 1996; Martinet al., 1997; Nieuwstadt et al., 1998; Martin et al., 2000; Rudolph, 2003; Sira-Ramirez and Agrawal, 2004). Originally, theconcept was formulated for finite–dimensional nonlinear systems but many aspects have been successfully generalized tocertain classes of infinite-dimensional systems. In this chapter, we provide an introduction to flatness–based techniques byfirst summarizing some results obtained for the finite–dimensional case. This is followed by illustrating different extensionsof this approach to infinite–dimensional systems governed by DPSs.

8.1 Finite–dimensional nonlinear control systems

Roughly speaking, flatness means, that there exists a so-called flat or basic output such that all system variables (states,inputs and outputs) can be parametrized in terms of this flat output and its time derivatives1. For the sake of simplicity letus consider a nonlinear system of the form

x = f (x ,u), t > 0, x(0) = x0, (8.1)

with the state x(t) ∈ X ⊂ Rn at time t ≥ 0 and the input u(t) ∈ U ⊂ Rm . The system (8.1) is called flat if there exists aso-called flat or basic output ξ such that (Fliess, J. Lévine, et al., 1995; Rothfuß, 1997; Rudolph, 2003)

(i) the components of ξ are functions of x and u and their time derivatives2

ξ=φ(x ,u, u, . . . ,u(γ)) , (8.2)

(ii) the components of ξ are not related by any differential or algebraic equation of the form3

ϕ(ξ, ξ, . . . ,ξ(δ))= 0 , (8.3)

(iii) and all system variables can be parametrized by the flat output and its time derivatives

x =ψ1(ξ, ξ, . . . ,ξ(β))

u =ψ2(ξ, ξ, . . . ,ξ(β+1)).

(8.4)

Example 8.1 (Linear time invariant system). A linear time-invariant system is flat if and only if it is controllable. Acontrollable linear system can be always transformed into the controllability normal form which for SISO systems is given

1As will be shown in subsequent sections for infinite–dimensional systems governed by PDEs an infinite number of time derivatives ofthe flat output may be required.

2Here and subsequently we will denote the γth time derivative of a function f (t ) in the form f (γ) = ∂γt f (t ) .3This condition is equivalent to dimu(t ) = dimξ(t ).

109

by

x1

x2

...

xn−1

xn

︸︷︷︸

x

=

0 1 0 . . . 0

0 0 1 . . . 0

......

. . .. . .

...

0 0 . . . 0 1

−a0 −a1 . . . −an−2 −an−1

︸︷︷︸

A

x1

x2

...

xn−1

xn

︸︷︷︸

x

+

0

0

...

0

1

︸︷︷︸

b

u (8.5)

with the coefficients a j , j = 1, . . . ,n, of the characteristic polynomial of the matrix A ∈Rn×n . Obviously, ξ= x1 serves as aflat output and the state and input parametrization according to (8.4) reads

x j = ξ( j−1), j = 1,2, . . . ,n (8.6)

u =n∑

j=0a j ξ

( j ) (8.7)

with an = 1 and ξ(0) = ξ.

In view of the analysis of DPSs some essential properties of flat finite–dimensional nonlinear control systems are summarizedbelow:

• Steady state analysis: The state and input parametrizations (8.4) can be similarly used to analyze the steady statebehavior, i.e. let ξ= ξs , ξ( j ) = 0, j ≥ 1 and consider 0 = f (x s ,us ), then

x s =ψ1(ξs ,0, . . . ,0) = ψ1(ξs )

us =ψ2(ξs ,0, . . . ,0) = ψ2(ξs ).

This property will play a substantial role for DPSs.

• Uniqueness of flat outputs: Flat systems may have more than a single flat output fulfilling the conditions (8.2)–(8.4).In this case (differential) relations exist, that allow to transfer one flat output into the other.

• Uncontrolled systems: Free (uncontrolled) systems x = f (x) are not flat. Assume for the moment, that there exists aflat output ξ with x =ψ1

(ξ, ξ, . . . ,ξ(β)). Then substitution of this expression into the differential equation yields a

differential equation of the form (8.3) which contradicts the assumption.

• Linear time invariant systems: For linear time invariant systems flatness and controllability are equivalent (seeExample 8.1).

• Existence of flat outputs: For the flatness of nonlinear SISO systems necessary and sufficient conditions existwhich are well-known from the theory of exact input–state linearization. In fact for SISO systems exact input–statelinearizability and flatness are equivalent. For MIMO systems also necessary and sufficient conditions are availablefor a nonlinear system to be exactly input–state linearizable, see, e.g., (Isidori, 1995; Nijmeier and Schaft, 1990). Inthe last years much effort has been made in finding conditions for a MIMO system to be flat, see, e.g., (J. Lévine,2004; Schlacher and Schöberl, 2007; Schöberl and Schlacher, 2014). However, if a MIMO system is exact input–statelinearizable, then it is also flat.

• Flatness–based feedforward control: Let ξ∗ ∈Cβ(R) denote a desired trajectory for the flat output. Then substitutionof ξ∗ into (8.4) directly yields the feedforward control u∗, that is required to realize x → x∗ in open–loop withoutintegration of any differential equation, i.e.

x∗ =ψ1(ξ∗, ξ∗, . . . ,ξ∗(β))

u∗ =ψ2(ξ∗, ξ∗, . . . ,ξ∗(β+1)).

(8.8)

The determined trajectories x and u can be in advance utilized to analyze the fulfillment of state and input constraints.In addition, x∗ can be considered as reference path to determine tracking controllers for (8.1).

• Flatness–based tracking control: Flat nonlinear systems can be exactly feedback linearized so that linear methodscan be considered for stabilization and control. Quasi–static or dynamic state–feedback control can be utilized forthe asymptotic stabilization of the tracking error e = ξ−ξ∗ or ex = x −x∗, respectively. By transferring (8.1) into theBrunovský normal form desired eigenvalues can be assigned for the tracking error dynamics.

It is obvious from the discussion, that flat systems have very pleasing properties to solve the trajectory tracking problem forfinite–dimensional nonlinear systems. Hence, the question arises if similar properties can be deduced when extendingflatness–based methods to DPSs. This is the main topic of this chapter.

110 Chapter 8 Flatness-based control

8.2 Distributed parameter control systems

The underlying idea of equivalence and flatness, i.e. the existence of a one–to–one correspondence between trajectories ofsystems, can be also adapted to systems governed by PDEs (Laroche et al., 2000; Rouchon, n.d.; Rudolph, 2003; Meurer,2005; Woittennek, 2007). Hence, the recent work on the flatness concept has mainly dealt with its extension to trajectoryplanning for boundary controlled linear and certain nonlinear DPSs in a single and multiple spatial coordinate(s).

Remark 8.1

It is nowadays rather common to refer to a basic output instead of a flat output when considering flatness–basedmethods for DPSs. Hence, in the following, both notations are used interchangeably.

