An Algorithm for Constructing Lyapunov Fn

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Electronic Journal of Differential Equations , Monograph 08, 2007, (101 pages).ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.eduftp ejde.math.txstate.edu (login: ftp)

AN ALGORITHM FOR CONSTRUCTINGLYAPUNOV FUNCTIONS

SIGURDUR FREYR HAFSTEIN

Abstract. In this monograph we develop an algorithm for constructing Lya-punov functions for arbitrary switched dynamical systems x = f (t, x ), pos-sessing a uniformly asymptotically stable equilibrium. Let x = f p ( t, x ), p P ,be the collection of the ODEs, to which the switched system corresponds. Thenumber of the vector elds f p on the right-hand side of the differential equa-tion is assumed to be nite and we assume that their components f p,i are C

2

functions and that we can give some bounds, not necessarily close, on theirsecond-order partial derivatives. The inputs of the algorithm are solely a nitenumber of the function values of the vector elds f p and these bounds. Thedomain of the Lyapunov function constructed by the algorithm is only limitedby the size of the equilibriums region of attraction. Note, that the concept of aLyapunov function for the arbitrary switched system x = f ( t, x ) is equivalentto the concept of a common Lyapunov function for the systems x = f p (t, x ), p P , and that if P contains exactly one element, then the switched systemis just a usual ODE x = f (t, x ). We give numerous examples of Lyapunovfunctions constructed by our method at the end of this monograph.

Contents

1. Introduction 22. Outline and Categorization 43. Preliminaries 83.1. Continuous dynamical systems 83.2. Arbitrary switched systems 93.3. Dini derivatives 123.4. Stability of arbitrary switched systems 143.5. Three useful lemmas 173.6. Linear programming 194. Lyapunovs Direct Method for Switched Systems 205. Converse Theorem for Switched Systems 24

5.1. Converse theorems 245.2. A converse theorem for arbitrary switched systems 256. Construction of Lyapunov Functions 33

2000 Mathematics Subject Classication. 35J20, 35J25.Key words and phrases. Lyapunov functions; switched systems; converse theorem;piecewise affine functions.c 2007 Texas State University - San Marcos.

Submitted August 29, 2006. Published August 15, 2007.1


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2 S. F. HAFSTEIN EJDE-2007/MON. 08

6.1. Continuous piecewise affine functions 346.2. The linear programming problem 396.3. Denition of the functions ,, and V Lya 446.4. Implications of the constraints (LC1) 446.5. Implications of the constraints (LC2) 456.6. Implications of the constraints (LC3) 466.7. Implications of the constraints (LC4) 466.8. Summary of the results and their consequences 486.9. The autonomous case 527. Constructive Converse Theorems 647.1. The assumptions 647.2. The assignments 657.3. The constraints (LC1) are fullled 667.4. The constraints (LC2) are fullled 677.5. The constraints (LC3) are fullled 68

7.6. The constraints (LC4) are fullled 687.7. Summary of the results 707.8. The autonomous case 718. An Algorithm for Constructing Lyapunov Functions 748.1. The algorithm in the nonautonomous case 748.2. The algorithm in the autonomous case 769. Examples of Lyapunov functions generated by linear programming 789.1. An autonomous system 789.2. An arbitrary switched autonomous system 809.3. A variable structure system 849.4. A variable structure system with sliding modes 889.5. A one-dimensional nonautonomous switched system 919.6. A two-dimensional nonautonomous switched system 93

10. Conclusion 96List of Symbols 98References 99

1. Introduction

Let P be a nonempty set and equip it with the discrete metric, let U R n bea domain containing the origin, and let be a norm on R n . For every p P assume that f p : R 0 U R n satises the local Lipschitz condition: for everycompact C

R 0

U there is a constant L p,C such that ( t, x ), (t, y )

C implies

f p(t, x) f p(t, y ) L p,C x y . Dene B ,R := {x R n : x < R } for everyR > 0. We consider the switched system x = f (t, x), where is an arbitrary right-continuous mapping R 0 P of which the discontinuity-points form a discrete set.In this monograph we establish the claims made in the abstract in the followingthree steps:

First, we show that the origin is a uniformly asymptotically stable equilibriumof the arbitrary switched system x = f (t, x), whenever there exists a commonLyapunov function for the systems x = f p(t, x), p P , and we show how to derive


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EJDE-2007/MON. 08 LYAPUNOV FUNCTIONS 3

a lower bound on the equilibriums region of attraction from such a Lyapunovfunction.

Second, we show that if

B ,R U is a subset of the region of attraction of the

arbitrary switched system x = f (t, x) and the vector elds f p, p P , satisfy theLipschitz condition: there exists a constant L such that for every p P and every(s, x), (t, y ) R 0 B ,R the inequality f p(s, x) f p(t, y) L(|s t|+ x y )holds; then for every 0 < R < R , there exists a common Lyapunov functionV : R 0 B ,R R for the systems x = f p(t, x), p P .Third, assuming that the set P is nite and that the second-order partial deriva-tives of the components of the vector elds f p are bounded, we write down a linearprogramming problem using the function values of the vector elds f p on a dis-crete set and bounds on the second-order partial derivatives of their components.Then we show how to parameterize a common Lyapunov function for the systemsx = f p(t, x), p P , from a feasible solution to this linear programming problem.We then use these results to give an algorithm for constructing such a commonLyapunov function for the systems x = f p(t, x), p P , and we prove that it alwayssucceeds in a nite number of steps if there exists a common Lyapunov functionfor the systems at all.

Let us be more specic on this last point. Consider the systems x = f p(t, x), p P , and assume that they possess a common Lyapunov function W : R 0 V R ,where V U is a domain containing the origin. That is, there exist functions , ,and of class K such that

( x ) W (t, x) ( x )and

x W (t, x) f p(t, x) + W

t (t, x) ( x )

for every p P , every t R 0 and every x V . The second inequality canequivalently be formulated aslimsup

h 0

W (t + h, x + hf p(t, x)) W (t, x)h ( x ).

Now, let t1 , t 2 R be arbitrary constants such that 0 t 1 < t 2 < + and let Cand D be arbitrary compact subsets of R n of positive measure such that the originis in the interior of D and D C V . We will prove that the algorithm will alwayssucceed in generating a continuous function V : [t1 , t 2]C R with the property,that there exist functions , , and of class K, such that( x ) V (t, x) ( x )

for all t [t1 , t 2] and all x C andlimsup

h 0

V (t + h, x + hf p(t, x)) V (t, x)h

( x )

for every p P , every t [t1 , t 2], and every x C\D. Note, that the last inequalityis not necessarily valid for x D, but because one can take D as small as one wishes,this is nonessential.It is reasonable to consider the autonomous case separately, because then there

exist common autonomous Lyapunov functions whenever the origin is an asymptot-ically stable equilibrium, and the algorithm is then also able to construct commonautonomous Lyapunov functions for the systems x = f p(x), p P .


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2. Outline and Categorization

This monograph is thematically divided into three parts. In the rst part,

which consists of the sections 3, 4, and 5, we develop a stability theory for ar-bitrary switched systems. In Section 3 we introduce switched dynamical systemsx = f (t, x) and discuss some elementary properties of their solutions. We furtherconsider the stability of isolated equilibria of arbitrary switched systems and weprove in Section 4 that if such a system possesses a Lyapunov function, then theequilibrium is uniformly asymptotically stable. These results are quite straightfor-ward if one is familiar with the Lyapunov stability theory for ordinary differentialequations (ODEs), but, because we consider Lyapunov functions that are merelycontinuous and not necessarily differentiable in this work, we establish the mostimportant results. Switched systems have gained much interest recently. For anoverview see, for example, [ 33] and [31].

In Section 5 we prove a converse theorem on uniform asymptotic stability forarbitrary switched nonlinear, nonautonomous, continuous systems. In the litera-ture there are numerous results regarding the existence of Lyapunov functions forswitched systems. A short non-exhaustive overview follows: In [48] Narendra andBalakrishnan consider the problem of common quadratic Lyapunov functions fora set of autonomous linear systems, in [15], in [46], in [32], and in [2] the resultswere considerably improved by Gurvits; Mori, Mori, and Kuroe; Liberzon, Hes-panha, and Morse; and Agrachev and Liberzon respectively. Shim, Noh, and Seoin [57] and Vu and Liberzon in [ 61] generalized the approach to commuting au-tonomous nonlinear systems. The resulting Lyapunov function is not necessarilyquadratic. Dayawansa and Martin proved in [41] that a set of linear autonomoussystems possesses a common Lyapunov function, whenever the corresponding arbi-trary switched system is asymptotically stable, and they proved that even in thissimple case there might not exist any quadratic Lyapunov function. The sameauthors generalized their approach to exponentially stable nonlinear, autonomoussystems in [ 7]. Mancilla-Aguilar and Garca used results from Lin, Sontag, andWang in [ 35] to prove a converse Lyapunov theorem on asymptotically stable non-linear, autonomous switched systems in [ 38].

In this work we prove a converse Lyapunov theorem for uniformly asymptoticallystable nonlinear switched systems and we allow the systems to depend explicitlyon the time t, that is, we work the nonautonomous case out. We proceed asfollows: Assume that the functions f p, of the arbitrary switched system x = f (t, x), : R 0 P , satisfy the Lipschitz condition: there exists a constant L suchthat f p(t, x) f p(t, y) 2 L x y 2 for all p P , all t 0, and all x, yin some compact neighborhood of the origin. Then, by combining a Lyapunovfunction construction method by Massera for ODEs, see, for example, [42] or Section5.7 in [60], with the construction method presented by Dayawansa and Martin in

[7], it is possible to construct a Lyapunov function V for the system. However,we need the Lyapunov function to be smooth, so we prove that if the functionsf p, p P , satisfy the Lipschitz condition: there exists a constant L such thatf p(t, x) f p(s, y) 2 L(|t s| + x y 2) for all p P , all s, t 0, and allx , y in some compact neighborhood of the origin in the state-space, then we cansmooth out the Lyapunov function to be innitely differentiable except possibly atthe origin.


