438 ieee transactions on automatic control on …/file/ieeeac0409.pdf · on a banach space and the...

438 IEEE TRANSACTIONS ON AUTOMATIC CONTROL

On Feedback Stabilizability of Linear SystemsWith State and Input Delays in Banach Spaces

Said Hadd and Qing-Chang Zhong, Senior Member, IEEE

Abstract—The feedback stabilizability of a general class of well-posed linear systems with state and input delays in Banach spacesis studied in this paper. Using the properties of infinite dimen-sional linear systems, a necessary condition for the feedback sta-bilizability of delay systems is presented, which extends the well-known results for finite dimensional systems to infinite dimensionalones. This condition becomes sufficient as well if the semigroup ofthe delay-free system is immediately compact and the control spaceis finite dimensional. Moreover, under the condition that the Ba-nach space is reflexive, a rank condition in terms of eigenvectorsand control operators is proposed. When the delay-free state spaceand control space are all finite dimensional, a very compact rankcondition is obtained. Finally, the abstract results are illustratedwith examples.

Index Terms—Banach spaces, feedback stabilizability, Hautuscriterion, rank condition, regular systems, time-delay systems.

I. INTRODUCTION

A. Literature Review and Motivation

D ELAY-DIFFERENTIAL systems arise in the studyof many problems with theoretical and practical im-

portance. The behavior of these systems was studied in theseventies, especially in the book by Hale [17], where thesemigroup theory was applied to reformulate delay systems asdistributed parameter systems (see also Bellman and Cooke[4] and Halanay [16] for an introduction to this approach).The recent well-developed theory for such systems is gatheredin several books e.g., Bátkai and Piazzera [2], Bensoussan etal. [3], and Hale and Verduyn Lunel [18]. More recently it isshown in [10], [11], [14] that delay systems form a subclass ofwell-posed and regular infinite dimensional linear systems inthe Salamon-Weiss sense [30]–[33], [37].

It has been recognized that the delay presence in the stateand/or input could induce bad performance (even instability)and complicate controller design and system analysis [39]. Thisfact makes the investigation of the stability of delay systems

Manuscript received June 22, 2007; revised March 14, 2008 and August 07,2008. Current version published March 11, 2009. This paper was presentedin part at the 46th IEEE Conference on Decision and Control, New Orleans,LA, December 2007 and at the 18th International Symposium on Mathemat-ical Theory of Networks and Systems (MTNS2008), July 2008. This work wassupported by the EPSRC, UK under Grant EP/C005953/1. Recommended byAssociate Editor G. Feng.

The authors are with the Department of Electrical Engineering and Elec-tronics, The University of Liverpool, Liverpool L69 3GJ, U.K. (e-mail: [email protected]; [email protected]).

Digital Object Identifier 10.1109/TAC.2009.2012969

very interesting and motivates many researchers to look for con-ditions guaranteeing stabilizability of such systems. By ana-lyzing the existing theory in this area one can note that the re-sults are obtained with some special techniques, which do notfit into a general theory as that developed for delay-free sys-tems. Also, most of the works have been devoted to feedbackstabilizability of state delay systems with a finite dimensionaldelay-free state space e.g., [4], [6], [20] or with an infinite di-mensional space [25]. The feedback stabilizability of state-inputdelay systems is investigated in [19], [22], [26]–[29], where thedelay-free state space is finite dimensional. It is shown by Ol-brot [26] that the feedback stabilizability of the state-input delaysystem

is equivalent to the condition

(1)where is the dimension of the delay-free system and

.We are interested in developing a more general approach to

the feedback stabilizability of state-input delay systems in Ba-nach spaces. We shall introduce a general class of feedback lawswhich stabilize the closed-loop system. This is based on the re-cent theory of regular linear systems [21], [31], [32], [37], andthe recent work on state-input delay systems in Banach spaces[11], [24], where it has been proved that systems with generaldistributed state, input and output delays can be reformulated asregular linear systems.

Notation and Problem Statement

Throughout this paper we use the following notation. For aBanach space , a real number , and a function

, the history function of isdefined as for and . We de-note by the space of all continuous functions from

to . For we denote by theLebesgue space of all -integrable functions[7], by the space of indefinite integrals of -in-tegrable functions, and by the Banach space of alllinear and bounded operators from a Banach space to anotherBanach space with .

In this paper we consider the state-input delay system

.(2)

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HADD AND ZHONG: FEEDBACK STABILIZABILITY OF LINEAR SYSTEMS 439

Here is the generator of a -semigroupon a Banach space and the delay operators

andare linear and bounded, where is a (control) Banachspace. The function is the history function of the func-tion and is the history function ofthe function . The initial conditions are

and , whichform the initial state of the system. If and are finitedimensional spaces then the operators and can take theexplicit form of Riemann-Stieltjes integrals

(3)

for and , whereand are functions

of bounded variations [3, p.249]. In this case the system (2) iswell-posed in the sense that it can be rewritten as a well-posedopen-loop system [11], [13].

In this paper is an arbitrary Banach space and the oper-ators and are not assumed to be of the form (3). In thiscase, it is shown in [11] that the system (2) can be reformu-lated as a well-posed control system on the state space

in the sense of [35] ifand are observation operators of regular linear systems gov-erned by the left shift semigroup (see [13] for more details onsuch systems). These conditions on and are always satis-fied if and are finite dimensional, see [13].