8.2.1 Trajectory planning for PDE systems

For parabolic and biharmonic PDEs the application of operational calculus using basically the Laplace transform or formalpower series yields the state and input parametrization in terms of fractional differentiation operators or infinite powerseries representations4. The arising series coefficients depend on successive time derivatives of the basic output. Thisrequires to restrict the basic output to a certain (Gevrey) class to ensure uniform convergence of the series. Examplesconcern trajectory planning for the linear heat equation (Laroche et al., 2000) and for the linear diffusion equation withspatially dependent coefficients (Laroche et al., 2000; Lynch and Rudolph, 2002) in several state variables (Fliess, Mounier,et al., 1998; Meurer, 2005). In addition, certain semi– and quasi–linear diffusion–convection–reaction systems modelingtubular reactors are considered, e.g., in (Lynch and Rudolph, 2002; Meurer and Zeitz, 2005; Meurer, 2005; Meurer and Zeitz,2008; Utz et al., 2010), while a moving boundary problem (Stefan problem) is studied in (Dunbar et al., 2003; Rudolph et al.,2005). A further generalization for semi–linear PDEs is considered in (Schörkhuber et al., 2013), where formal integrationis used to determine the state and input parametrization. Besides parabolic PDEs, results on the trajectory planning forhyperbolic systems exhibiting wave dynamics are available (Petit and Rouchon, 2001; Rudolph, 2003; Woittennek, 2007;Wagner et al., 2008).

Solutions to the trajectory planning problem for PDEs defined on higher–dimensional domains are provided, e.g., in(Rudolph, 2003) for the control of the temperature evolution inside a cylinder. By exploiting the rotational symmetry ofthe domain the problem is thereby reduced to two decoupled 1–dimensional systems. The motion of a fluid representedby linearized wave equations under the shallow water approximation inside a moving tank being subject to controlledtranslations and rotations is analyzed in (Petit and Rouchon, 2002). The solution to the trajectory planning problem isobtained by in principle superimposing the solution of two decoupled 1–dimensional problems. First computations withentire functions are suggested in (Rouchon, 2005) for the 2– and 3–dimensional wave equation with a finite–dimensionalcontrol acting simultaneously on all of the domain’s boundary. It is, however, shown that the resulting flatness–basedparametrizations diverge in general.

By considering Riesz spectral systems (see Section 4.1.3) a rather generic approach for flatness–based trajectory planningfor DPSs is developed in (Meurer, 2011; Meurer, 2013), that covers both boundary and in–domain control. Here, a particularre–formulation of the resolvent operator is used to systematically construct a basic output. Convergence of the differentialstate and input parametrizations is analyzed by making use of entire function theory and essentially relies on the distributionof the eigenvalues of the system operator. For boundary controlled diffusion–convection–reaction systems with spatiallyand time varying parameters defined on a parallelepiped domain a solution to the trajectory planning problem is providedin (Meurer and Kugi, 2009) by considering a formal integration of the PDE.

Based on this short survey of the available techniques in the following selected techniques are summarized and evaluatedfor benchmark examples.

8.3 Operational calculus

Application of operational calculus, i.e., integral transformations such as the Laplace transform or the Mikusinski calculus,enables to transfer linear initial boundary value problems into ordinary differential equations. As is shown subsequently fordifferent examples, this allows to determine a basic output for the original distributed parameter formulation.

4This introductory literature review is a condensed version of the respective section in (Meurer, 2013).

8.2 Distributed parameter control systems 111

Remark 8.2

It should be mentioned, that the main focus of this section is the development of the underlying ideas. For the gener-alization of the design approach based on operational calculus to rather generic formulations of linear distributedparameter control problems the reader is referred to the literature cited at respective positions in the text.

8.3.1 Flatness–based trajectory planning for the linear heat equation

The temperature distribution x(z, t ) for the heated rod shown below with ideal insulation at z = 0 and boundary input u atz = 1 is considered.

x(z, t )

-

z0 1

u

@@@@@@@@@@@@@@

The spatial–temporal evolution of x(z, t ) is governed by the linear heat equation

∂t x(z, t ) = k∂2z x(z, t ), z ∈ (0,1), t > 0 (8.9a)

∂z x(0, t ) = 0, x(1, t ) = u, t > 0 (8.9b)

x(z,0) = x0(z), z ∈ [0,1] (8.9c)

with k =λ/(ρcp ). For the sake of simplicity it is assumed, that all variables are dimensionless which can be easily achievedby proper normalization. Let x0(z) = 0, then the application of the Laplace transform to (8.9) results in the boundary valueproblem

∂2z x(z, s) = s

kx(z, s), ∂z x(0, s) = 0, x(1, s) = u(s) (8.10)

which admits the solution

x(z, s) = g (z, s)u(s), g (z, s) = cosh(µ(s)z)

cosh(µ(s))(8.11)

with µ(s) =ps/k. Based on the transfer function g (z, s) we verify the existence of a basic output. Let u(s) = cosh(µ(s))ξ,

then (8.11) implies

x(z, s) = cosh(µ(s)z)ξ, u(s) = cosh(µ(s))ξ. (8.12)

Due to the formal relation with (8.3) the quantity ξ can be considered a basic output in the operational domain. For thedetermination of the time–domain parametrizations the series expansion of the cosh–function is considered, i.e.,

cosh(µ(s)z) =∞∑

n=0

(µ(s)z)2n

(2n)!=

∞∑n=0

( s

k

)n z2n

(2n)!.

This yields with (8.12) the state and input parametrizations

x(z, t ) =∞∑

n=0

z2n

kn (2n)!∂n

t ξ(t ), u(t ) =∞∑

n=0

1

kn (2n)!∂n

t ξ(t ). (8.13)

In addition, the introduced formal quantity ξ or ξ, respectively, admits the interpretation ξ= x(0, t ) which can be verified bydirect substitution in (8.12).

8.3.1.1 Convergence analysis

The state and input parametrizations (8.13) impose certain restrictions on any admissible trajectory for the basic output. Inparticular it is required, that


• ξ is infinitely often continuously differentiable, i.e., it is a smooth function and

• the growth of the derivatives of ξ is sufficiently bounded so that convergence of the series (8.13) can be ensured.

To address both issues the notion of a Gevrey class function is required (Hua and Rodino, 1996).

Definition 8.1: Gevrey class

The function ξ :Ω→R is in GD,α(Ω), the class of Gevrey functions of order α, if ξ ∈C∞(Ω) and there exists a positiveconstant D such that

supt∈Ω

|∂nt ξ| ≤ Dn+1(n!)α (8.14)

for all n ∈N∪ 0.

Recall the Cauchy–Hadamard theorem for the radius of convergence % of the power series∑

n an zn , i.e.,

% =

0 if limn→∞ |an |

1n →∞

∞ if limn→∞ |an |1n = 0

1

limsupn→∞ |an |1n

else.

Herein, limsupn→∞ |an |1n denotes the largest accumulation point of the sequence (|an |

1n )n . With these preliminaries it is a

rather easy task to proof the following result.

Proposition 8.1. Let ξ ∈GD,α(R), then the series

x(z, t ) =∞∑

n=0

z2n

kn (2n)!∂n

t ξ(t ) (8.15)

converges uniformly with infinite radius of convergence % in z if α< 2 and with radius of convergence % = 2p

k/D if α= 2.

Proof. Since ξ is by definition a Gevrey class function of order α and hence fulfills (8.14), the series (8.15) for k > 0 andz ∈ [0,1] can be bounded according to

|x(z, t )| =∣∣∣ ∞∑

n=0

z2n

kn (2n)!∂n

t ξ(t )∣∣∣≤ ∞∑

n=0

Dn+1

kn(n!)α

(2n)!z2n = D

∞∑n=0

(n!)α

(2n)!

( Dz2

k

)n.

Let η= Dz2/k, then the last expression can be interpreted as a power series in η. Taking into account the Cauchy–Hadamardtheorem its radius of convergence %η in the variable η can be determined as

%η =

∞, α< 2

4, α= 2

0, α> 2.

To verify this result the Stirling formula can be used which provides the asymptotic relation n! ∼p2πnn+ 1

2 /en for n À 1. In

view of η= Dz[2]/k the radius of convergence in z follows as %z =√

k%η/D and proves the claim.