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separately, because in this case it is possible to parameterize an autonomous Lya-punov function with a more simple linear programming problem, which is denedin Denition 6.12. These results are generalizations of former results by the author,presented in [39, 40, 17, 16, 18].

In Section 7 we prove, that if we construct a linear programming problem asin Denition 6.8 for a switched system that possesses a uniformly asymptoticallystable equilibrium, then, if the boundary conguration of the function space CPWAis sufficiently closely meshed, there exist feasible solutions to the linear program-ming problem. There are algorithms, for example the simplex algorithm, thatalways nd a feasible solution to a linear programming problem, provided thereexists at least one. This implies that we have reduced the problem of constructinga Lyapunov function for the arbitrary switched system to a more simple prob-lem of choosing an appropriate boundary conguration for the CPWA space. If the systems x = f p(t, x), p P , are autonomous and exponentially stable, thenit was proved in [ 17] that it is even possible to calculate the mesh-sizes directlyfrom the original data, that is, the functions f p. This, however, is much more re-strictive than necessary, because a systematic scan of boundary congurations isconsiderably more effective, will lead to success for merely uniformly asymptoticallystable nonautonomous systems, and delivers a boundary conguration that is morecoarsely meshed. Just as in Section 5 we consider the more simple autonomouscase separately.

In Section 8 we use the results from Section 7 to dene our algorithm in Procedure8.1 to construct Lyapunov functions and we prove in Theorem 8.2 that it alwaysdelivers a Lyapunov function, if the arbitrary switched system possesses a uniformlyasymptotically stable equilibrium. For autonomous systems we do the same inProcedure 8.3 and Theorem 8.4. These procedures and theorems are generalizationsof results presented by the author in [17, 16, 18].

In the last decades there have been several proposals of how to numerically con-

struct Lyapunov functions. For comparison to the construction method presentedin this work some of these are listed below. This list is by no means exhaustive.

In [59] Vandenberghe and Boyd present an interior-point algorithm for construct-ing a common quadratic Lyapunov function for a nite set of autonomous linearsystems and in [34] Liberzon and Tempo took a somewhat different approach todo the same and introduced a gradient decreasing algorithm. Booth methods arenumerically efficient, but unfortunately, limited by the fact that there might existLyapunov functions for the system, non of which is quadratic. In [23], [24], and[22] Johansson and Rantzer proposed construction methods for piecewise quadraticLyapunov functions for piecewise affine autonomous systems. Their constructionscheme is based on continuity matrices for the partition of the respective state-space. The generation of these continuity matrices remains, to the best knowledgeof the author, an open problem. Further, piecewise quadratic Lyapunov functionsseem improper for the following reason:

Let V be a Lyapunov function for some autonomous system. Ex-panding in power series about some y in the state-space gives

V (x) V (y ) + V (y ) (x y) + 12

(x y)T H V (y )(x y),where H V (y ) is the Hessian matrix of V at y, as a second-orderapproximation. Now, if y is an equilibrium of the system, then


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necessarily V (y ) = 0 and V (y ) = 0 and the second-order approx-imation simplies toV (x) 12(x y)T H V (y )(x y),

which renders it very reasonable to make the Lyapunov functionansatz

W (x) = ( x y)T A(x y)for some square matrix A and then try to determine a suitablematrix A. This will, if the equilibrium is exponentially stable, de-liver a function that is a Lyapunov function for the system in some(possibly small) vicinity of the equilibrium.

However, if y is not an equilibrium of the system, then V (y ) = 0and V (y ) = 0 and using several second-order power series expan-sions about different points in the state-space that are not equilib-ria, each one supposed to be valid on some subset of the state-space,becomes problematic. Either one has to try to glue the local approx-imations together in a continuous fashion, which causes problemsbecause the result is in general not a Lyapunov function for thesystem, or one has to consider non-continuous Lyapunov functions,which causes even more problems.

Brayton and Tong in [ 4, 5], Ohta, Imanishi, Gong, and Haneda in [49], Michel,Sarabudla, and Miller in [45], and Michel, Nam, and Vittal in [44] reduced the Lya-punov function construction for a set of autonomous linear systems to the designof a balanced polytope fullling certain invariance properties. Polanski in [ 52] andKoutsoukos and Antsaklis in [30] consider the construction of a Lyapunov functionof the form V (x) := W x , where W is a matrix, for autonomous linear systemsby linear programming. Julian, Guivant, and Desages in [27] and Julian in [25]presented a linear programming problem to construct piecewise affine Lyapunovfunctions for autonomous piecewise affine systems. This method can be used forautonomous, nonlinear systems if some a posteriori analysis of the generated Lya-punov function is done. The difference between this method and our (non-switchedand autonomous) method is described in Section 6.2 in [ 40]. In [21] Johansen useslinear programming to parameterize Lyapunov functions for autonomous nonlin-ear systems. His results are, however, only valid within an approximation error,which is difficult to determine. P. Parrilo in [51] and Papachristodoulou and Prajnain [50] consider the numerical construction of Lyapunov functions that are pre-sentable as sums of squares for autonomous polynomial systems under polynomialconstraints. Recently, Giesl proposed in [ 11, 12, 13, 14] a method to constructLyapunov functions for autonomous systems with exponentially stable equilibriumby solving numerically a generalized Zubov equation (see, for example, [68] and foran extension to perturbed systems [ 6])

V (x) f (x) = p(x), (2.1)where usually p(x) = x 2 . A solution to the partial differential equation ( 2.1)is a Lyapunov function for the system. He uses radial basis functions to nd anumerical approximation to the solution of ( 2.1).

In the third part of this monograph in Section 9 we give several examples of Lya-punov functions generated by the linear programming problems from Denition 6.8


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for every t0 R 0 and every U (see, for example, Theorems VI and IX inIII.10 in [63]) and we denote this solution by t (t, t 0 , ) and we say that isthe solution to the state equation ( 3.1).For this reason we will, in this monograph, only consider continuous dynamical

systems, of which the dynamics are modelled by an ODE

x = f (t, x), (3.3)

where f : R 0 U R n satises a local Lipschitz condition as in ( 3.2).Two well known facts about ODEs that we will need in this monograph are givenin the next two theorems:

Theorem 3.1. Let U R n be a domain, f : R 0 U R n satisfy a local Lipschitz condition as in ( 3.2 ), and be the solution to the state equation x = f (t, x). Let m N 0 and assume that f [Cm (R 0 U )]n , that is, every component f i of f is in Cm (R 0 U ), then , [Cm (dom( ))]n .Proof. See, for example, the Corollary at the end of III.

13 in [63].

Theorem 3.2. Let I R be a nonempty interval, be a norm on R n , and let U be a domain in R n . Let f , g : I U R n be continuous mappings and assume that there exists a constant L R such that f satises the Lipschitz condition: there exists a constant L such that

f (t, x) f (t, y ) L x y for all t I and all x , y U .Let t0 I and let , U and denote the (unique) global solution to the initial value problem

x = f (t, x ), x(t0) := ,by y : I y R n and let z : I z R n be any solution to the initial value problem

x = g(t, x), x(t0) = .

Set

J :=

I y I z and let and be constants, such that and f (t, z(t)) g(t, z(t))

for all t J . Then the inequality y (t) z(t) eL | t t 0 | +

L

(eL | t t 0 | 1)holds for all t J .Proof. See, for example, Theorem III. 12.V in [63]. 3.2. Arbitrary switched systems. A switched system is basically a family of dynamical systems and a switching signal, where the switching signal determineswhich system in the family describes the dynamics at what times or states. As weare concerned with the stability of switched systems under arbitrary switchings,

the following denition of a switching signal is sufficient for our needs.Denition 3.3 (Switching signal) . Let P be a nonempty set and equip it withthe discrete metric d( p, q ) := 1 if p = q . A switching signal : R 0 P is aright-continuous function, of which the discontinuity-points are a discrete subset of R 0 . The discontinuity-points are called switching times. For technical reasons itis convenient to count zero with the switching times, so we agree upon that zero isa switching time as well. We denote the set of all switching signals R 0 P byS P .


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With the help of the switching signal in the last denition we can dene theconcept of a switched system and its solution.

Denition 3.4 (Solution to a switched system) . Let U R n be a domain, let P be a nonempty set, and let f p : R 0 U R n , p P , be a family of mappings,of which each f p, p P , satises a local Lipschitz condition as in ( 3.2). For everyswitching signal S P we dene the solution t (t,s, ) to the initial valueproblemx = f (t, x), x(s) = , (3.4)

in the following way:Denote by t0 , t 1 , t 2 , . . . the switching times of in an increasing order. If there

is a largest switching time tk we set tk+1 := + and if there is no switchingtime besides zero we set t1 := + . Let s R 0 and let k N 0 be such thattk s < t k +1 . Then is dened by gluing together the trajectories of the systems

x = f p(t, x), p P ,using p := (s) between s and tk+1 , p := (tk +1 ) between tk+1 and tk+2 , and ingeneral p := (t i ) between ti and ti +1 , i k + 1. Mathematically this can beexpressed inductively as follows:

Forward solution:(i) (s,s, ) = for all s R 0 and all U .(ii) Denote by y the solution to the initial value problem

x = f (s ) (t, x), x(s) = ,

on the interval [ s, t k+1 [, where k N 0 is such that tk s < t k+1 . Thenwe dene (t,s, ) on the domain of t y(t) by (t,s, ) := y(t).If the closure of graph( y ) is a compact subset of [ s, t k +1 ] U , then thelimit lim t t k +1 y(t) exists and is in U and we dene (tk+1 , s, ) :=limt t k +1 y(t).

(iii) Assume (t i , s, ) U is dened for some integer i k + 1 and denote byy the solution to the initial value problemx = f ( t i ) (t, x), x(t i ) = (t i , s, ),

on the interval [ t i , t i +1 [ . Then we dene (t,s, ) on the domain of t y(t) by (t,s, ) := y(t). If the closure of graph( y ) is a compact subset of [t i , t i +1 ]U , then the limit lim t t i +1 y(t) exists and is in U and we dene (t i +1 , s, ) := lim t t i +1 y(t).

