INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND … · 2010-09-28 · The importance of the control of partial differential equations and delay differential ... The recent well-developed

Feedback Stabilization of Abstract DelaySystems on Banach Lattices

Tomoaki Hashimoto

Abstract— In this paper, we examine the stabilization prob-lem of systems described by partial differential equations anddelay differential equations. The control of a partial differentialequation with a time delay is a challenging problem withmany applications that include physical, chemical, biological,economic, thermal, and fluid systems. The semigroup methodis a unified approach to addressing systems that includeordinary differential equations, partial differential equations,and delay differential equations. Using semigroup theory, weintroduce the concept of an abstract delay system that can beused to characterize the behavior of a wide class of dynamicalsystems. This paper examines the stabilization problem of anabstract delay system on a Banach lattice on the basis ofsemigroup theory. To tackle this problem, we take advantageof the properties of a non-negativeC0 semigroup on a Banachlattice. The objective of this paper is to propose a stabilizationmethod for an abstract delay system on a Banach lattice. Wederive a sufficient condition under which an abstract delaysystem is delay-independently stabilizable. Furthermore, weprovide illustrative examples to verify the effectiveness of theproposed method.

Keywords— Stabilization, Partial differential equation,Time delay,C0 semigroup, Banach Lattice, Abstract CauchyProblem, Infinite dimensional system

I. INTRODUCTION

PARTIAL differential equations arise from manyphysical, chemical, biological, thermal, and

fluid systems which are characterized by both spatialand temporal variables [1]-[5]. Fig. 1 illustratesthat the time evolution of a solution of a partialdifferential equation is depending on both spatialand temporal variables.

Time delays also arise in many dynamical sys-tems because, in most instances, physical, chemical,biological, and economic phenomena naturally de-pend not only on the present state but also on somepast occurrences [6]-[8]. Fig. 2 illustrates that the

T. Hashimoto is with the Department of Systems Innovation,Graduate School of Engineering Science, Osaka University,1-3 Machikaneyama, Toyonaka, Osaka 560-8531, Japan.Corresponding author’s email: [email protected] [email protected]

Fig. 1. A schematic view of the time response of a partial differentialequation.

time evolution of the solution of a delay differentialequation is depending on both the present andpast solutions. The importance of the control ofpartial differential equations and delay differentialequations is well recognized in a wide range ofapplications. Hence, this paper examines the sta-bilization problem of partial differential equationswith time delays.

Partial differential equations and delay differen-tial equations are known to be infinite-dimensionalsystems, while ordinary differential equations arefinite-dimensional systems. The control of infinite-dimensional systems is a challenging problem at-tracting considerable attention in many researchfields. It has been recognized that semigroups havebecome important tools in infinite-dimensional con-trol theory over the past several decades. The semi-group method is a unified approach to addressingsystems that include ordinary differential equations,partial differential equations, and delay differen-tial equations. The behaviors of many dynamicalsystems including infinite-dimensional systems andfinite-dimensional systems can be characterized bysemigroup theory. The recent well-developed theory

INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES

Issue 3, Volume 4, 2010 212

Fig. 2. A schematic view of the time response of a delay differentialequation.

in such a framework has been accumulated in sev-eral books [9]-[13]. In this paper, using semigrouptheory, we introduce the concept of an abstractdelay system that can be used to describe thebehavior of a wide class of dynamical systems.Fig. 3 shows that an abstract delay system is ageneralized model that includes ordinary differentialequations, partial differential equations, and delaydifferential equations. From this point of view, thecontrol method proposed here for an abstract delaysystem is advantageous for its applicability to a wideclass of dynamical systems.

The linear quadratic control problem for an ab-stract delay system has been studied in [15]-[17].Furthermore, theH1 control problem for such asystem has been examined in [18]. The problemsaddressed in [15]-[18] have been reduced to findinga solution of the corresponding operator Riccatiequation in Hilbert spaces. The feedback stabiliz-ability of an abstract delay system on a Banachspace has been investigated in [19]. The analytic

Fig. 3. An abstract delay system is a generalized model.

approach in [19] is based on the compactness ofBanach spaces, while the problem in [15]-[18] isformulated in Hilbert spaces to make use of theproperties of the inner product.