8.3.1.2 Admissible trajectories for the basic output

As has been shown before the flatness–based approach essentially relies on planning admissible trajectories for the flatoutput taking into account the conditions formulated in Proposition 8.1. In addition, two goals have to be distinguished:

• Finite time transitions between steady states: On rather common control task is the realization of transitions be-tween steady states within a desired time interval t ∈ [0,T ]. For the heat equation (8.9) this requires the determinationof the feedforward control

u∗, t ∈ [0,T ] : x0(z) = x∗(z,0) → x(z,T ) = x∗T (z). (8.16)

8.3 Operational calculus 113

Here, the initial and final profile x0(z) and xT (z), respectively, are assumed to be solutions xs (z;us ) of the boundaryvalue problem associated with (8.9), i.e.,

k∂2z xs (z) = 0, z ∈ (0,1)

∂z xs (0) = 0, xs (1) = us

with x∗0 (z) = xs (z;u∗

0 ) and x∗T (z) = xs (z;u∗

T ).

Due to the flatness property the transition can be alternatively formulated in terms of flat output ξ= x(0, t ). Let ξ∗0and ξ∗T denote desired initial and final values, then steady state solutions xs (z;ξs ) fulfill the boundary value problem

k∂2z xs (z) = 0, z ∈ (0,1)

∂z xs (0) = 0, xs (0) = ξs .

Hence, the transition can be formulated as finding the trajectory ξ∗ ∈GD,α(R) connecting ξ∗0 and ξ∗T , i.e.,

ξ∗, t ∈ [0,T ] : ξ∗0 = ξ∗(0) → ξ∗(T ) = ξ∗T with ∂nt ξ

∗|t∈0,T = 0. (8.17)

For the explicit realization of ξ∗ satisfying the specification (8.17) and hence x∗ as well as u∗ by evaluating (8.13) it isrequired, that ξ∗ is infinitely often differentiable in t but locally non–analytic at t = 0 and t = T . As is shown belowthis restricts the Gevrey order α of ξ∗ to the interval α ∈ (1,2).

To fulfill these restrictions in the following we make use of the Gevrey class function

ξ∗ = ξ∗0 + (ξ∗T −ξ∗0

)Θω,T (t ) (8.18)

with

Θω,T (t ) =

0, t ≤ 0

1, t ≥ T∫ t0 θω,T (τ)dτ∫ T0 θω,T (τ)dτ

, t ∈ (0,T )

(8.19)


0 0.2 0.4 0.6 0.8 1 1.20

0.2

0.4

0.6

0.8

1

1.2

t

yω=1.1ω=2.0

Figure 8.1: Trajectory y = ξ∗ defined in (8.18) with (8.19), (8.20) for ξ∗0 = 0, ξ∗T = 1, T = 1, and ω ∈ 1.1,2.

and

θω,T (t ) =

0, t 6∈ (0,T )

exp(− ([

1− tT

] tT

)−ω), t ∈ (0,T ).

(8.20)

In particular,Θω,T (t ) is of Gevrey orderα= 1+1/ω (Hua and Rodino, 1996; Lynch and Rudolph, 2002). The parameterω and the transition time T can be used to adjust the slope of ξ∗. Figure 8.1 shows ξ∗ defined in (8.18) for twodifferent values of ω to illustrate this effect.

• Finite time transitions between arbitrary states: If the initial and final profiles x∗(z,0) and x∗(z,T ) do not corre-spond to steady state profiles of the considered DPS, then trajectory planning is significantly complicated. Explicitcomputations are not provided here but the interested reader is referred to (Laroche et al., 2000; Dunbar et al., 2003),where the projection of the initial and final profile onto the basis spanned by power series of the underlying functionspace is proposed. Another projection technique is suggested in (Meurer, 2013) based on the parametrizationapproach introduced in Section 8.4 below.

8.3.1.3 Simulation results

To illustrate the theoretical concepts simulation results are presented for the feedforward control of the heat equation (8.9).The boundary input is determined by evaluating the input parametrization in (8.13) with the desired trajectory ξ∗ for thebasic output constructed as described before, i.e., by making use of (8.18). The series for u∗ is cut off after 21 addends.Numerical results are shown in Figure 8.2 for ξ∗ of Gevrey order α= 2 and in Figure 8.3 when reducing the Gevrey orderto α= 1.5. The transition time is assigned as T = 1. The numerical evaluation reveals that a reduction in α results in anincrease of the slope of ξ∗ and thus in an increase in input amplitude.

0

0.5

1

0

0.5

1

1.50

0.5

1

1.5

2

zt

x

0 0.5 1 1.50

0.5

1

1.5

2

t

u*

Figure 8.2: Feedforward boundary control of the heat equation (8.9) for Gevrey order α= 2 and k = 1: State evolution x(z, t )(left), feedforward control u∗ (right).


0

0.5

1

0

0.5

1

1.50

1

2

3

4

zt

x

0 0.5 1 1.50

0.5

1

1.5

2

2.5

3

3.5

4

t

u*

Figure 8.3: Feedforward boundary control of the heat equation (8.9) for Gevrey order α = 1.5 and k = 1: State evolutionx(z, t ) (left), feedforward control u∗(t ) (right).

8.3.2 Flatness–based trajectory planning for the linear wave equation

The wave equation is the prototype of so–called hyperbolic PDEs, that exhibit wave dynamics with finite speed of propaga-tion. This classification is based on the so–called characteristics or characteristic curves (see Section 3.2 and, e.g., (John,1982; Whitham, 1999)).

In the following the implications of the finite speed of propagation are illustrated by developing flatness–based trajectoryplanning for the linear wave equation with boundary control, i.e.,

∂2t x(z, t ) = c2∂2

z x(z, t ), z ∈ (0,1), t > 0 (8.21a)

∂z x(0, t ) = u, ∂z x(1, t ) = 0, t > 0 (8.21b)

x(z,0) = x0(z), ∂t x(z,0) = x1(z) z ∈ [0,1] (8.21c)

with c denoting the phase velocity. Assuming zero ICs, the application of the Laplace transform transfers (8.21) into aboundary value problem, whose solution in the operator domain can be determined as

x(z, s) =− c cosh(µ(s)(1− z))

s sinhµ(s)u(s) (8.22)

with µ(s) = s/c. Proceeding as in Section 8.3.1 the basic output ξ = x(1, s) can be introduced in the operational domainwhich yields the state and input parametrizations

x(z, s) = cosh(µ(s)(1− z))ξ(s) (8.23)

u(s) =− s

csinh(µ(s))ξ(s). (8.24)

These equations of the inverse system can be transferred into the time domain by taking into account the shifting propertywhich yields

x(z, t ) = 1

2

[ξ

(t + 1− z

c

)σ

(t + 1− z

c

)+ξ

(t − 1− z

c

)σ

(t − 1− z

c

)], (8.25a)

u(t ) =− 1

2c

[ξ

(t + 1

c

)σ

(t + 1

c

)− ξ

(t − 1

c

)σ

(t − 1

c

)]. (8.25b)

Differing from the heat equation example delayed and advanced arguments arise, that also need to enter the assignment ofadmissible trajectories ξ∗ for the basic output ξ. In addition it should be mentioned, that it is sufficient to assign ξ∗ ∈C 0(R)which, by (8.25b), implies a (piecewise) continuous input u∗.