Backward solution:(i) (s,s, ) = for all s R 0 and all U .

(ii) Denote by y the solution to the initial value problemx = f ( t k ) (t, x), x(s) = ,

on the interval ] tk , s ], where k N 0 is such that tk s < t k+1 . Thenwe dene (t,s, ) on the domain of t y(t) by (t,s, ) := y(t). If the closure of graph( y ) is not empty and a compact subset of [ tk , s ] U ,then the limit lim t t k + y(t) exists and is in U and we dene (tk , s, ) :=limt t k + y(t).


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(iii) Assume (t i , s, ) U is dened for some integer i, 0 < i k and denoteby y the solution to the initial value problemx = f ( t i ) (t, x), x(t i ) = (t i , s, ),

on the interval ] t i 1 , t i ]. Then we dene (t,s, ) on the domain of t y(t) by (t,s, ) := y(t). If the closure of graph( y ) is a compact subset of [t i 1 , t i ]U , then the limit lim t t i 1 + y(t) exists and is in U and we dene (t i 1 , s, ) := lim t t i 1 + y(t).

Thus, for every S P we have dened the solution to the differential equationx = f (t, x).

Note, just as one usually suppresses the time dependency of x in (3.1), thatis, writes x = f (t, x) instead of x (t) = f (t, x (t)), one usually suppresses the timedependency of the switching signal too, that is, writes x = f (t, x) instead of x (t) = f ( t ) (t, x(t)).

Now, that we have dened the solution to ( 3.4) for every S P , we can denethe switched system and its solution.Switched System 3.5. Let U R n be a domain, let P be a nonempty set, and let f p : R 0 U R n , p P , be a family of mappings, of which each f p, p P ,satises a local Lipschitz condition as in ( 3.2 ).

The arbitrary switched system

x = f (t, x), S P ,is simply the collection of all the differential equations

{x = f (t, x) : S P },whose solutions we dened in Denition 3.4. The solution to the arbitrary switched system is the collection of all the solutions to the individual switched systems.

Because the trajectories of the Switched System 3.5 are dened by gluing togethertrajectory-pieces of the corresponding continuous systems, they inherit the followingimportant property: For every S P , every s R 0 , and every U the closureof the graph of t (t,s, ), t s, is not a compact subset of R 0 U and theclosure of the graph of t (t,s, ), t s, is not a compact subset of R > 0 U .Further, note that if , S P , = , then in general (t, t 0 , ) is not equal to (t, t 0 , ) and that if the Switched System 3.5 is autonomous, that is, none of thevector elds f p, p P , does depend on the time t , then

(t, t , x ) = (t t , 0, ), where (s) := (s + t ) for all s 0,for all t t 0 and all U . Therefore, we often suppress the middle argumentof the solution to an autonomous system and simply write (t, ).

We later need the following generalization of Theorem 3.2 to switched systems.

Theorem 3.6. Consider the Switched System 3.5, let be a norm on R n , and assume that the functions f p satisfy the Lipschitz condition: there exists a constant L such that

f p(t, x) f p(t, y) L x y


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for all t 0, all x, y U , and all p P . Let t0 0, let , U , let , S P ,and assume there is a constant 0 such that f ( t ) (t, x) f ( t ) (t, x)

for all t 0 and all x U .Denote the solution to the initial value problem x = f (t, x), x(s0) = ,

by y : I y R n and denote the solution to the initial value problem x = f (t, x), x(s0) = ,

by z : I z R n . Set J := I y I z and set := . Then the inequality y(t) z(t) eL | t s 0 | +

L

(eL | t s 0 | 1) (3.5)holds for all t J .Proof. We prove only inequality ( 3.5) for t s0 , the case t < s 0 follows similarly.Let s1 be the smallest real number larger than s0 that is a switching time of ora switching time of . If there is no such a number, then set s1 := sup xJ x. ByTheorem 3.2 inequality (3.5) holds for all t, s0 t < s 1 . If s1 = sup xJ x weare nished with the proof, otherwise s1 J and inequality ( 3.5) holds for t = s1too. In the second case, let s2 be the smallest real number larger than s1 that is aswitching time of or a switching time of . If there is no such a number, then sets2 := sup xJ x. Then, by Theorem 3.2,

y (t) z(t) e L (s 1 s 0 ) + L

(eL (s 1 s 0 ) 1) eL (t s 1 ) + L

(eL (t s 1 ) 1)= eL (t s 0 ) + L e

L (t s 0 ) L eL (t s 1 ) + L eL (t s 1 ) L= eL (t s 0 ) +

L

(eL (t s 0 ) 1)for all t such that s1 t < s 2 . As this argumentation can, if necessary, be repeatedad innitum, inequality ( 3.5) holds for all t s0 such that t J . 3.3. Dini derivatives. The Italian mathematician Ulisse Dini introduced in 1878in his textbook [8] on analysis the so-called Dini derivatives. They are a gener-alization of the classical derivative and inherit some important properties from it.Because the Dini derivatives are point-wise dened, they are more suited for ourneeds than some more modern approaches to generalize the concept of a derivativelike Sobolev Spaces (see, for example, [ 1]) or distributions (see, for example, [64]).The Dini derivatives are dened as follows:

Denition 3.7 (Dini derivatives) . Let I R , g : I R be a function, and y I .(i) Assume y is a limit point of I ]y, + [. Then the right-hand upper Diniderivative D + of g at the point y is dened by

D + g(y) := limsupx y+

g(x) g(y)x y

:= lim0+

supx I ]y, + [

0


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and the right-hand lower Dini derivative D + of g at the point y is denedby

D+ g(y) := liminf x y+

g(x) g(y)x y := lim

0+ inf

x I ]y, + [0


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a continuous function. Let D {D + , D + , D , D } be a Dini derivative and let J be an interval, such that Dg(x) J for all x I \C. Then

g(x) g(y)x y J for all x, y I , x = y.Proof. See, for example, Theorem 12.24 in [62].

The previous theorem has an obvious corollary.

Corollary 3.10. Let I be an interval of strictly positive measure in R , let C be a countable subset of I , let g : I R be a continuous function, and let D{D + , D + , D , D } be a Dini derivative. Then:

Dg(x) 0 for all x I \C, implies that g is a monotonically increasing function on I .

D

g(x) > 0 for all x I \C, implies that g is a strictly monotonically increasing function on I .Dg(x) 0 for all x I \C, implies that g is a monotonically

decreasing function on I .Dg(x) < 0 for all x I \C, implies that g is a strictly monotonically

decreasing function on I .3.4. Stability of arbitrary switched systems. The concepts equilibrium pointand stability are motivated by the desire to keep a dynamical system in, or atleast close to, some desirable state. The term equilibrium or equilibrium point of a dynamical system, is used for a state of the system that does not change in thecourse of time, that is, if the system is at an equilibrium at time t = 0, then it will

stay there for all times t > 0.Denition 3.11 (Equilibrium point) . A state y in the state-space of the SwitchedSystem 3.5 is called an equilibrium or an equilibrium point of the system, if andonly if f p(t, y ) = 0 for all p P and all t 0.

If y is an equilibrium point of Switched System 3.5, then obviously the initialvalue problem

x = f (t, x), x(0) = yhas the solution x(t) = y for all t 0 regardless of the switching signal S . Thesolution with y as an initial value in the state-space is thus a constant vector andthe state does not change in the course of time. By a translation in the state-spaceone can always reach that y = 0 without affecting the dynamics. Hence, there is noloss of generality in assuming that a particular equilibrium point is at the origin.

A real-world system is always subject to some uctuations in the state. Thereare some external effects that are unpredictable and cannot be modelled, somedynamics that have (hopefully) very little impact on the behavior of the systemare neglected in the modelling, etc. Even if the mathematical model of a physicalsystem would be perfect, which hardly seems possible, the system state wouldstill be subject to quantum mechanical uctuations. The concept of local stabilityin the theory of dynamical systems is motivated by the desire, that the systemstate stays at least close to an equilibrium point after small uctuations in the


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only if there exists a KL such that for every S P , every t t0 0,and every N the following inequality holds (t, t 0 , ) ( , t t0). (3.6)

(iii) The origin is said to be a uniformly exponentially stable equilibrium pointof the Switched System 3.5 on the neighborhood N U of the origin, if andonly if there exist constants k > 0 and > 0, such that for every S P ,every t t0 0, and every N the following inequality holds

(t, t 0 , ) ke ( t t 0 ) .The stability denitions above imply, that if the origin is a uniformly exponen-

tially stable equilibrium of the Switched System 3.5 on the neighborhood N , thenthe origin is a uniformly asymptotically stable equilibrium on N as well, and, if theorigin is a uniformly asymptotically stable equilibrium of the Switched System 3.5on the neighborhood N , then the origin is a uniformly stable equilibrium on N .

If the Switched System 3.5 is autonomous, then the stability concepts presentedabove for the systems equilibria are uniform in a canonical way, that is, independentof t0 , and the denitions are somewhat more simple.

Denition 3.14. (Stability concepts for equilibria of autonomous systems) As-sume that the origin is an equilibrium point of the Switched System 3.5, denote by the solution to the system, let be an arbitrary norm on R n , and assume thatthe system is autonomous.

(i) The origin is said to be a stable equilibrium point of the autonomousSwitched System 3.5 on a neighborhood N U of the origin, if and onlyif there exists an K such that for every S P , every t 0, and every N the following inequality holds

(t, )

( ).

(ii) The origin is said to be an asymptotically stable equilibrium point of theautonomous Switched System 3.5 on the neighborhood N U of the origin,if and only if there exists a KL such that for every S P , every t 0,and every N the following inequality holds

(t, ) ( , t ).(iii) The origin is said to be an exponentially stable equilibrium point of the

Switched System 3.5 on the neighborhood N U of the origin, if and onlyif there exist constants k > 0 and > 0, such that for every S P , everyt 0, and every N the following inequality holds (t, ) ke t .