In this paper, we study the stabilization problemof an abstract delay system on a Banach lattice,which is a Banach space supplied with an orderrelation. To tackle this problem, we take advantageof the properties of a non-negativeC0 semigroup ona Banach lattice. The objective of this paper is topropose a stabilization method for an abstract delaysystem on a Banach lattice. We derive a sufficientcondition under which an abstract delay system isdelay-independently stabilizable. Furthermore, weprovide illustrative examples to verify the effective-ness of the proposed method.

This paper is organized as follows. Some notationand terminology are given in Sec. II. The systemconsidered here is defined in Sec. III. Moreover,Sec. III is devoted to the introduction of a stabilitycriterion for an abstract delay system on a Banachlattice. The main results are provided in Sec. IV.In Sec. IV, we study the control design problemfor an abstract delay system on the basis of thestability criterion provided in Sec. III. Then, wederive a sufficient condition for the stabilization ofan abstract delay system under the assumption thatthe system has a non-negative delay operator. Fur-thermore, we provide illustrative examples to verifythe effectiveness of the proposed method. Finally,some concluding remarks are given in Sec. V.

II. NOTATION AND TERMINOLOGY

Let R andR+ denote the sets of real numbersand non-negative real numbers, respectively. LetN+denote the set of positive integers. LetX be aBanach space endowed with the operator normk�k.Let L(X;Y ) denote the set of all bounded linearoperators from a Banach spaceX to another BanachspaceY . Let L(X) be defined byL(X;X). LetId 2 L(X) denote the identity operator onX.

Definition 1: A family (T (t))t�0 of bounded lin-ear operators on a Banach spaceX is called aC0semigroup if all the following properties hold:

(i) T (0) = Id.(ii) T (t+ s) = T (t)T (s) for all t; s 2 R+.

(iii) The orbit mapst 7! T (t)x are continuousmaps fromR+ into X for everyx 2 X.



Definition 2: Let (T (t))t�0 be aC0 semigroup ona Banach spaceX and letD(A) be the subspace ofX defined asD(A) := �x 2 X : limh&0 1h(T (h)x� x) exists

� :For everyx 2 D(A), we defineAx := limh&0 1h(T (h)x� x):The operatorA : D(A) � X ! X is calledthe generator of the semigroup(T (t))t�0. In thefollowing, let (A;D(A)) denote the operatorA withdomainD(A).

Definition 3: Let (A;D(A)) be the generator ofa C0 semigroup(T (t))t�0.!0(A) := inff! 2 R : 9M > 0 such thatkT (t)k �Me!t;8t 2 R+gis called the semigroup’s growth bound.

Definition 4: Let (A;D(A)) be a closed operatoron a Banach spaceX. The set�(A) := f� 2 C : �Id �A is bijectivegis called the resolvent set ofA, and the set�(A) := C n�(A)is called the spectrum ofA. For � 2 �(A),R(�;A) := (�Id �A)�1is called the resolvent ofA at �.s(A) := sup fReal part of � : � 2 �(A)gis called the spectral bound ofA.

Definition 5: A C0 semigroup (T (t))t�0 withgenerator(A;D(A)) is said to be uniformly expo-nentially stable if!0(A) < 0.

Definition 6: A Banach spaceX is called a Ba-nach lattice ifX is supplied with an order relationsuch that all the following conditions hold:

(i) f � g ) f + h � g + h for all f; g; h 2 X.(ii) f � 0 ) �f � 0 for all f 2 X and� 2 R+.

(iii) jf j � jgj ) kfk � kgk for all f; g 2 X.Definition 7: A C0 semigroup (T (t))t�0 on a

Banach latticeX is said to be non-negative if0 � x 2 X ) 0 � T (t)x; for all t � 0:An operatorT (x) 2 L(X) on a Banach latticeX isalso said to be non-negative ifT (x) � 0 whenever0 � x 2 X.