To illustrate this consider the desired triangular–shaped trajectory ξ∗ of width 2a depicted in Figure 8.4. Causality requiresξ∗(t ≤ 1/c) = 0. The corresponding feedforward control u∗ determined from (8.25b) is shown in 8.5 for a pulse width of2a < 2/c (top) and 2a > 2/c (bottom). The effect of the feedforward control u∗ on the spatial–temporal dynamics of thewave equation is illustrated in Figure 8.6 in the (z, t , x)–plane. The first contribution of the feedforward control (8.25b), i.e.,

u∗(1)(t ) =− 1

2cξ

(t + 1

c

)σ

(t + 1

c

)(8.26)


-

6 ξ∗(t )

ξ∗(t )

t

@

@@@@@

1a

1

− 1a

01c

1c +a 1

c +2a

Figure 8.4: Triangular–shaped desired trajectory for the basic output ξ∗ and its derivative ξ∗.

-

-

6

6

0

0

u∗(t )

u∗(t )

t

t

1c

a 2c

2a 2c +a

2c +2a

a 1c

2a 2c

2c +a

2c +2a

− 12ac

12ac

1ac

− 12ac

12ac

Figure 8.5: Feedforward control u∗ for the desired trajectory ξ∗ of Figure (8.4) for a < 1/c (top) and a > 1/c (bottom).

induces a triangular impulse, that propagates along the characteristic curves in the (z, t )–plane to right border z = 1. Thetime of arrival of the first signal is t(1) = 1/c. Reflection at the free end implies amplitude doubling at z = 1 and traveling ofan image of the initial impulse to the left border z = 0. The time of arrive of the first backward–traveling signal is t(2) = 2/c.This wave superposes with the (second) contribution imposed by the feedforward control u∗, i.e.,

u∗(2)(t ) = 1

2cξ

(t − 1

c

)σ

(t − 1

c

)(8.27)

which results in the extinction of the incoming wave.

Summarizing these results it has to be pointed out, that trajectory planning for hyperbolic PDEs has to take into account thefinite speed of propagation. This corresponds to a minimal control time which, e.g., does not arise for the parabolic heatequation, whose speed of propagation is infinite.


-

6

0

x(z, t )

z1

t

a

2a

2c

2c +a

2c +2a

1c

1c +a

1c +2a

:

:

:

``````

``````

``````

````

``````

``````

````

``````

``````

``````

`

y

y

y

12

AAAAAA

CCCCCCCCCCC

6

?

1

AAAAAA

Reflected wave

HHH Control ∂z x(0,t )=u∗(t )

EEEEEEEE

Incomming waveby reflection

Extinction bysuperposition

Figure 8.6: Feedforward boundary control for the wave equation (8.21) to realize trajectory tracking x(1, t ) = ξ→ ξ∗ with ξ∗from Figure 8.4 and pulse width a > 1/c.

8.4 Riesz spectral operators

The spectral analysis of a finite– or infinite–dimensional linear operator is a well–established and powerful mathematicaltool for stability analysis and feedback control design. The dynamic system properties are thereby determined based on theeigenvalue distribution and the respective set of eigenvectors. For infinite–dimensional systems governed by PDEs certainrestrictions apply that are in particular related to the possible existence of continuous spectra.

However, as pointed out in Chapter 4 a wide class of physically important systems including, e.g., diffusion–convection–reaction, wave, Euler–Bernoulli, and Timoshenko beam equations, yields Riesz spectral operators (see Section 4.1.3).These have the favorable property of a discrete eigenvalues distribution with the respective eigenvectors and adjointeigenvectors spanning an orthogonal basis for the underlying function space. On the one hand this significantly simplifiesthe analysis of structural properties such as controllability and observability which can be performed rather similar tothe finite–dimensional case (Curtain and Zwart, 1995). On the other hand Riesz spectral operators satisfy the so–calledspectrum determined growth assumption which implies, that stability can be directly evaluated based on the eigenvaluedistribution (Curtain and Zwart, 1995; Luo et al., 1999).

8.4.1 Flatness–based state and input parametrization

In the following, DPSs in abstract form governed by

dx

dt= Ax +Bu, t > 0 (8.28a)

x(0) = x0 ∈D(A) (8.28b)

are considered for x ∈ X . Moreover, we assume that

(i) A is a (Riesz) spectral operator according to Definition 4.10 with non–zero eigenvalues λk , k ∈N(ii) the initial state x0 is a steady state satisfying Ax0 = 0, x0 ∈D(A) such that without loss of generality we can consider

x0 = 0


In addition, in view of the following examples we

(iii) restrict the analysis to input operators of the form

Bu =m∑

l=1bl ul (t ) (8.29)

with the spatial input characteristics bl = bl (z) and refer to (Meurer, 2013) for a general treatise and

(iv) assume that (8.28) is approximately controllable, i.e.,

rank[⟨b1,ψk ⟩X . . . ⟨bm ,ψk ⟩X

]= 1

for all k ∈N.

8.4.1.1 Construction of a basic output in the operational domain

The resolvent operator corresponds to the Laplace transform of the C0–semigroup generated by A, i.e.,

x( · , s) = (sI − A

)−1B u(s), s ∈ ρ(A) with s > supk∈N

ℜλk ,

where s ∈C denotes the Laplace variable and x and u the Laplace transforms of x and u. By recalling (4.84) this yields for alls ∈ ρ(A) with s > supk∈Nℜλk , that

x( · , s) = ∑k∈N

1

s −λk⟨B u(s),ψk ⟩X φk =− ∑

k∈N1

λk

1

1− sλk

⟨B u(s),ψk ⟩X φk . (8.30)

The above expression serves as basis for the construction of a basic output for the linear system (8.28). For this, we re–writethe resolvent by sufficiently extending numerator and denominator

x( · , s) =− ∑k∈N

1

λk

∏j∈N, j 6=k

(1− s

λ j

)∏

j∈N(1− s

λ j

) ⟨B u(s),ψk ⟩X φk

and observe, that in view of (8.29) we have

⟨B u(s),ψk ⟩X =m∑

l=1⟨bl ,ψk ⟩X ul (s).

Substitution into the previous expression yields

x =− ∑k∈N

1

λk

m∑l=1

⟨bl ,ψk ⟩X φk∏

j∈N, j 6=k

(1− s

λ j

) ul (s)∏j∈N

(1− s

λ j

) .

Formally introducing the variables ξl (s), l = 1, . . . ,m by

ul (s) = ∏j∈N

(1− s

λ j

)︸︷︷︸

Du (s)

ξl (s) (8.31a)

provides

x( · , s) =− ∑k∈N

1

λk

m∑l=1

⟨bl ,ψk ⟩X φk∏

j∈N, j 6=k

(1− s

λ j

)︸︷︷︸

Dxk (s)

ξl (s). (8.31b)

Expressions (8.31) can be interpreted as state and input parametrizations in the operational (Laplace) domain in terms ofξl (s). Hence, in accordance with the previous results ξl (s), l = 1, . . . ,m is called a basic output in the operational domain.

8.4 Riesz spectral operators 119

8.4.1.2 Time domain representation and convergence analysis

The state and input parametrizations (8.31) are so far only formal since their (uniform) convergence has to be ensured.Convergence obviously relies on the properties of the basic output and as such reduces to a problem of assigning suitabletrajectories to ξl (t ). In addition it is decisive to note, that the introduced operators Du (s) and Dx

k (s) define entire functionsin the complex domain s ∈C. With this, the main result reads as follows.