The set of those points in the state-space of a dynamical system, that are at-tracted to an equilibrium point by the dynamics of the system, is of great impor-tance. It is called the region of attraction of the equilibrium. Some authors preferdomain , basin , or even bassin instead of region . For nonautonomous systems itmight depend on the initial time.

Denition 3.15 (Region of attraction) . Assume that y = 0 is an equilibriumpoint of the Switched System 3.5 and let be the solution to the system. For


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every t0 R 0 the set

Rt 0Att :=

{

U : lim sup

t +

(t, t 0 , ) = 0 for all

S P

}is called the region of attraction with respect to t0 of the equilibrium at the origin.The region of attraction RAtt of the equilibrium at the origin is dened by

RAtt :=t 0 0

Rt 0Att .

Thus, for the Switched System 3.5, RAtt implies lim t + (t, t 0 , ) = 0for all S P and all t0 0.3.5. Three useful lemmas. It is often more convenient to work with smoothrather that merely continuous functions and later on we need estimates by convex

C K functions. The two next lemmas state some useful facts in this regard.Lemma 3.16. Let f :

R> 0

R 0 be a monotonically decreasing function. Then there exists a function g : R > 0 R > 0 with the following properties:

(i) g C (R > 0).(ii) g(x) > f (x) for all x R > 0 .(iii) g is strictly monotonically decreasing.(iv) limx 0+ g(x) = + and limx + g(x) = lim x + f (x).(v) g is invertible and g 1 C (g(R > 0)) .

Proof. We dene the function h : R > 0 R > 0 by,h(x) := f 1n +1 + 1x , if x [ 1n +1 , 1n [ for some n N > 0 ,f (n) + 1x , if x [n, n + 1[ for some n N > 0 ,

and the function h : R > 0

R > 0 by

h(x) := h(x tanh( x)) .Then h is a strictly monotonically decreasing measurable function and because his, by its denition, strictly monotonically decreasing and larger than f , we have

h(x + tanh( x)) = h(x + tanh( x) tanh( x + tanh( x))) > h(x) > f (x)for all x R > 0 .Let C (R ) such that (x) 0 for all x R , supp( ) ] 1, 1[, and R (x)dx = 1. We claim that the function g : R > 0 R > 0 ,

g(x) := x+tanh( x )x tanh( x ) x ytanh( x) h(y)tanh( x) dy = 1 1 1(y)h(x y tanh( x))dy,fullls the properties (i)(v).

Proposition (i) follows from elementary Lebesgue integration theory. Proposition(ii) follows from

g(x) = 1 1 (y)h(x y tanh( x))dy> 1 1 (y)h(x + tanh( x))dy


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> 1 1 (y)f (x)dy = f (x).To see that g is strictly monotonically decreasing let t > s > 0 and consider that

t y tanh( t) > s y tanh( s) (3.7)for all y in the interval [ 1, 1]. Inequality ( 3.7) follows from

t y tanh( t) [s y tanh( s)] = t s y[tanh( t) tanh( s)]= t s y(t s)(1 tanh 2(s + t,s (t s))) > 0,

for some t,s [0, 1], where we used the Mean-value theorem. But thenh(t y tanh( t)) < h (s y tanh( s))

for all y [1, 1] and the denition of g implies that g(t) < g (s). Thus, proposition(iii) is fullled.Proposition iv) is obvious from the denition of g. Clearly g is invertible and by

the chain rule

[g 1] (x) = 1

g (g 1(x)),

so it follows by mathematical induction that g 1 C (g(R > 0)), that is, proposition(v). Lemma 3.17. Let K. Then, for every R > 0, there is a function R K,such that:

(i) R is a convex function.(ii) R restricted to R > 0 is innitely differentiable.

(iii) For all 0

x

R we have R (x)

(x).

Proof. By Lemma 3.16 there is a function g, such that g C (R > 0), g(x) > 1/ (x)for all x > 0, limt 0+ g(x) = + , and g is strictly monotonically decreasing. Thenthe function R : R 0 R 0 , dened through R (x) :=

1R x0 d g( ) ,

has the desired properties. First, R (0) = 0 and for every 0 < x R we have R (x) =

1R x0 d g( ) 1g(x) < (x).

Second, to prove that R is a convex K function is suffices to prove that the secondderivative of R is strictly positive. But this follows immediately because for everyx > 0 we have g (x) < 0, which implies

d 2 Rdx2

(x) = g (x)R[g(x)]2

> 0.

The third existence lemma is the well known and very useful Masseras lemma[42].


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4. Lyapunovs Direct Method for Switched Systems

The Russian mathematician and engineer Alexandr Mikhailovich Lyapunov pub-

lished a revolutionary work in 1892 on the stability of motion, where he introducedtwo methods to study the stability of general continuous dynamical systems. AnEnglish translation of this work can be found in [ 36].

The more important of these two methods, known as Lyapunovs second method or Lyapunovs direct method , enables one to prove the stability of an equilibriumof (3.3) without integrating the differential equation. It states, that if y = 0 is anequilibrium point of the system, V C1(R 0 U ) is a positive denite function ,that is, there exist functions 1 , 2 K such that

1( x 2) V (t, x) 2( x 2)for all x U and all t R 0 , and is the solution to the ODE ( 3.3). Then theequilibrium is uniformly asymptotically stable, if there is an K such that theinequality

ddt

V (t, (t, t 0 , )) = [x V ](t, (t, t 0 , )) f (t, (t, t 0 , )) + V

t (t, (t, t 0 , ))

( (t, t 0 , ) 2)(4.1)

holds for all (t, t 0 , ) in an open neighborhood N U of the equilibrium y. Inthis case the equilibrium is uniformly asymptotically stable on a neighborhood,which depends on V , of the origin. The function V satisfying ( 4.1) is said tobe a Lyapunov function for (3.3). The direct method of Lyapunov is covered inpractically all modern textbooks on nonlinear systems and control theory. Somegood examples are [ 19, 20, 28, 54, 60, 3, 65].

We will prove, that if the time-derivative in the inequalities above is replacedwith a Dini derivative with respect to t, then the assumption V C1(R 0 U ) canbe replaced with the less restrictive assumption, that V is merely continuous. Thesame is done in Theorem 42.5 in [ 19], but a lot of details are left out. Further, wegeneralize the results to arbitrary switched systems.

Before we state and prove the direct method of Lyapunov for switched systems,we prove a lemma that we use in its proof.

Lemma 4.1. Assume that the origin is an equilibrium of the Switched System 3.5 and let be a norm on R n . Further, assume that there is a function K, such that for all S P and all t t0 0 the inequality

(t, t 0 , ) ( ) (4.2)holds for all in some bounded neighborhood N U of the origin.Under these assumptions the following two propositions are equivalent:

(i) There exists a function

L, such that

(t, t 0 , ) ( ) (t t0) for all S P , all t t0 0, and all N .(ii) For every > 0 there exists a T > 0, such that for every t0 0, every S P , and every N the inequality (t, t 0 , )

holds for all t T + t0 .


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Proof. Let R > 0 be so large that N B ,R and set C := max {1, (R)}.Note that Proposition (i) implies proposition (ii): For every > 0 we set T :=

1

(/ (R)) and proposition (ii) follows immediately.Proposition (ii) implies proposition (i): For every > 0 dene T () as theinmum of all T > 0 with the property, that for every t0 0, every S P , andevery N the inequality (t, t 0 , )

holds for all t T + t0 .Then T is a monotonically decreasing function R > 0 R 0 and, because of (4.2),T () = 0 for all > (R). By Lemma 3.16 there exists a strictly monotonicallydecreasing C (R > 0) bijective function g : R > 0 R > 0 , such that g() > T () forall > 0. Now, for every pair t > t 0 0 set := g 1(t t0) and note that becauset = g( ) + t0

T ( ) + t0 we have

g 1(t

t0) =

(t, t 0 , ) .

But then

(s) := 2C C/g (1) s, if s [0, g(1)], Cg 1(s), if s > g (1),is an L function such that (t, t 0 , ) (t t0),for all t t0 0 and all N , and therefore

(t, t 0 , ) ( ) (t t0).We come to the main theorem of this section: The Lyapunovs direct method

for arbitrary switched systems.Theorem 4.2. Assume that the Switched System 3.5 has an equilibrium at the origin. Let be a norm on R n and let R > 0 be a constant such that the closure of the ball B ,R is a subset of U . Let V : R 0 B ,R R be a continuous function and assume that there exist functions 1 , 2 K such that

1( ) V (t, ) 2( ) for all t 0 and all B ,R . Denote the solution to the Switched System 3.5 by and set d := 12 ( 1(R)) . Finally, let D {D + , D + , D , D } be a Dini derivative with respect to the time t , which means, for example with D= D + , that

D + [V (t, (t, t 0 , ))] := lim suph 0+

V (t + h, (t + h, t 0 , )) V (t, (t, t 0 , ))h

.

Then the following propositions are true:(i) If for every S P , every U , and every t t0 0, such that (t, t 0 , ) B ,R , the inequality

D[V (t, (t, t 0 , ))] 0 (4.3)holds, then the origin is a uniformly stable equilibrium of the Switched Sys-tem 3.5 on B ,d .


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(ii) If there exists a function K, with the property that for every S P ,every U , and every t t0 0, such that (t, t 0 , ) B ,R , the inequality D[V (t, (t, t 0 , ))] ( (t, t 0 , ) ) (4.4)

holds, then the origin is a uniformly asymptotically stable equilibrium of the Switched System 3.5 on B ,d .