III. PRELIMINARIES

In this section, we introduce the concept of anabstract delay system that can be used to describethe behavior of a wide class of dynamical systems.For a Banach spaceY and a constant� 2 R+,let C([��; 0℄; Y ) denote the set of all continuousfunctions with domain[��; 0℄ and rangeY . For aBanach spaceX := C([��; 0℄; Y ), let� 2 L(X;Y )be a delay operator, and let(B;D(B)) be the genera-tor of aC0 semigroup onY . With these notations, anabstract delay system is described by the followingequation with an initial function' : [��; 0℄! Y :� _x(t) = Bx(t) + �(x(t� � )) for t � 0;x0 = ' 2 X: (1)

A continuous functionx : [��;1) ! Y is called asolution of (1) if all the following properties hold:

(i) x(t) is right-sided differentiable att = 0 andcontinuously differentiable for allt > 0.

(ii) x(t) 2 D(B) for all t � 0.(iii) x(t) satisfies (1).

Let Cr be the set of allr-times continuously dif-ferentiable functions. Let(A;D(A)) be the corre-sponding delay differential operator onX definedby Af := _f , (2)D(A) := ff 2 C1([��; 0℄; Y ) :f(0) 2 D(B)

and _f (0) = Bf(0) + �(f(�� ))g:Lemma 1 ([9]): The operator(A;D(A)) in (2)

generates aC0 semigroup(T (t))t�0 on X.Lemma 2 ([9]): If ' 2 D(A), then the functionx : [��;1)! Y defined byx(t) := � '(t) if � � � t � 0;[T (t)'℄ (0) if 0 < t;

is the unique solution of (1).In the subsequent discussion, we assume that each

Banach spaceX, Y in (1) is a Banach lattice.Lemma 3 ([9]): If B generates a non-negativeC0 semigroup onY and the delay operator� 2L(X;Y ) is non-negative, then theC0 semigroup(T (t))t�0 generated by(A;D(A)) in (2) is also non-

negative, and the following equivalence holds:s(A) < 0, s(B + �) < 0:Lemma 4 ([9]): Assume that(T (t))t�0 is a non-

negativeC0 semigroup with generator(A;D(A))on X. Then, s(A) = !0(A):



The following proposition directly follows fromLemmas 3 and 4.

Proposition 1: Under the assumption thatB gen-erates a non-negativeC0 semigroup onY and thedelay operator� 2 L(X;Y ) is non-negative, theC0semigroup(T (t))t�0 generated by(A;D(A)) in (2)is uniformly exponentially stable if and only if thespectral bounds(B + �) < 0.

Note that the equality in Lemma 4 might nothold in general. This means that aC0 semigroup(T (t))t�0 generated by(A;D(A)) is not neces-sarily uniformly exponentially stable even if thespectral bound is negative, i.e.,s(A) < 0. It canbe seen from Proposition 1 that the non-negativityassumption enables us to determine the stability ofan abstract delay system simply by examining thespectral bound.

IV. STABILIZATION OF ABSTRACTDELAY SYSTEMS

Let (B;D(B)) be the generator of aC0 semigroupon a Banach latticeY . For a Banach latticeX :=C([��; 0℄; Y ), let � 2 L(X;Y ) be a delay operator.In this section, we consider the stabilization problemof an abstract delay system described by� _x(t) = Bx(t) + �(x(t� � )) + Cu(t);x0 = ' 2 X; (3)

whereu(t) : t 2 R+ ! Y is the control input, and(C;D(C)) is the generator of aC0 semigroup onY .Assumption 1:� is assumed to be non-negative.

Next, we consider the feedback stabilization prob-lem of (3). Letu(t) be given byu(t) = Kx(t); (4)

where(K;D(K)) is the generator of aC0 semigroupon Y .

Definition 8: System (3) is said to be delay-independently stabilizable if there existsu(t) in (4)such that the equilibrium pointx = 0 of the result-ing closed-loop system is uniformly exponentiallystable.Now, we state the following theorem.