Theorem 8.1

Let (λk )k∈N be the sequence of disjoint eigenvalues of the Riesz spectral operator A. Assume that (λk )k∈N is ofconvergence exponent γ and genus g s . Then Du (s) is an entire function of order ρ = γ and admits a MacLaurin seriesexpansion

Du (s) = ∑n∈N

cn sn , c1 = 1. (8.32)

If Du (s) is in addition of finite type τ, then

f (t ) = Du (∂t )ξ(t ) = ∑n∈N

cn∂nt ξ(t ) (8.33)

converges uniformly for ξ ∈GD,α(R) with α≤ 1/ρ and ‖ f ‖∞ is bounded.

For a proof of Theorem 8.1 consult (Meurer, 2013). Identical properties can be deduced for Dxk (s) based on the results for

Du (s). As a result of the previous analysis the time–domain representation of (8.31) follows immediately by taking intoaccount the MacLaurin series expansion, i.e.,

ul (t ) = Du (∂t )ξl (t ) = ∑n∈N

cn∂nt ξ(t ) (8.34a)

x( · , t ) =− ∑k∈N

1

λkDx

k (∂t )ξl (t )m∑

l=1⟨bl ,ψk ⟩X φk . (8.34b)

Remark 8.3

The convergence of (8.34b) imposes an additional condition, that is omitted but is derived in (Meurer, 2013). This isdue to the summation over k ∈Nweighting the coefficients of the basis functionsφk so that the L2–convergence ofthis Fourier series has to be ensured.

In view of the convergence result, the assignment of suitable desired trajectories ξl ,∗(t ) for the basic output ξl (t ) followsalong the lines of Section 8.3.1.2.

To apply these results obviously certain notions from the theory of entire functions are required, that are summarized inthe following remark (see (Meurer, 2013) for a comprehensive overview of the required notions and properties in view offlatness–based methods).

Remark 8.4

The so–called maximal modulus M(η) of an entire function f (s) is defined as

M(η) = max|s|=η | f (s)|. (8.35)

Obviously, M(η) enables to characterize the growth of the entire function f (s). Thereby, two properties are essential,namely type and order:

• The entire function f (s) is of finite order if M(η) <as exp(ηk ) for some k > 0. The order ρ is the infimum ofthose k for which the asymptotic inequality <as is fulfilled. With this, we have

eηρ−ε <n M(η) <as eη

ρ+ε


and by taking the logarithm twice we conclude

ρ = limsupη→∞

loglog M(η)

logη.

• The function f (s) has a finite type if for some A > 0 the inequality M(η) <as exp(Aηρ) holds. The type τ is theinfimum of those A for which the asymptotic inequality <as is satisfied. Moreover, this implies the inequalities

e(τ−ε)ηρ <n M(η) <as e(τ+ε)ηρ

and hence

τ= limsupη→∞

log M(η)

ηρ. (8.36)

If for a given ρ the type of f (s) is infinite, then the function is of maximal type. If 0 < τ<∞, then the type isnormal while for τ= 0 the type is minimal.

Given a non–decreasing sequence (an )n∈N, an ∈C the so–called counting function N(η) is

N(η) = #an , n ∈N : |an | ≤ η. (8.37)

and its order ρ1 (Boas, 1954, Theorem 2.5.8) can be defined as

ρ1 = limsupη→∞

logN(η)

logη. (8.38)

The counting function is a particularly useful tool to deduce properties of entire functions.

Given a sequence (an )n∈N, an ∈Cwith an 6= 0, limn→∞ an →∞ the convergence exponent is defined as the infimumof positive numbers γ for which the series∑

n∈N1

|an |γ(8.39)

converges. The relationship between the convergence exponent of a sequence and its counting function is given inthe following lemma (Levin, 1996, Section 3.2).

Lemma 8.1

Given a sequence (an )n∈N, an ∈Cwith an 6= 0, limn→∞ an →∞, then γ= ρ1.

Denote by g s +1 the smallest positive integer γ for which (8.39) converges. Then the integer g s is called the genus ofthe sequence (an )n∈N. The genus g s is not necessarily equal to the genus of the entire function f (s) but there is aclass of entire functions, where equality holds. Assume that the sequence (an )n∈N is of genus g s . With this, considernow the infinite product

Π(s) = ∏n∈N

G

(s

an, g s

), (8.40)

with the so–called Weierstrass primary factors G(s, g s) defined as

G(s, g s)=

1− s, g s = 0

(1− s)expF(s, g s) , g s > 0

, F(s, g s)= g s∑

i=1

si

i. (8.41)

The infinite product (8.40) converges absolutely and uniformly in every disk s ∈ C : |s| ≤ R < ∞ (Levin, 1996,Section 4.1) andΠ(s) is called the Weierstrass canonical product of genus g s . Following Boas (1954, Theorem 2.6.5),Π(s) defines an entire function of order equal to the convergence exponent of its zeros.

With these preliminary considerations one of the main theorems in the theory of entire functions can formulated,which provides a general representation formula for entire functions of finite order (Levin, 1996, Section 4.2).


Theorem 8.2: Hadamard theorem

An entire function f (s) of finite order ρ may be represented in the form

f (s) = sm ePq (s)∞∏

n=1G

(s

an, g s

), (8.42)

where the sequence (an )n∈N of genus g s includes all nonzero roots of the function f (s), g s ≤ ρ, Pq (s) is apolynomial in s of degree q ≤ ρ, and m is the multiplicity of the root at the origin.

The product representation is useful to connect the growth of an entire function and the distribution of its zeros.

Theorem 8.3

The convergence exponent γ of the zero set of an entire function f (s) of non–integer order is equal to the orderof growth ρ of f (s).

For a proof, see, e.g., Levin (1996, Section 5.1). If the entire function is of integer order, then no such simple result isavailable since the order might be larger than the distribution of its zeros might indicate. As an example considerexp(s), which has no (finite) zeros but is of order 1. Let aρ denote the coefficient of sρ in the polynomial Pq (s) in theHadamard representation (8.42).

8.4.2 Application to the linear heat and wave equation with in–domain control

In the following, we consider the application of the spectral design approach for the linear heat and wave equation definedon the line z = [0,1]. In–domain control in terms of b(z1)u is assumed with the spatial characteristic

b(z) =σ(z −a)−σ(z −b)

for 0 < a < b < 1.

It is a straightforward task (co. Section 4.1) to determine the eigenvalues and eigenvectors of the respective (self–adjoint)operators A. For the heat equation, we obtain

λ(heat )k =−(kπ)2 φ(heat )

k =ψ(heat )k =p

2sin(kπz), k ∈N (8.43)

while for the wave equation eigenvalues and eigenvectors follow as

λ(w ave)k = ıkπ φ(w ave)

k =ψ(w ave)k =

1

λ(w ave)k

Fk sin(kπz) k ∈Z\ 0 (8.44)

with Fk = 1/(kπ). Both sets of eigenvectors φ(heat )k k∈N and φ(w ave)

k k∈Z\0 form orthonormal bases for the respective

spaces X (heat ) = L2(0,1) and X (w ave) = H10 (0,1)×L2(0,1) and hence Riesz bases. Thus the operators A(heat ) and A(w ave)

are Riesz spectral operators or scalar operators and the flatness–based state and input parametrizations can be directlyobtained from the results above.

8.4.2.1 Heat equation

The evaluation of (8.31) yields

Du (s) = ∏n∈N

(1− s

λ(heat )n

)= sinh(

ps)p

s(8.45)

Dxk (s) = ∏

n∈N,n 6=k

(1− s

λ(heat )n

)=−

λ(heat )k

s −λ(heat )k

sinh(p

s)ps

. (8.46)

Moreover, we have bk = ⟨b,φ(heat )k

⟩X (heat ) =

p2/(kπ)(cos(akπ)−cos(bkπ)) and we assume, that a, b are chosen so that

bk 6= 0 for all k ∈Nwhich guarantees the approximate controllability.