Proof. Proposition (i): Let t0 0, B ,d , and S P all be arbitrary but xed.By the note after the denition of Switched System 3.5 either (t, t 0 , ) B ,Rfor all t t0 or there is a t > t 0 such that (s, t 0 , ) B ,R for all s [t0 , t[and (t, t 0 , ) B ,R . Assume that the second possibility applies. Then, byinequality ( 4.3) and Corollary 3.10 1(R) V (t, (t, t 0 , )) V (t0 , ) 2( ) < 2(d),

which is contradictory to d = 12 ( 1(R)). Therefore (t, t 0 , ) B ,R for allt t0 .But then it follows by inequality ( 4.3) and Corollary 3.10 that

1( (t, t 0 , ) ) V (t, (t, t 0 , )) V (t0 , ) 2( ),for all t t0 , so

(t, t 0 , ) 11 ( 2( ))for all t t0 . Because 11 2 is a class K function, it follows, because t0 0, B ,d , and S P were arbitrary, that the equilibrium at the origin is auniformly stable equilibrium point of the Switched System 3.5 on B ,d .Proposition (ii): Inequality ( 4.4) implies inequality ( 4.3) so Lemma 4.1 appliesand it suffices to show that for every > 0 there is a nite T > 0, such that

t T + t0 implies (t, t 0 , ) (4.5)for all t0 0, all B ,d , and all S P . To prove this choose an arbitrary > 0 and set

:= min {d, 12 ( 1())} and T := 2(d)( )

.

We rst prove that for every S P the following proposition: B ,d and t0 0 implies (t, t 0 , ) < (4.6)

for some t [t0 , T + t0]. We prove ( 4.6) by contradiction. Assume that

(t, t 0 , ) (4.7)for all t [t0 , T + t0]. Then

0 < 1( ) 1( (T + t0 , t 0 , ) ) V (T + t0 , (T + t0 , t 0 , )) . (4.8)By Theorem 3.9 and the assumption ( 4.7), there is an s

[t

0, T + t

0], such that

V (T + t0 , (T + t0 , t 0 , )) V (t0 , )T [D

V ](s, (s, t 0 , ))]

( (s, t 0 , ) ) ( ),

that is

V (T + t0 , (T + t0 , t 0 , )) V (t0 , ) T ( )


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2( ) T ( )< 2(d) T ( )= 2(d)

2(d)( )

( ) = 0 ,

which is contradictory to ( 4.8). Therefore proposition (4.6) is true.Now, let t be as in (4.6) and let t > T + t0 be arbitrary. Then, because

s V (s, (s, t 0 , )) , s t0 ,is strictly monotonically decreasing by inequality ( 4.4) and Corollary 3.10, we getby (4.6), that

1( (t, t 0 , ) ) V (t, (t, t 0 , )) V (t, (t, t 0 , )) 2( (t, t 0 , ) )< 2(

)= min { 2(d), 1()} 1(),

and we have proved (4.5). The proposition (ii) follows.

The function V in the last theorem is called a Lyapunov function for the SwitchedSystem 3.5.

Denition 4.3 (Lyapunov function) . Assume that the Switched System 3.5 hasan equilibrium at the origin. Denote the solution to the Switched System 3.5 by and let be a norm on R n . Let R > 0 be a constant such that the closureof the ball

B ,R is a subset of

U . A continuous function V : R 0

B ,R

R

is called a Lyapunov function for the Switched System 3.5 on B ,R , if and only if there exists a Dini derivative D {D + , D + , D , D } with respect to the time tand functions 1 , 2 , K with the properties that:(L1)

1( ) V (t, ) 2( )for all t 0 and all B ,R .(L2)

D[V (t, (t, t 0 , ))] ( (t, t 0 , ) )for every S P , every U , and every t t0 0,such that (t, t 0 , ) B ,R .

The Direct Method of Lyapunov (Theorem 4.2) can thus, by Denition 4.3, be

rephrased as follows:Assume that the Switched System 3.5 has an equilibrium point atthe origin and that there exists a Lyapunov function dened on theball B ,R , of which the closure is a subset of U , for the system.Then there is a d, 0 < d < R , such that the origin is a uniformlyasymptotically stable equilibrium point of the system on the ball

B ,d (which implies that B ,d is a subset of the equilibriumsregion of attraction). If the comparison functions 1 and 2 in the


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condition (L1) for a Lyapunov function are known, then we cantake d = 12 ( 1(R)).

5. Converse Theorem for Switched Systems

In the last section we proved, that the existence of a Lyapunov function V for the Switched System 3.5 is a sufficient condition for the uniform asymptoticstability of an equilibrium at the origin. In this section we prove the converse of this theorem. That is, if the origin is a uniformly asymptotically stable equilibriumof the Switched System 3.5, then there exists a Lyapunov function for the system.

Later, in Section 8, we prove that our algorithm always succeeds in construct-ing a Lyapunov function for a switched system if the system possesses a Lyapunovfunction, whose second-order derivatives are bounded on every compact subset of the state-space that do not contain the origin. Thus, it is not sufficient for our pur-poses to prove the existence of a merely continuous Lyapunov function. Therefore,we prove that if the origin is a uniformly asymptotically stable equilibrium of theSwitched System 3.5, then there exists a Lyapunov function for the system that isinnitely differentiable with a possible exception at the origin.

5.1. Converse theorems. There are several theorems known, similar to Theo-rem 4.2, where one either uses more or less restrictive assumptions regarding theLyapunov function than in Theorem 4.2. Such theorems are often called Lyapunov-like theorems. An example for less restrictive assumptions is Theorem 46.5 in [19]or equivalently Theorem 4.10 in [ 28], where the solution to a continuous system isshown to be uniformly bounded, and an example for more restrictive assumptions isTheorem 5.17 in [54], where an equilibrium is proved to be uniformly exponentiallystable. The Lyapunov-like theorems all have the form:

If one can nd a function V for a dynamical system, such that

V satises the properties X , then the system has the stabilityproperty Y .A natural question awakened by any Lyapunov-like theorem is whether its converseis true or not, that is, if there is a corresponding theorem of the form:

If a dynamical system has the stability property Y , then thereexists a function V for the dynamical system, such that V satisesthe properties X .

Such theorems are called converse theorems in the Lyapunov stability theory. Fornonlinear systems they are more involved than Lyapunovs direct method and theresults came rather late and did not stem from Lyapunov himself. The conversetheorems are covered quite thoroughly in Chapter VI in [19]. Some further generalreferences are Section 5.7 in [60] and Section 4.3 in [ 28]. More specic references

were given here in Section 2.About the techniques to prove such theorems W. Hahn writes on page 225 in hisbook Stability of Motion [19]:

In the converse theorems the stability behavior of a family of mo-tions p(t, a , t 0)1 is assumed to be known. For example, it mightbe assumed that the expression p (t, a , t 0) is estimated by known

1In our notation p (t, a , t 0 ) = (t, t 0 , a ).


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comparison functions (secs. 35 and 36). Then one attempts to con-struct by means of a nite or transnite procedure, a Lyapunovfunction which satises the conditions of the stability theorem un-der consideration.

In this section we prove a converse theorem on uniform asymptotic stabilityof an arbitrary switched systems equilibrium, where the functions f p, p P , of the systems x = f p(t, x), satisfy the common Lipschitz condition: there exists aconstant L such that

f p(s, x) f p(t, y ) L(|s t|+ x y )for all p P , all s, t R 0 , and all x, y B ,R . To construct a Lyapunov functionthat is merely Lipschitz in its state-space argument, it suffices that the functionsf p, p P , satisfy the common Lipschitz condition:

f p(t, x) f p(t, y ) Lx x y (5.1)for all p P , all t R 0 , and all x , y B ,R , as shown in Theorem 5.2. However,our procedure to smooth it to a C function everywhere except at the origin, as donein Theorem 5.4, does not necessarily work if ( s, t ) f p(s, x) f p(t, x) / |s t|,s = t, is unbounded. Note, that this additional assumption does not affect thegrowth of the functions f p, p P , but merely excludes innitely fast oscillationsin the temporal domain. The Lipschitz condition ( 5.1) already takes care of that

f p(t, x) Lx x Lx R for all t 0 and all x B ,R because f p(t, 0) = 0.5.2. A converse theorem for arbitrary switched systems. The constructionhere of a smooth Lyapunov function for the Switched System 3.5 is quite longand technical. We therefore arrange the proof in a few denitions, lemmas, andtheorems. First, we use Masseras lemma (Lemma 3.18) to dene the functionsW ,

S P , and them in turn to dene the function W , and after that we prove

that the function W is a Lyapunov function for the Switched System 3.5 used inits construction.

Denition 5.1 (The functions W and W ). Assume that the origin is a uniformlyasymptotically stable equilibrium point of the Switched System 3.5 on the ball

B ,R U , where is a norm on R n and R > 0, and let KL be such that (t, t 0 , ) ( , t t0) for all S P , all B ,R , and all t t0 0.Assume further, that there exists a constant L for the functions f p, such thatf p(t, x) f p(t, y) L x y

for all t 0, all x , y B ,R , and all p P . By Masseras lemma (Lemma 3.18)there exists a function g C1(R 0), such that g, g K, g is innitely differentiableon R > 0 ,

+ 0 g( (R, ))d < + , and + 0 g ( (R, ))eL d < + .(i) For every S P we dene the function W for all t 0 and all B ,Rby

W (t, ) := + t g( (,t, ) )d.


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(ii) We dene the function W for all t 0 and all B ,R byW (t, ) := sup

S PW (t, ).

Note, that if the Switched System 3.5 is autonomous, then W does notdepend on t , that is, it is time-invariant.

The function W from the denition above (Denition 5.1) is a Lyapunov functionfor the Switched System 3.5 used in its construction. This is proved in the nexttheorem.