Theorem 1:If there existsK such that(B+ CK)generates a non-negativeC0 semigroup ands(B + CK + �) < 0 (5)

is satisfied, then system (3) is delay-independentlystabilizable.

Proof of Theorem 1: Under the assumption that� is non-negative and(B + CK) generates a non-negativeC0 semigroup, we see from Proposition 1that the resulting closed-loop system is uniformlyexponentially stable ifs(B + CK + �) < 0 holds.

An illustrative example is shown below. Letbe a constant. We consider the following partialdifferential equation with a time delay, defined fort � 0; x 2 [0; `℄; s 2 [��; 0℄, as�z(x; t)�t = �2z(x; t)�x2 � d(x)z(x; t)+ b(x)z(x; t� � ) + u(x; t); (6)

with the Dirichlet boundary conditionz(0; t) = z(`; t) = 0 for all t � 0; (7)

and with the initial conditionz(x; s) = h(x; s): (8)

This equation can be interpreted as a model forthe growth of a population in[0; `℄. z(x; t) is thepopulation density at timet and spacex. Theterm�2z(x; t)=�x2 describes the internal migration.Moreover, the continuous functionsd(x) and b(x)represent space-dependent death and birth rates,respectively.� is the delay due to pregnancy. Letd(x) andb(x) be given as follows:d(x) = 1 + os(8�x=`); (9)b(x) = 1 + 2 sin(�x=`): (10)

Let u(x; t) be given byu(x; t) = �k(x)z(x; t): (11)

To rewrite system (6) as an abstract delay system,we introduce the spacesY := C[0; `℄ and X :=C([��; 0℄; Y ). Moreover, we define the followingoperators� := d2dx2 ; (12)D(�) := �f 2 C2[0; `℄ : f(0) = f(`) = 0 ; (13)B := ��Md �Mk; D(B) := D(�); (14)� := Mb�� 2 L(X;Y ); (15)

whereMd, Mk, andMb are the multiplication oper-ators induced byd(x), k(x), andb(x), respectively.�� : X ! Y denotes the point evaluation int 2 [��; 0℄. We see from (14) and (15) that system(6) can be rewritten as an abstract delay system (1).



It is shown in [9] that� generates a non-negativeC0 semigroup. Sincee�t(Md+Mk) is non-negative,we see from the Trotter product formula [9] thatB in (14) generates a non-negativeC0 semigroup.Moreover, we see from (10) and (15) that� is anon-negative operator. Consequently, it follows fromLemmas 3 and 4 that!0(�+Mb�Md �Mk) = s(�+Mb�Md�Mk):

In the following, we designK such thats(B + �) < 0is satisfied. LetÆ be defined byÆ := infx2[0;`℄ (d(x) + k(x)� b(x)) :If Æ > 0, then the operator(�+Mb�Md�Mk+Æ)is dissipative. Hence, we obtain!0(� +Mb �Md �Mk) < �Æ:This condition shows that ifb(x)� d(x)� k(x) < 0; for all x 2 [0; `℄;then a solution of (6) is uniformly exponentiallystable. For example, if we designk(x) = 2� d(x) + b(x); (16)

then system (6) is delay-independently stable.The effectiveness of controller (16) is verified by

numerical simulations. To solve equation (6) usinga numerical algorithm, we must discretize equation(6) into the finite difference equation. The Crank-Nicolson method [21] is a finite difference methodused for numerically solving a partial differentialequation. It is a second-order method in time andspace, and is numerically stable. For the sake ofcompleteness, a brief description of the Crank-Nicolson method applied to this problem is providedin the subsequent discussion.