0 0.2 0.4 0.60

10

20

30

40

50

t

u∗ξ∗

00.5

1 00.2

0.40.6

0

0.2

0.4

0.6

0.8

1

tz1

x

x0

xT

Figure 8.7: Feedforward control u∗(t ) and basic output ξ∗(t ) (left); numerical solution of the heat equation with in–domaincontrol when applying u∗(t ) (right). ©2012, Springer

It can be rather easily verified, that Du (s) is an entire function of finite type and finite order ρ = 1/2. With Theorem 8.1 theconvergence of the formal parametrizations for the heat equation follows immediately provided, that ξ is a Gevrey classfunction of order α< 2. The input u(t ) = Du (∂t )ξ is determined using (8.45) and results in the series

u(t ) =∞∑

n=0

∂nt ξ(t )

(2n +1)!. (8.47)

The basic output is subsequently chosen to realize a finite time transition between steady states which are governed by

xs (us ) =(z1

∫ 1

0

∫ η

0b(ξ)dξdη−

∫ z1

0

∫ η

0b(ξ)dξdη

)us . (8.48)

Noting us = ξs in steady state conditions (see (8.45)) the desired trajectory ξ∗ for the basic output is assigned as

ξ∗ = ξs,0 + (ξs,T −ξs,0)Θω,T (t )

with Θω,T (t) from (8.19) for ω > 1. Substitution into (8.47) provides the feedforward control u∗(t) required to achievethe transition starting at the steady state xs (z;ξs,0) to the final steady state xs (z;ξs,T ) within the time interval t ∈ [0,T ].Consistency with the zero initial state implies ξs,0 = 0.

Simulation results are shown in Figure 8.7 for ξs,T = 19.5, T = 0.5, and ω= 2 and the spatial characteristic being restricted toz ∈ (1/2,3/4).

8.4.2.2 Wave equation

Evaluating (8.31) for the wave equation provides

Du (s) = ∏n∈Z\0

(1− s

λ(w ave)n

)= ∏

n∈N

(1+ s2

λ(w ave)n λ(w ave)

k

)= ∏

n∈N

(1+ s2

(kπ)2

)= sinh(s)

s(8.49)

Dxk (s) = ∏

n∈Z\0,n 6=k

(1− s

λ(w ave)n

)=−

λ(w ave)k

s −λ(w ave)k

sinh(s)

s. (8.50)

The following analysis holds true if bk = ∫ 10 b(z)ψ(w ave)

2,k c(z)dz = Fkλ(w ave)k /(kπ)(cos(akπ)− cos(bkπ)) 6= 0 for all k ∈ N

which implies the approximate controllability of the system.

We can rather easily verify, that Du (s) as introduced in (8.50) is entire, of finite type and of order ρ = 1. Convergenceaccording to Theorem 8.1 requires the basic output ξ to be of Gevrey order α≤ 1. However, as is shown in Section 8.3.1.2,this does not allow to realize finite time transitions between steady states. To resolve this issue note

Du (s)ξ= sinh(s)

sξ(s) = es −e−s

2sξ r b 1

2

(ξ(t +1)σ(t +1)− ξ(t −1)σ(t −1)

)= u, (8.51)

with5 ξ(t) = ∫ t0− ξ(p)dp. Obviously, the parametrization exactly recovers the wave dynamics with a finite speed of wave

propagation so that the feedforward control involves advanced and delayed arguments.

5Since a basic output is not necessarily unique either ξ(t ) or ξ(t ) can be considered here.


The wave dynamics also implies the existence of a minimal transition time, here Tmin = 2, which corresponds to twice thewave speed.

Since the operator (8.51) does not induce any differentiability requirement the realization of a finite time transition betweenan initial zero steady state and a final non–zero steady state can be achieved for the basic output trajectory being even adiscontinuous function of time. For simulation, we make use of

ξ∗ = ξs,0δ(t )+ (ξs,T −ξs,0)σ(t −1) (8.52)

or

ξ∗(t ) = ξs,0 + (ξs,T −ξs,0)(t −1)σ(t −1),

respectively. The shift t −1 is introduced for causality purposes to guarantee u∗(t ) = 0 for t < 0.

Simulation results are shown in Figure 8.8 for ξs,0 = 0, ξs,T = 19.5 and T = 2 with the spatial characteristic being restricted toz ∈ (1/2,3/4).

0 0.5 1 1.5 2 2.50

5

10

15

20

25

30

t

u∗

ξ∗

00.5

1 01

2

0

0.2

0.4

0.6

0.8

1

tz1

x

x0

xT

Figure 8.8: Feedforward control u∗(t ) and basic output ξ∗(t ) (left); numerical solution of the wave equation with in–domaincontrol when applying u∗(t ) (right). ©2012, Springer

The above methods can in principle be extended to nonlinear problems, as shown, e.g., in (Schörkhuber et al., 2012).

References

Boas, R. (1954). Entire functions. Academic Press, New York (cit. on p. 121).Curtain, R. F. and H. Zwart (1995). An introduction to infinite-dimensional linear systems theory. Springer

(cit. on p. 118).Dunbar, W., N. Petit, P. Rouchon, and P. Martin (2003). „Motion Planning for a nonlinear Stefan Problem“. In:

ESAIM Contr Optim Ca 9, pp. 275–296 (cit. on pp. 111, 115).Fliess, M., J. Lévine, P. Martin, and P. Rouchon (1995). „Flatness and defect of non–linear systems: introductory

theory and examples“. In: Int J Control 61, pp. 1327–1361 (cit. on p. 109).Fliess, M., P. Lévine, P. Martin, and P. Rouchon (1999). „A Lie-Bäcklund Approach to Equivalence and Flatness of

Nonlinear systems“. In: IEEE Trans Automatic Control 44(5), pp. 922–937 (cit. on p. 109).Fliess, M., H. Mounier, and P. Rouchon J. Rudolph (1998). „A distributed parameter approach to the control of a

tubular reactor: a multi–variable case“. In: Proc. 37th Conference on Decision and Control, Tampa, FL, USA,pp. 439–442 (cit. on p. 111).

Hua, C. and L. Rodino (1996). „General theory of PDE and Gevrey classes“. In: General Theory of PDEs andMicrolocal Analysis. Ed. by Q. Min-You and L. Rodino. Addison Wesley, pp. 6–81 (cit. on pp. 113, 115).

Isidori, A. (1995). Nonlinear Control Systems. Springer-Verlag, London (cit. on p. 110).John, F. (1982). Partial Differential Equations. 4th. Springer–Verlag, New York (cit. on p. 116).Laroche, B., P. Martin, and P. Rouchon (2000). „Motion planning for the heat equation“. In: Int. J. Robust

Nonlinear Control 10, pp. 629–643 (cit. on pp. 111, 115).Levin, B. (1996). Lectures on Entire Functions. American Mathematical Society, Providence, Rhode Island (cit. on

pp. 121, 122).Lévine, J. (2004). „On flatness necessary and sufficient conditions“. In: Proc. 6th IFAC Symposium ”Nonlinear

Control Systems” (NOLCOS 2004), Stuttgart (D), pp. 125–130 (cit. on p. 110).


Luo, Z.-H., B.-Z. Buo, and O. Morgul (1999). Stability and Stabilization of Infinite Dimensional Systems withApplications. Springer–Verlag, London (cit. on p. 118).