Theorem 5.2 (Converse theorem for switched systems) . The function W in Deni-tion 5.1 is a Lyapunov function for the Switched System 3.5 used in its construction.Further, there exists a constant LW > 0 such that

|W (t, ) W (t, )| LW (5.2) for all t 0 and all , B ,R , where the norm and the constant R are the same as in Denition 5.1.Proof. We have to show that the function W complies to the conditions (L1) and(L2) of Denition 4.3. Because

(u,t, ) = + ut f ( ) (, (,t, ))d,and f (s ) (s, y ) LR for all s 0 and all y B ,R , we conclude (u,t, ) (u t)LR for all u t 0, B ,R , and all S P . Therefore,

(u,t, ) 2

whenever t u t +

2LR,

which implies

W (t, ) :=

+

t

g( (,t, ) )d

2LR

g( / 2)

and then 1( ) W (t, ) for all t 0 and all B ,R , where 1(x) :=x/ (2LR )g(x/ 2) is a K function.By the denition of W ,W (t, ) t+ ht g( (,t, ) )d + W (t + h, (t + h,t, ))

(reads: supremum over all trajectories emerging from at time t is not less than over any particular trajectoryemerging from at time t)

for all B ,R , all t 0, all small enough h > 0, and all S P , from whichlimsuph 0+

W (t + h,

(t + h,t, ))

W (t, )

h g( )follows. Because g K this implies that the condition (L2) from Denition 4.3holds for the function W .

Now, assume that there is an LW > 0 such that inequality ( 5.2) holds. ThenW (t, ) 2( ) for all t 0 and all B ,R , where 2(x) := LW x is aclass K function. Thus, it only remains to prove inequality ( 5.2). However, as thisinequality is a byproduct of the next lemma, we spare us the proof here.


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The results of the next lemma are needed in the proof of our converse theorem onuniform asymptotic stability of a switched systems equilibrium and as a convenientside effect it completes the proof of Theorem 5.2.

Lemma 5.3. The function W in Denition 5.1 satises for all t s 0, all , B ,R , and all S P the inequality W (t, ) W (s, ) C (t,s, ) ts g( (,s, ) )d, (5.3)

where

C := + 0 g ( (R, ))eL d < + .Especially,

|W (t, ) W (t, )| C (5.4) for all t 0 and all , B ,R .The norm

, the constants R, L , and the functions and g are, of course, the

same as in Denition 5.1.

Proof. By the Mean-value theorem and Theorem 3.6 we have

W (t, ) W (s, ) (5.5)= + t g( (,t, ) )d + s g( (,s, ) )d + t g( (,t, ) ) g( (,s, ) ) d ts g( (,s, ) )d = + t g( (,t, ) ) g( (,t, (t,s, )) ) d ts g( (,s, ) )d

+

t g ( (R, t)) (,t, ) (,t, (t,s, )) d t

s g( (,s, ) )d

+ t g ( (R, t))eL ( t ) (t,s, ) d ts g( (,s, ) )d = C (t,s, ) ts g( (,s, ) )d.

We now show that we can replace W (t, ) W (s, ) by W (t, ) W (s, ) onthe leftmost side of inequality ( 5.5) without violating the relations. That this ispossible might seem a little surprising at rst sight. However, a closer look revealsthat this is not surprising at all because the rightmost side of inequality (5.5) onlydepends on the values of (z) for s z t and because W (t, ) W (s, ) W (t, ) W (s, ), where the left-hand side only depends on the values of (z)for z t, .To rigidly prove the validity of this replacement let > 0 be an arbitrary constantand choose a S P , such that

W (t, ) W (t, ) < 2

, (5.6)

and a u > 0 so small that

ug( ( , 0)) + 2 CR (eu 1) < 2

. (5.7)


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Theorem 5.4 (Smooth converse theorem for switched systems) . Assume that the origin is a uniformly asymptotically stable equilibrium point of the Switched System 3.5 on the ball

B ,R U , R > 0, where

is a norm on R n . Assume further,

that the functions f p, p P , satisfy the common Lipschitz condition: there exists a constant L > 0 such that f p(s, x) f p(t, y ) L(|s t|+ x y ) (5.11)

for all s, t 0, all x , y B ,R , and all p P .Then, for every 0 < R < R , there exists a Lyapunov function V : R 0 B ,R R for the switched system, that is innitely differentiable at every point (t, x) R 0 B ,R , x = 0.Further, if the Switched System 3.5 is autonomous, then there exists a time-invariant Lyapunov function V : B ,R R for the system, that is innitely differentiable at every point x B ,R , x = 0.Proof. The proof is long and technical, even after all the preparation we havedone, so we split it into two parts. In part I we introduce some constants andfunctions that we will use in the rest of the proof and in part II we dene a functionV C (R 0 B ,R \{0}) and prove that it is a Lyapunov function for thesystem.Part I: Because the assumptions of the theorem imply the assumptions made inDenition 5.1, we can dene the functions W : R 0 B ,R R 0 , S P , andW : R 0 B ,R R 0 just as in the denition. As in Denition 5.1, denote byg be the function from Masseras lemma 3.18 in the denition of the functions W ,and set

C := + 0 g ( (R, ))eL d,where, once again, is the same function as in Denition 5.1.

Let m, M > 0 be constants such thatx 2 m x and x M x 2

for all x R n and let a be a constant such thata > 2m and set y :=

mRa

.

Dene

K := g(y)

a L C m(1 + M )R + mR

43

LR + M + g(4R/ 3)mR ,

and set

:= min a

3g(y), a(R R)

Rg(y) ,

a2mRLg (y)

, 1K

. (5.12)

Note that is a real-valued constant that is strictly larger than zero. We dene thefunction : R 0 R 0 by

(x) := xa0 g(z)dz. (5.13)The denition of implies

(x) g(x/a )xa

a3g(y) g(x/a )

xa

x3

(5.14)


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for all 0 x mR and (x) =

a

g(x/a ) (5.15)

for all x 0.Dene the function by (x) := g(2x/ 3) g(x/ 2) for all x 0. Then (0) = 0and for every x > 0 we have (x) =

23

g (2x/ 3) 12

g (x/ 2) > 0

because g K, that is K.Part II: Let C (R ) be a nonnegative function with supp( ) ] 1, 0[ and R (x) = 1 and let C (R n ) be a nonnegative function with supp( ) B 2 ,1and R n (x)dn x = 1. Extend W on R R n by setting it equal to zero outside of R 0 B ,R . We claim that the function V : R 0 B ,R R 0 , V (t, 0) := 0for all t 0, and

V (t, ) := R R n t ( 2) y( 2) W [, y]n +1 ( 2) dn yd = R R n ( ) (y )W [t ( 2), ( 2)y]dn yd

for all t 0 and all B ,R \ {0}, is a C (R 0 B ,R \{0}) Lyapunovfunction for the switched system. Note, that if the Switched System 3.5 in questionis autonomous, then W is time-invariant, which implies that V is time-invarianttoo.

Because, for every y 2 1 and every < R , we have by (5.14) and (5.12),that

( 2)y

1 +

( 2)

2

1 + g( 2 /a )

2 2a

< 1 + a(R R)g(y)

Rg(y)aR

= R,

so V is properly dened on R 0B ,R . But then, by construction, V C (R 0B ,R \{0}). It remains to be shown that V fullls the conditions (L1) and (L2)in Denition 4.3 of a Lyapunov function.

By Theorem 5.2 and Lemma 5.3 there is a function 1 K and a constantLW > 0, such that

1( )

W (t, )

L

W

for all t 0 and all B ,R . By inequality ( 5.14) we have for all B ,R andall y 2 1, that ( 2)y

23

2

= 23

, (5.16)

( 2)y + 23

2

= 43

. (5.17)


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Hence

1(2 / 3) =

R

R n

( ) (y )1(2 / 3)dn yd

R R n ( ) (y )1( ( 2)y )dn yd R R n ( ) (y )W [t ( 2), ( 2)y ]dn yd = V (t, )

R R n ( ) (y )LW ( 2)y dn yd

4LW 3 ,

(5.18)

and the function V fullls the condition (L1) .We now prove that V fullls the condition (L2) . To do this let t

0,

B ,R ,

and S P be arbitrary, but xed throughout the rest of the proof. Denote by I the maximum interval in R 0 on which s (s,t, ) is dened and setq (s, ) := s ( (s,t, ) 2)

for all s I and all 1 0 and deneD(h, y , ) := W [q (t + h, ), (t + h,t, ) ( (t + h,t, ) 2)y ]

W [q (t, ), ( 2)y ]for all h such that t + h I , all y 2 1, and all 1 0. Then

V (t + h, (t + h,t, )) V (t, ) = R R n ( ) (y )D (h, y , )dn yd for all h such that t + h

I , especially this equality holds for all h in an interval of

the form [0 , h [, where 0 < h + .We are going to show thatlimsup

h 0+

D (h, y , )h ( ). (5.19)

If we can prove that ( 5.19) holds, then, by Fatous lemma,

limsuph 0+

V (t + h, (t + h,t, )) V (t, )h

R R n ( ) n (y)limsuph 0+ D (h, y , )h dn yd ( ),

and we would have proved that the condition (L2) is fullled.To prove inequality ( 5.19) observe that q (t, ) 0 for all 1 0 and, forevery s > t that is smaller than any switching-time (discontinuity-point) of largerthan t , and because of ( 5.12) and ( 5.11), we have

dq ds

(s, ) = 1 g( (s,t, ) 2 /a )

a (s,t, ) (s,t, ) 2 f (s ) (s, (s,t, ))

1 g(y)LmR

a


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12

,

so q (t + h, )

q (t, )

0 for all small enough h

0.

Now, denote by the constant switching signal (t) in S P , that is (s) := (t)for all s 0, and consider that by Lemma 5.3D (h, y , )

h C h

(t + h,t, ) ( (t + h,t, ) 2)y

(q (t + h, ), q (t, ), ( 2)y )

1h q( t + h, )q( t, ) g( (s, q (t, ), ( 2)y) )ds

= C (t + h,t, )

h ( (t + h,t, ) 2) ( 2)

h y

(q (t + h, ), q (t, ), ( 2)y ) [ ( 2)y ]

h

1h q( t + h, )q( t, ) g( (s, q (t, ), ( 2)y) )ds.

For the next calculations we need s q (s, ) to be differentiable at t . If it is not,which might be the case if t is a switching time of , we replace with S P where(s) :=

(t), if 0 s t,(s), if s t.