We divide the space and time intoM 2 N+steps andN 2 N+ steps, respectively. This meansthat each step size is given by�x := `=(M � 1)and �t := tf=(N � 1), where tf denotes theterminal time of the simulation. By means of thediscretization,z(x; t); (0 � t � tf ) can be describedas zi;j (i = 1; � � � ;M; j = 1; � � � ; N), where thesubscriptsi andj denote the space and time, respec-tively. For other variables, we adopt such notation

without explanation. Note that equation (6) can bediscretized as follows:zi;j+1 � zi;j�t = 12 �zi+1;j+1 � 2zi;j+1 + zi�1;j+1�x2+ zi+1;j � 2zi;j + zi�1;j�x2 � � fd(xi) + k(xi)g zi;j+ b(xi)zi;j��=�t (17)

Taking Dirichlet boundary condition (7) into ac-count, we see that (17) yields the following equa-tion:Azj+1 = (C �D�K)zj +Bzj��=�t; (18)

where letr be defined byr := �t2�x2 ;and letA, B, C, D 2 RM�M, and zj 2 RM bedefined asA := 2666666664 1 + 2r �r 0 0 � � � 0�r 1 + 2r �r 0 . . .

...0 �r . . . . . . . . . 00 . . . . . . . . . �r 0...

. . . 0 �r 1 + 2r �r0 � � � 0 0 �r 1 + 2r 3777777775 ;C := 2666666664 1� 2r r 0 0 � � � 0r 1 � 2r r 0 . . ....0 r . . . . . . . . . 00 . . . . . . . . . r 0

.... . . 0 r 1� 2r r0 � � � 0 0 r 1 � 2r 3777777775 ;D := �t2666666664 d(x1) 0 0 0 � � � 00 d(x2) 0 0 . . .

...0 0 . . . . . . . . . 00 . . . . . . . . . 0 0...

. . . 0 0 . . . 00 � � � 0 0 0 d(xM ) 3777777775 ;K := �t2666666664 k(x1) 0 0 0 � � � 00 k(x2) 0 0 . . ....0 0 . . . . . . . . . 00 . . . . . . . . . 0 0

.... . . 0 0 . . . 00 � � � 0 0 0 k(xM) 3777777775 ;



B := �t2666666664 b(x1) 0 0 0 � � � 00 b(x2) 0 0 . . ....0 0 . . . . . . . . . 00 . . . . . . . . . 0 0

.... . . 0 0 . . . 00 � � � 0 0 0 b(xM) 3777777775 ;zj := 266664 z1;jz2;j

...zM�1;jzM;j 377775 :Consequently, it follows from (18) thatzj+1 = A�1f(C�D�K)zj +Bzj��=�tg: (19)

Therefore, we see thatzj(t)(j=1;�� ;N) is calculatedrecursively by (19), for a given initial stateh(x; s).The parameters employed in the numerical simula-tions are listed in Table I.M 101 [steps]N 101 [steps]` 10 [m]tf 10 [sec]�x 0:1 [m]�t 0:1 [sec]� 1 [sec]h(x;s) j sin(2�x=`)j for all s 2 [��; 0℄

TABLE ITHE PARAMETERS EMPLOYED IN THE NUMERICAL SIMULATIONS.

Fig. 4 shows that the initial stateh(x; s) is givenby h(x; s) = j sin(2�x=`)j for all s 2 [��; 0℄. Fig. 5shows the space-dependent death and birth rates,d(x) andb(x), respectively. Moreover, we see fromFig. 5 that the conditionb(x)� d(x)� k(x) < 0 issatisfied for allx 2 [0; `℄.

The simulation results with the proposed methodare shown in Figs. 6-7. Fig. 6 shows the freeresponse ofz(x; t) without control. We see that thepopulation density increases over time. Fig. 7 showsthe time response ofz(x; t) in which the controller(16) is employed. We see that the population densityis uniformly stabilized atz = 0. The figures revealthe effectiveness of controller (16).