Lynch, A. F. and J. Rudolph (2002). „Flatness-based boundary control for a class of quasilinear parabolicdistributed parameter systems“. In: Int. J. Cont. 75 (15), pp. 1219–1230 (cit. on pp. 111, 115).

Martin, P., R. Murray, and P. Rouchon (1997). „Flat systems“. In: 4th European Control Conference (ECC),Brussels, Belgium, Plenary Lectures and Mini–Courses, pp. 211–264 (cit. on p. 109).

— (2000). „Flat Systems: Open Problems, Infinite Dimensional Extension, Symmetries and Catalog“. In: Ad-vances in the Control of Nonlinear Systems, LNCIS 264, pp. 33–57 (cit. on p. 109).

Meurer, T. (2005). Feedforward and Feedback Tracking Control of Diffusion–Convection–Reaction Systemsusing Summability Methods. Vol. 1081. Fortschr.–Ber. VDI Reihe 8. VDI Verlag, Düsseldorf (cit. on p. 111).

— (2011). „Flatness–based Trajectory Planning for Diffusion–Reaction Systems in a Parallelepipedon — ASpectral Approach“. In: Automatica 47(5), pp. 935–949 (cit. on p. 111).

— (2013). Control of Higher Dimensional PDEs. Communication and Control Engineering. Springer (cit. onpp. 111, 115, 119, 120).

Meurer, T. and A. Kugi (2009). „Trajectory Planning for Boundary Controlled Parabolic PDEs With VaryingParameters on Higher-Dimensional Spatial Domains“. In: IEEE Trans. Autom. Control 54 (8), pp. 1854–1868(cit. on p. 111).

Meurer, T. and M. Zeitz (2005). „Feedforward and feedback tracking control of nonlinear diffusion–convection–reactionsystems using summability methods“. In: Ind Eng Chem Res 44, pp. 2532–2548 (cit. on p. 111).

— (2008). „Model inversion of boundary controlled parabolic partial differential equations using summabilitymethods“. In: Math Comp Model Dyn Sys (MCMDS) 14(3), pp. 213–230 (cit. on p. 111).

Nieuwstadt, M. van, M. Rathinam, and R. Murray (1998). „Differential Flatness and Absolute Equivalence ofNonlinear Control Systems“. In: SIAM J Control Optim 36(4), pp. 1225–1239 (cit. on p. 109).

Nijmeier, H. and A. van der Schaft (1990). Nonlinear dynamical control systems. Springer Verlag, New York(cit. on p. 110).

Petit, N. and P. Rouchon (2001). „Flatness of Heavy Chain Systems“. In: SIAM J Control Optim 40(2), pp. 475–495(cit. on p. 111).

— (2002). „IEEE Trans Automatic Control“. In: 47(4), pp. 594–609 (cit. on p. 111).Rothfuß, R. (1997). Anwendung der flachheitsbasierten Analyse und Regelung nichtlinearer Mehrgrößensysteme.

Vol. 664. Fortschr.–Ber. VDI Reihe 8. VDI Verlag, Düsseldorf (cit. on p. 109).Rothfuß, R., J. Rudolph, and M. Zeitz (1996). „Flatness–based control of a nonlinear chemical reactor model“.

In: Automatica 32, pp. 1433–1439 (cit. on p. 109).Rouchon, P. (2005). „Flatness–based control of oscillators“. In: Z Angew Math Mech 85(6), pp. 411–421 (cit. on

p. 111).— (n.d.). „Motion planning, equivalence, and infinite dimensional systems“. In: Int J Appl Math Comp Sc 11 (),

pp. 165–188 (cit. on p. 111).Rudolph, J. (2003). Flatness Based Control of Distributed Parameter Systems (cit. on pp. 109, 111).Rudolph, J., J. Winkler, and F. Woittennek (2005). „Flatness Based Approach to a Heat Conduction Problem in a

Crystal Growth Process“. In: Control and Observer Design for Nonlinear Finite and Infinite DimensionalSystems, LNCIS. Ed. by et al Meurer T. 322. Springer–Verlag, Berlin, pp. 387–401 (cit. on p. 111).

Schlacher, K. and M. Schöberl (2007). „Construction of flat outputs by reduction and elimination“. In: Proc. 7thIFAC Symposium ”Nonlinear Control Systems” (NOLCOS 2007), Pretoria (SA), pp. 666–671 (cit. on p. 110).

Schöberl, M. and K. Schlacher (2014). „On an implicit triangular decomposition of nonlinear control systemsthat are 1–flat — a constructive approach“. In: Automatica 50(6), pp. 1649–1655 (cit. on p. 110).

Schörkhuber, B., T. Meurer, and A. Jüngel (2012). „Flatness–based trajectory planning for semilinear parabolicPDEs“. In: Proc. IEEE Conference on Decision and Control (CDC), Maui (HI), USA, pp. 3538–3543 (cit. onp. 124).

— (2013). „Flatness of Semilinear Parabolic PDEs—A Generalized Cauchy–Kowalevski Approach“. In: IEEETrans. Autom. Control 58 (9), pp. 2277–2291 (cit. on p. 111).

Sira-Ramirez, H. and S. Agrawal (2004). „Differentially Flat Systems“. In: (cit. on p. 109).Utz, T., T. Meurer, and A. Kugi (2010). „Trajectory planning for quasilinear parabolic distributed parameter

systems based on finite-difference semi-discretizations“. In: Int. J. Cont. 83 (6), pp. 1093–1106 (cit. on p. 111).Wagner, M., T. Meurer, and A. Kugi (2008). „Feedforward control design for the inviscid Burger equation using

formal power series and summation methods“. In: Proc. 17th IFAC World Congress, Seoul (KR), pp. 8743–8748 (cit. on p. 111).

Whitham, G. (1999). Linear and Nonlinear Waves. John Wiley & Sons, New York (cit. on p. 116).Woittennek, F. (2007). Beiträge zum Steuerungsentwurf für lineare, örtlich verteilte Systeme mit konzentrierten

Stelleingriffen. Berichte aus der Steuerungs– und Regelungstechnik. Shaker–Verlag, Aachen (cit. on p. 111).

REFERENCES 125


Appendix A

Basic results from functional analysis

In the following some basid results from functional analysis and operator theory are summarized that are important for thestudy of PDE control systems. In these notes these results are overall related to the considerations in Chapters 4 and 5.

A.1 The Hille–Yosida theorem

The Hille–Yosida theorem enables a characterization of the infinitesimal generators of C0–semigroups (curtain_zwart:95;Engel and R.Nagel, 2006; Jüngel, 2001). With Definition 4.8 the following relation between the resolvent and the growthbound of a semigroup can be established.

Theorem 1.1

Let T (t) be a C0–semigroup with infinitesimal generator A and growth bound ω. Let ℜλ >ω1 >ω. It follows thatλ ∈ ρ(A) and it holds true that ∀x ∈X ,

R(λ, A)x = (λI − A)−1x =∫ ∞

0e−λt T (t )xd t and ‖R(λ, A)‖ ≤ M

ℜλ−ω1.

The proof of this theorem can be found, e.g., in (Curtain and Zwart, 1995). With these preliminaries the Hille–Yosidatheorem for the characterization of a generator A can be formulated (Curtain and Zwart, 1995).