Note that this does not affect the numerical value

limsuph 0+

D (h, y , )h

because (t + h) = (t + h) for all h 0. Hence, with p := (t), and by (5.11),the chain rule, ( 5.16), and ( 5.17),limsup

h 0+

D (h, y , )h

C f p(t, ) f p(q (t, ), ( 2)y ) dq dt

(t , )t = t

( 2) ddt

(t , t, ) 2t = t

y

g( (q (t, ), q (t, ), ( 2)y ) ) dq dt

(t , )t = t

= C f p(t, )

f p(q (t, ),

( 2)y ) 1

( 2)

2 f p(t, )

( 2)[ 2 f p(t, )]y

g( ( 2)y ) 1 ( 2) 2 f p(t, )

C f p(t, ) f p(q (t, ), ( 2)y )+ C ( 2) f p(t, ) 2 f p(q (t, ), ( 2)y ) + y


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g(2 / 3) + g(4 / 3) ( 2) f p(t, ) 2 CL |t q (t, )|+ ( 2) y ) ]

+ C ( 2)mLR L ( 2)y ) + M y 2g(2 / 3) + g(4 / 3) ( 2)mLR

C L(1 + M )( 2) + ( 2)mLR L43

+ M

g(2 / 3) + g(4 / 3) ( 2)mLR.Therefore, by ( 5.14), (5.15), and ( 5.12), and with x := , we can further simplify,

limsuph 0+

D (h, y , )h

g(2x/ 3) + a

g(mx/a )L C m(1 + M )x + mR43

Lx + M + g(4x/ 3)mR

g(2x/ 3) + K g(x/ 2)

(x),and because t 0, B ,R , and S P were arbitrary, we have proved that V is a Lyapunov function for the system.

Now, we have proved the main theorem of this section, our much wanted conversetheorem for the arbitrary Switched System 3.5.

6. Construction of Lyapunov Functions

In this section we present a procedure to construct Lyapunov functions for theSwitched System 3.5. After a few preliminaries on piecewise affine functions we givean algorithmic description of how to derive a linear programming problem from the

Switched System 3.5 (Denition 6.8), and we prove that if the linear programmingproblem possesses a feasible solution, then it can be used to parameterize a Lya-punov function for the system. Then, in Section 8 and after some preparation inSection 7, we present an algorithm that systematically generates linear program-ming problems for the Switched System 3.5 and we prove, that if the switchedsystem possesses a Lyapunov function at all, then the algorithm generates, in anite number of steps, a linear programming problem that has a feasible solution.Because there are algorithms that always nd a feasible solution to a linear pro-gramming problem if one exists, this implies that we have derived an algorithm forconstructing Lyapunov functions, whenever one exists. Further, we consider thecase when the Switched System 3.5 is autonomous separately, because in this caseit is possible to parameterize a time-independent Lyapunov function for the system.Let us be a little more specic on these points before we start to derive the results:

To construct a Lyapunov function with a linear programming problem, one needsa class of continuous functions that are easily parameterized. That is, we need aclass of functions that is general enough to be used as a search-space for Lyapunovfunctions, but it has to be a nite-dimensional vector space so that its functionsare uniquely characterized by a nite number of real numbers. The class of thecontinuous piecewise affine functions CPWA is a well suited candidate.

The algorithm for parameterizing a Lyapunov function for the Switched System3.5 consists roughly of the following steps:


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(i) Partition a neighborhood of the equilibrium under consideration in a familyS of simplices.

(ii) Limit the search for a Lyapunov function V for the system to the class of continuous functions that are affine on any S S .(iii) State linear inequalities for the values of V at the vertices of the simplicesin S , so that if they can be fullled, then the function V , which is uniquelydetermined by its values at the vertices, is a Lyapunov function for thesystem in the whole area.

We rst partition R n into n-simplices and use this partition to dene the functionspaces CPWA of continuous piecewise affine functions R n R . A function inCPWA is uniquely determined by its values at the vertices of the simplices in S .Then we present a linear programming problem, algorithmically derived from theSwitched System 3.5, and prove that a CPWA Lyapunov function for the system canbe parameterized from any feasible solution to this linear programming problem.Finally, in Section 7, we prove that if the equilibrium of the Switched System3.5 is uniformly asymptotically stable, then any simplicial partition with smallenough simplices leads to a linear programming problem that does have a feasiblesolution. Because, by Theorem 4.2 and Theorem 5.4, a Lyapunov function exists forthe Switched System 3.5 exactly when the equilibrium is uniformly asymptoticallystable, and because it is always possible to algorithmically nd a feasible solutionif at least one exists, this proves that the algorithm we present in Section 8 canparameterize a Lyapunov function for the Switched System 3.5 if the system doespossess a Lyapunov functions at all.

6.1. Continuous piecewise affine functions. To construct a Lyapunov functionby linear programming, one needs a class of continuous functions that are easilyparameterized. Our approach is a simplicial partition of R n , on which we dene the

nite dimensionalR

-vector space CPWA of continuous functions, that are affine onevery of the simplices. We rst discuss an appropriate simplicial partition of R nand then dene the function space CPWA. The same is done in considerable moredetail in Chapter 4 in [40].

The simplices S , where Perm[{1, 2, . . . , n }], will serve as the atoms of ourpartition of R n . They are dened in the following way:Denition 6.1 (The simplices S ). For every Perm[{1, 2, . . . , n }] we denethe n-simplex

S := {y R n : 0 y (1) y (2) y (n ) 1},where y ( i ) is the (i)-th component of the vector y. An equivalent denition of the n-simplex S is

S = con n

j =1

e ( j ) ,n

j =2

e ( j ) , . . . ,n

j = n +1

e ( j )

=n +1

i =1

in +1

j = i

e ( j ) : 0 i 1 for i = 1 , 2, . . . , n + 1 andn +1

i =1

i = 1 ,

where e ( i ) is the (i)-th unit vector in R n .


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For every Perm[{1, 2, . . . , n }] the set S is an n-simplex with the volume1/n ! and, more importantly, if , Perm[{1, 2, . . . , n }], thenS S = con {x R

n: x is a vertex of S and x is a vertex of S }. (6.1)Thus, we can dene a continuous function p : [0, 1]n R that is affine on every S , Perm[{1, 2, . . . , n }], by just specifying it values at the vertices of the hypercube[0, 1]n . That is, if x S , then

x =n +1

i =1

in +1

j = i

e ( j )

where 0 i 1 for i = 1 , 2, . . . , n + 1 and n +1i =1 i = 1, Then we set p(x) = p

n +1

i =1

in +1

j = i

e ( j ) =n +1

i =1

i pn +1

j = i

e ( j ) .

The function p is now well dened and continuous because of ( 6.1). We could nowproceed by partitioning R n into the simplices ( z + S )zZ n ,Perm[ {1,2,...,n }], butwe prefer a simplicial partition of R n that is invariable with respect to reectionsthrough the hyperplanes ei x = 0, i = 1 , 2, . . . , n , as a domain for the functionspace CPWA. We construct such a partition by rst partitioning R n 0 into thefamily ( z + S )zZ n 0 , Perm[ {1,2,...,n }] and then we extend this partition on R

n byuse of the reection functions R J , where J P ({1, 2, . . . , n }).Denition 6.2 (Reection functions R J ). For every J P ({1, 2, . . . , n }), wedene the reection function R J : R n R n ,

R J (x) :=n

i =1

(1) J ( i ) x i e ifor all x

R n , where

J :

{1, 2, . . . , n

} {0, 1

} is the characteristic function of

the set J .Clearly R J , where J := { j1 , j 2 , . . . , j k}, represents reections through the hy-perplanes e j 1 x = 0, e j 2 x = 0 , . . . , e j k x = 0 in succession.The simplicial partition of R n that we use for the denition of the function spaces

CPWA of continuous piecewise affine functions is(R J (z + S )) zZ n 0 , J P ({1,2,...,n }) , Perm[ {1,2,...,n }].

Similar to ( 6.1), this partition has the advantageous property, that from

S, S RJ (z + S ) : z Z n 0 , J P ({1, 2, . . . , n }), Perm[{1, 2, . . . , n }]

follows, that S S is the convex hull of the vertices that are common to S andS . This leads to the following theorem:Theorem 6.3. Let (q z )zZ n be a collection of real numbers. Then there is exactly one continuous function p : R n R with the following properties:

(i) p(z) = q z for every z Z n .(ii) For every J P ({1, 2, . . . , n }), every Perm[{1, 2, . . . , n }], and every z Z n 0 , the restriction of the function p to the simplex R J (z + S ) is affine.Proof. See, for example, Corollary 4.12 in [40].


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A CPWA space is a set of continuous affine functions from a subset of R n intoR with a given boundary conguration. If the subset is compact, then the bound-ary conguration makes it possible to parameterize the functions in the respectiveCPWA space with a nite number of real-valued parameters. Further, the CPWAspaces are vector spaces over R in a canonical way. They are thus well suitedas a foundation, in the search of a Lyapunov function with a linear programmingproblem.

We rst dene the function spaces CPWA for subsets of R n that are the unionsof n -dimensional cubes.

Denition 6.4 (CPWA function on a simple grid) . Let Z Z n , Z = , be suchthat the interior of the set N :=

zZ (z + [0 , 1]n ),

is connected. The function space CPWA[

N ] is then dened as follows.

A function p : N R is in CPWA[ N ], if and only if:(i) p is continuous.(ii) For every simplex R J (z + S ) N , where z Z n 0 , J P ({1, 2, . . . , n }),

and Perm[{1, 2, . . . }], the restriction p|R J (z + S ) is affine.We will need continuous piecewise affine functions, dened by their values on

grids with smaller grid steps than one, and we want to use grids with variable gridsteps. We achieve this by using images of Z n under mappings R n R n , of whichthe components are continuous and strictly increasing functions R R , affine onthe intervals [ m, m + 1] for all integers m, and map the origin on itself. We callsuch R n R n mappings piecewise scaling functions .Note that if yi,j , i = 1 , 2, . . . , n and j Z , are real numbers such that yi,j < y i,j +1and yi, 0 = 0 for all i = 1 , 2, . . . , n and all j

Z , then we can dene a piecewise

scaling function PS : R n R n by PS i( j ) := yi,j for all i = 1 , 2, . . . , n and all j Z .