In the following, we consider system (6) with theNeumann boundary condition�z(0; t)�x = �z(`; t)�x = 0 for all t � 0;

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

x

z(x,s

),(−

τ≤

s≤

0)

Fig. 4. The initial conditionh(x; s).0 2 4 6 8 10

−2

0

2

4

6

x

d(x)b(x)b(x)−d(x)−k(x)

Fig. 5. The space-dependent death and birth rates.

and with the same initial condition as (8). In thiscase, we can also rewrite the system as an abstractdelay system by introducing the following opera-tors:� := d2dx2 ;D(�) := �f 2 C2[0; `℄ : dfdx(0) = dfdx(`) = 0� ;B := ��Md �Mk; D(B) := D(�);� := Mb�� 2 L(X;Y );Similarly, we see that ifb(x)� d(x)� k(x) < 0; for all x 2 [0; `℄is satisfied, then the system is uniformly exponen-tially stable. To perform numerical simulations for



05

10

0

5

100

100

200

300

tx

z(x,

t)

Fig. 6. Free response ofz(x; t) without control with the Dirichletboundary condition.

05

10

0

5

100

0.5

1

tx

z(x,

t)

Fig. 7. Time history ofz(x; t) controlled by (16) with the Dirichletboundary condition.

system (6) with the Neumann boundary condition,we also obtain the discretized equation as in (19),whereA andC are changed as follows:A = 2666666664 1 + 2r �2r 0 0 � � � 0�r 1 + 2r �r 0 . . .

...0 �r . . . . . . . . . 00 . . . . . . . . . �r 0...

. . . 0 �r 1 + 2r �r0 � � � 0 0 �2r 1 + 2r 3777777775 ;

05

10

0

5

100

100

200

300

tx

z(x,

t)

Fig. 8. Free response ofz(x; t) without control with the Neumannboundary condition.

05

10

0

5

100

0.5

1

tx

z(x,

t)

Fig. 9. Time history ofz(x; t) controlled by (16) with the Neumannboundary condition.C = 2666666664 1 � 2r 2r 0 0 � � � 0r 1� 2r r 0 . . .

...0 r . . . . . . . . . 00 . . . . . . . . . r 0...

. . . 0 r 1 � 2r r0 � � � 0 0 2r 1� 2r 3777777775 :Likewise,zj(t)(j=1;�� ;N) is calculated recursively by(19), for a given initial stateh(x; s).

The simulation results with the Neumann bound-ary condition are shown in Figs. 8-9. Fig. 8 showsthe free response ofz(x; t) without control. Wesee that the population density increases over time.



Fig. 9 shows the time response ofz(x; t) in whichthe controller (16) is employed. We see that thepopulation density is uniformly stabilized atz = 0.The figures also reveal the effectiveness of controller(16).

V. CONCLUSION

In this study, the stabilization problem of a partialdifferential equation with a time delay was exam-ined using semigroup theory. We first introducedthe concept of an abstract delay system that canbe used to characterize the behavior of a wideclass of dynamical systems. Next, we examined thestabilization problem of an abstract delay systemon a Banach lattice on the basis of the propertiesof a non-negativeC0 semigroup. In Sec. IV, wederived a sufficient condition for the stabilizationof an abstract delay system under the assumptionthat the system has a non-negative delay operator.An abstract delay system is a generalized modelthat includes ordinary differential equations, partialdifferential equations, and delay differential equa-tions. From this point of view, the control methodproposed here for an abstract delay system is ad-vantageous for its applicability to a wide class ofdynamical systems. Illustrative examples revealedthat the stabilization method proposed here is use-ful for designing a controller to stabilize a partialdifferential equation with a time delay.

ACKNOWLEDGEMENT

This research was partially supported by theMinistry of Education, Culture, Sports, Science andTechnology, Grant-in-Aid for Young Scientists (B),22760318.

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Tomoaki Hashimoto received the B. Eng., M.Eng., and D. Eng. degrees from the TokyoMetropolitan Institute of Technology, Tokyo,Japan, in 2003, 2004, and 2007, respectively,all in aerospace engineering. He is currentlyan Assistant Professor at the Department ofSystems Innovation, Graduate School of Engi-neering Science, Osaka University, Japan.



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INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND … · 2010-09-28 · The importance of the control of partial differential equations and delay differential ... The recent well-developed