Theorem 1.2: Hille–Yosida theorem

A linear operator A is the infinitesimal generator of a C0–semigroup T ( · ) with ‖T (t )‖ ≤ Meωt , M ,ω ∈R if and only if

a) A is closed with D(A) =X and

b) for all λ>ω ∈∈Rwith λ>ω with λ ∈ ρ(A) it holds true that ‖R(λ, A)r ‖ ≤ M(λ−ω)r , ∀r ∈N.

It should be noted that the Hille-Yosida theorem exists in different (equivalent) formulations, where in particular condition(ii) allows different representations (see, e.g., (Pazy, 1983)).

A.2 The Lumer–Phillips theorem

Besides the Hille–Yosida theorem, the Lumer–Phillips theorem allows an alternative characterizatioin of infinitesimalgenerators of a contracting C0–semigroup. For this purpose the notion of a dissipative operator is introduced (see, e.g.,(cazenave_haraux:98; liu:99)).

Definition 1.1: Dissipative operator

Let X be a real or complex Hilbert space with inner product ⟨ · , ·⟩ and induced norm ‖ ·‖X . Let A be a linear operator

127

with D(A) dense in X . Then the operator A is called dissipative, if for all x ∈D(A) it holds that

ℜ⟨Ax , x⟩ ≤ 0. (A.1)

An alternative option definition of dissipative operators requires the introduction of the concept of the dual space.

Definition 1.2: Linear mapping, linear functional, dual space

Let (X ,‖ ·‖X ) and (Y ,‖ ·‖Y ) be half–normed spaces over a field K (e.g., R or C). A mapping Abb : X −→ Y is calledlinear if

∀α,β ∈K ∀x , y(t ) ∈X : Abb(αx +βy(t )) =αAbb(x)+βAbb(y(t )).

A linear mapping is called an operator, or in the case of Y = K also a linear functional. In the following L (X ,Y )denotes the set of all linear continous mappings from X to Y . The set X ′ of all linear continuous functionals L (X ,K)from X to the fieldK is called the dual space.

Let x ′ ∈X ′. For any y(t) ∈X there exists exactly one x ∈X , such that x ′(y(t)) = ⟨x , y(t)⟩. Here x ′(y(t)) is called the dualproduct. In particular it holds that x ′(x) = ⟨x , x⟩ = ‖x‖2. With these preliminaries the notion of dissipative operator can bedefined (cp. (Pazy, 1983)).

Definition 1.3: Dissipative operator

A linear operator A is called dissipative, if for any x ∈D(A) there exists an x ′ ∈ F (x) = x ′| x ′ ∈X ′ with x ′(x) = ‖x‖2 =‖x ′‖2 ⊆X ′, such that ℜx ′(Ax) ≤ 0.

With this notion the following Theorem for the characterization of dissipative operators can be formulated (Pazy, 1983).

Theorem 1.3

A linearer operator A is dissipative, if and only if

‖(λI − A)x‖ ≥λ‖x‖ for all x ∈D(A) and λ> 0.

This result offers an inmediate approach for analyzing the dissipativity of an operator by direct evaluation of the condition(A.1). The relation between the dissipativity of an operator and the existence of an associated semigroup of contractions isestablished in the theorem of Lumer–Phillips (Pazy, 1983).

Theorem 1.4: Lumer–Phillips theorem

Let A be a linear operator with domain D(A) that is dense in X . It holds that:

a) 1If A is dissipative and there exist a λ0 > 0, so that the range of λ0I −A is identical to X , then A is the infinitesimalgenerator of a C0–semigroup of contractions on X .

b) If A is the infinitesimal generator of a C0–semigroup of contraction on X , then the range of λI − A is identical toX for all λ> 0 and A is dissipative. In addition, for x ∈D(A) and all x ′ ∈ F (x) it holds true that that ℜx ′(Ax) ≤ 0.

References

Curtain, R. F. and H. Zwart (1995). An introduction to infinite-dimensional linear systems theory. Springer(cit. on p. 127).

Engel, K.J. and R.Nagel (2006). A short course on operator semigroups. Springer-Verlag New York (cit. on p. 127).Jüngel, A. (2001). Eine Einführung in die Halbgruppentheorie. Vorlesungsskript, Univ. Konstanz (cit. on p. 127).Pazy, A. (1983). Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer,

New York (cit. on pp. 127, 128).

1Under the conditions of the theorem, (i) is equivalent to the requirement that A is dissipative and 0 ∈ ρ(A) according to (liu:99), i.e.,A−1 exists and is bounded. In applications this condition can be evaluated by a direct calculation of the inverse.

128 Chapter A Basic results from functional analysis

Appendix B

Properties and correspondences of theLaplace transform

The following list summarizes the most important properties and rules for the Laplace–transform:

Laplace transform L f (t ) = f (s) = ∫ ∞0 f (t )e−st dt (ℜs > γ)

Inverse transform L−1 f (s) = f (t )|t>0 = 1

2πi

∫ c+i∞

c−i∞f (s)est ds (c > γ)

Linearity L k1 f1(t )+k2 f2(t ) = k1L f1(t )+k2L f2(t ), (k1,2 arbitrary)L−1k1F1(s)+k2F2(s) = k1L−1F1(s)+k2L−1F2(s)

Scaling L f (at ) = 1

af( s

a

)(a ∈R+)

Time shift L σ(t −b) f (t −b) = e−bs f (s) (b ∈R+)

Frequency shift L−1 f (s ± c) = e∓ct f (t ) (c ∈C)

Integration L ∫ t

0 . . .∫ t

0 f (τ)dτn = 1

sn f (s)

Ordinarydifferentiation

L f (n)(t ) = sn f (s)− sn−1 f (0+)− . . .− f (n−1)(0+)

Generalizeddifferentiation

L Dn f (t ) = sn f (s)− sn−1 f (0−)− . . .− f (n−1)(0−)

Differentiation ofthe image

L−1F (n)(s) = (−1)n t n f (t )

Convolution L−1 f1(s) f2(s) = ∫ t0 f1(t −τ) f2(τ)dτ= f1(t )? f2(t )

Limit value theorems • Initial–value theorem

limσ→∞ s f (s) = lim

t→0+ f (t )

• Final value theorem, condition: ∃ f (t →∞)

lims→0

s f (s) = limt→∞ f (t )

Table B.1: Properties and rules for the Laplace transform.

In addition, table B.2 provides some important correspondences of the Laplace transform, which are important for differentapplications, in particular for solving differential equations. Additional correspondences, which are in particular moredirectly related to transcedental functions can be found, e.g., in (Roberts and Kaufman, 1966) and references therein.

129

Nr. f (t ) = 12π j

∫ c+ j∞c− j∞ f (s)est ds, t > 0, c ≥ γ f (s) = ∫ ∞

0 f (t )e−st dt , ℜs =σ> γ

1 δ(t ) 1

2 σ(t ) und 11

s

3 t n , n = 1,2,3, . . .n!

sn+1

4 t n e−at n!

(s +a)n+1

5 cosω0ts

s2 +ω20

6 sinω0tω0

s2 +ω20

7 e−at cosω0ts +a

(s +a)2 +ω20

8 e−at sinω0tω0

(s +a)2 +ω20

9 t cosω0ts2 −ω2

0

(s2 +ω20)2

10 t sinω0t2ω0s

(s2 +ω20)2

11 1−e−at a

s(s +a)

12 e−at +at −1a2

s2(s +a)

Table B.2: Correspondences of the Laplace transform.

References

Roberts, G. and H. Kaufman (1966). Table of Laplace Transforms. W.B. Saunders Company (cit. on p. 129).

130 Chapter B Properties and correspondences of the Laplace transform

Documents

Control of PDE systems - Uni Kiel