Moreover, the piecewise scaling functions R n R n are exactly the functions, thatcan be constructed in this way.In the next denition we use piecewise scaling functions to dene general CPWA

spaces.

Denition 6.5 (CPWA function, general) . Let PS : R n R n be a piecewisescaling function and let Z Z n , Z = , be such that the interior of the set N :=

zZ (z + [0 , 1]n )

is connected. The function space CPWA[ PS , N ] is dened asCPWA[ PS ,

N ] :=

{ p

PS 1 : p

CPWA[

N ]

}and we denote by S [PS , N ] the set of the simplices in the family(PS (R J (z + S ))) zZ n 0 , J P ({1,2,...,n }) , Perm[ {1,2,...,n }]

that are contained in the image PS ( N ) of N under PS .Clearly

{x R n : x is a vertex of a simplex in S [PS , N ]} = PS ( N Z n )


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+ 12

n

r,s =1[er (y i y n +1 )][e s (y i yn +1 )]

2f x r x s

(y i, x ) .

Further, because a simplex is a convex set, the vectors y x and y 1,x , y2,x , . . . , y n, xare all in S . But then

f (y ) n +1

i=1

i f (y i )

12

n +1

i =1

in

r,s =1|er (y i y n +1 )|(|e s (y y n +1 )|+ |e s (y i yn +1 )|) B rs

= 12

n +1

i =1

in

r,s =1B rs Ar,i (|e s (y y n +1 )|+ As,i )

and because

|e s (y y n +1 )| n +1

i =1

i |e s (y i yn +1 )| |es (y 1 yn +1 )| = As, 1it follows that

f (y ) n +1

i =1

i f (y i ) 12

n +1

i =1

in

r,s =1B rs Ar,i (As, 1 + As,i ) =

n +1

i =1

i E i .

An affine function p, dened on a simplex S R n and with values in R , hasthe algebraic form p(x) = w x + q , where w is a constant vector in R n and q isconstant in R . Another characterization of p is given by specifying its values atthe vertices as stated. The next lemma gives a formula for the components of thevector w when the values of p at the vertices of S are known and S is a simplex inS [PS , N ].Lemma 6.7. Let PS : R n R n be a piecewise scaling function, let z Z n 0 , let J P ({1, 2, . . . , n }), let Perm[{1, 2, . . . , n }], and let p(x) := w x + q be an affine function dened on the n -simplex with the vertices

y i := PS (R J (z +n

j = i

e ( j ) )) , i = 1 , 2, . . . , n .

Then

w =n

i =1

p(y i ) p(y i +1 )e ( i )

(y i

y i +1 )

e ( i ) .

Proof. For any i {1, 2, . . . , n } we have p(y i ) p(y i +1 ) = w (y i y i +1 )

=n

k=1

w (k ) [e (k ) (y i y i +1 )]= w ( i ) [e ( i ) (y i y i +1 )]


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because the components of the vectors y i and y i+1 are all equal with except of the(i)-th one. But then

w ( i ) = p(y i )

p(y i+1 )

e ( i ) (y i y i +1 )and we have nished the proof.

Now, that we have dened the function spaces CPWA we are ready to stateour linear programming problem, of which every feasible solution parameterizes aCPWA Lyapunov function for the Switched System 3.5 used in the derivation of its linear constraints.

6.2. The linear programming problem. We come to the linear programmingproblem, of which every feasible solution parameterizes a Lyapunov function forthe Switched System 3.5. The Lyapunov function is of class CPWA. We rstdene the linear programming problem in Denition 6.8. In the denition thelinear constraints are grouped into four classes, (LC1), (LC2), (LC3), and (LC4),for linear constraints 1, 2, 3, and 4 respectively. Then we show how the variablesof the linear programming problem that fulll these constraints can be used toparameterize functions that meet the conditions (L1) and (L2) of Denition 4.3,the denition of a Lyapunov function. Then we state and discuss the results inSection 6.8. Finally, we consider a more simple linear programming, dened inDenition 6.12, for autonomous systems and we show that it is equivalent to thelinear programming problem in Denition 6.8 with additional constraints that forcethe parameterized CPWA Lyapunov function to be time-invariant.

The next denition plays a central role in this work. It is generalization of thelinear programming problems presented in [ 40], [39], [17], and [16] to serve thenonautonomous Switched System 3.5.

Denition 6.8. (Linear programming problem LP ({f p : p P}, N , PS , t , D, ))Consider the Switched System 3.5 where the set P has a nite number of elements.Let T and T be constants such that 0 T < T and let PS : R n R n be apiecewise scaling function and N U be such that the interior of the set

M :=zZ n , PS (z +[0 ,1]n )N

PS (z + [0 , 1]n )

is a connected set that contains the origin. Let

D := PS ( ]d1 , d+1 []d2 , d+2 [ . . . ]dn , d+n [ )be a set, of which the closure is contained in the interior of M, and either D = or di and d+i are integers such that di 1 and 1 d+i for every i = 1 , 2, . . . , n .Finally, let be an arbitrary norm on R n and t := ( t0 , t 1 , . . . , t M ) R M +1 ,M N > 0 , be a vector such that T =: t0 < t 1 < < t M := T .

We assume that the components of the f p, p P , have boundedsecond-order partial derivatives on [ T , T ](M\D).Before we go on, it is very practical to introduce an alternate notation for the

vectors ( t, x) R R n , because it considerably shortens the formulae in the linearprogramming problem. We identify the time t with the zeroth component x0 of thevector

x := ( x0 , x1 , . . . , xn )


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and x with the components 1 to n, that is t := x0 and x i := x i for all i = 1 , 2, . . . , n .Then, the systems

x = f p

(t, x), p

P ,

can be written in the equivalent formd

dx0x = f p(x ), p P ,

where

f p(x ) := f p,0(x ), f p,1(x ), f p,2(x ), . . . , f p,n (x )

:= 1, f p,1(t, x), f p,2(t, x), . . . , f p,n (t, x) ,

(Recall that f p,i denotes the i-th component of the function f p.) that is, f p,0 := 1and f p,i (x ) := f p,i (t, x), where x = ( t, x), for all p P and all i = 1 , 2, . . . , n .Further, let PS 0 : R R be a piecewise scaling function such that PS 0(i) := t ifor all i = 0 , 1, . . . , M and dene the piecewise scaling function

PS : R R n R R nthrough

PS (x) := PS 0(x0), PS 1(x1), . . . , PS n (xn )) ,that is,

PS (x) = PS 0(t), PS (x) ,where x = ( t, x).

We will use the standard orthonormal basis in R n +1 = R R n , but start theindexing at zero (use e0 , e1 , . . . , en ), that is,x :=

n

i =0

x i e i = te0 +n

i =1

x i e i .

Because we do not have to consider negative time-values t = x0

< 0, it is moreconvenient to use reection functions that do always leave the zeroth-component of x = ( t, x) unchanged. Therefore, we dene for every reection function R J : R n R n , where J {1, 2, . . . , n }, the function R

J : R R n R R n throughR J (x ) := x0 , R J (x) := te0 +

n

i =1

(1) J ( i ) x i e i .We dene the seminorm : R R n R 0 through

(x0 , x1 , . . . , xn ) := (x1 , x2 , . . . , xn ) .Then, obviously, x = x for all x = ( t, x) R R n . The linear programmingproblem LP ({f p : p P}, N , PS , t , D, ) is now constructed in the following way:

(i) Dene the sets

G := {x R R n : x PS (Z Zn ) [T , T ](M \D) }

and

X := {x : x PS (Z n ) M}.The set G is the grid, on which we will derive constraints on the valuesof the CPWA Lyapunov function, and X is the set of distances of allrelevant points in the state-space to the origin with respect to the norm

.


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(ii) Dene for every Perm[{0, 1, . . . , n }] and every i = 0 , 1, . . . , n + 1 thevectorx i :=

n

j = ie (j ) ,

where, of course, the empty sum is interpreted as 0 R R n .(iii) Dene the set Z through: The tuple ( z, J ), where z := ( z0 , z1 , . . . , z n ) Z 0 Z n 0 and J P ({1, 2, . . . , n }, is an element of Z , if and only if PS (R J (z + [0 , 1]n +1 )) [T , T ] M\D.

Note that this denition implies that

(z ,J )Z

PS (R J (z + [0 , 1]n +1 )) = [ T , T ] M \D.

iv) For every ( z, J ) Z , every Perm[{0, 1, . . . , n }], and everyi = 0 , 1, . . . , n + 1 we set

y (z ,J ),i := PS (RJ (z + xi )).

The vectors y (z ,J ),i are the vertices of the simplices in our simplicial partitionof the set [T , T ] M \ D. The position of the simplex is given by(z, J ), where z0 species the position in time and ( z1 , z2 , . . . , z n ) speciesthe position in the state-space. Further, species the simplex and ispecies the vertex of the simplex.

v) Dene the set

Y := y(z ,J ),k , y

(z ,J ),k +1 } Perm[{0, 1, . . . , n }], (z, J ) Z ,

and k {0, 1, . . . , n } .The set Y is the set of all pairs of neighboring grid points in the grid G.

(vi) For every p P , every ( z, J ) Z , and every r, s = 0 , 1, . . . , n let B(z ,J )

p,rsbe a real-valued constant, such that

B (z ,J ) p,rs maxi =1 ,2,...,n supx fPS ( eR J (z +[0 ,1]n +1 )) 2 f p,i xr xs

(x ) .

The constants B (z ,J ) p,rs are local bounds on the second-order partial deriva-tives of the components of the functions f p, p P , with regard to the in-nity norm , similar to the constants B rs in Lemma 6.6. Note, thatbecause f p,0 := 1, the zeroth-components can be left out in the denitionof the B (z ,J ) p,rs because they are identically zero anyways. Further, f

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