B. Overview of the Major Contributions

After discussing left shift semigroups, transforming thesystem (2) into a well-posed open-loop system on andbased on [11] and characterizing the structure of feedbackstabilizable operators, we introduce a necessary condition forthe stabilizability of the system (2), using the generalization ofthe Hautus criterion for the stabilizability of distributed-param-eter linear systems [38, Proposition 3.5]. This generalizes thecondition (1) to the case of general Banach spaces with generaldelay operators (see Theorem 7). This necessary conditiondoes not require any regularity on the semigroup or on thegeometry of the state and control spaces. However, to showthe sufficiency of this condition, we need to assume that thecontrol space is finite dimensional and the semigroupis compact for , i.e., immediately compact. Under theseconditions, the delay system (2) is feedback stabilizable if andonly if

holds for all with , which belongs to the finite un-stable spectrum of the operator , where

is defined aswith . Furthermore, a rank condition, de-

rived from the above condition, is given in terms of eigenvectorsand control operators in the case with a reflexive Banach space

. This extends the result in [25] where a rank condition wasgiven for systems with state delays only. When the delay-freestate space and control space are finite dimensional, a compact

rank condition equivalent to that of Olbrot [26] is obtained. Weillustrate our abstract results with examples involving a singledelay, multiple delays, distributed delays and elliptic operatorsin bounded domains of the Euclidean space.

C. Organization of the Paper

The organization of the paper is as follows. In Section IIwe recall some preliminaries about well-posed and regularinfinite dimensional systems. In Section III after we discussleft shift semigroups, state the main assumptions and refor-mulate the delay system (2) to an infinite dimensional openloop system, we present the definition of feedback stabiliz-ability and characterize the structure of feedback stabilizableoperators. Section IV is devoted to state and prove the mainresults on feedback stabilizability of the delay system (2)and in Section V we present some examples. Conclusions aremade in Section VI, followed by two appendices, in which thefrequently-cited known results are gathered.

II. PRELIMINARIES: WELL-POSED AND REGULAR INFINITE

DIMENSIONAL LINEAR SYSTEMS

Here, we briefly recall the framework of infinite dimensionalwell-posed and regular linear systems in the Salamon-Weisssense from [31], [32], [34]–[36], [33], [37].

A. Basic Concepts

Let be the generator of a -semigroupon a Banach space . We denote by

the resolvent set of , i.e., the set of all suchthat is invertible. The spectrum of is by definition

. We define the resolvent operator of as. The domain endowed

with the graph norm , is aBanach space. We also define the normfor some . The completion of with respect to thenorm is a Banach space denoted by , which is calledthe extrapolation space associated with and . Moreover, thecontinuous injection holds. The semigroup can benaturally extended to a strongly continuous semigroup

on , of which the generatoris the extension of from to ; see [8] for more details.

The pair is called a controlsystem on , if is a -semigroup on and

, is a linear bounded operator satis-fying, for and ,

(4)

See [35], [36] for more details on the maps . By the rep-resentation theorem due to Weiss [35, Theorem 3.9], there ex-ists a unique operator , called an admissiblecontrol operator for (or ), such that for any and

,

(5)

where the integral exists in . We say that is repre-sented by the control operator . Each control system



with the representing control operator is completely deter-mined by an abstract differential equation of the form

(6)

It is well-posed in the sense that it has a unique strong solution,called the state trajectory with initial state , given by

(7)

An operator is called an admissible obser-vation operator for (or ) if

(8)

holds for any and for some constants and. For simplicity, denote

(9)

For , the linear system

(10)

is well-posed in the sense that the observation function canbe extended to a function in . In fact, from (8), themap

(11)

can be extended to a linear bounded operator from to, called the extended output map. Then we can set

for any and almost every . Ifwe define

(12)

for and , and 0 for , thenis a family of bounded linear operators from to .We say that is an observation systemon , , and the observation equation in (10) satisfies

for almost every (a.e.) and all. Note that is extended in the abstract way. In order

to obtain a pointwise representation in terms of the observationoperator , consider the Yosida extension[34] of with respectto defined as

(13)

Let be endowed with the graph norm. For ,

The right hand side goes to 0 when , according toLemma 3.4 in [8, p.73]. Hence, andfor any . Note that the definition of does not requirethat . However, if this is the case,and for all and a.e. , (see [34,Theorem 4.5]).

For , denote its convolution with the semi-group as

Then, we have the following result [9, Proposition 3.3].Proposition 1: Assume and .

Define

Then and

for and a constant , which is independent ofand approaches 0 as .

B. Well-Posed and Regular Systems

Here, we focus on a control system represented byand an observation system represented by . We say thatthe linear system

(14)

with the initial state , is well-posed on , , if thereexists a family of bounded linear operators from

to , satisfying

(15)

for , and . For moredetails on operators we refer to [36], [37]. In this case wealso say that the quadruple is a well-posedsystem on , , .

Let be the operator of truncation to , that isfor and zero otherwise. The op-

erators are compatible in the sense thatfor and zero otherwise. This property providesa unique operator ,called the extended input-output map, which satisfies

for . We recall from[36, Theorem 3.6] that there exist and a unique boundedand analytic functionsuch that

if . Here and are the Laplace transform of and, respectively. Here, is called the transfer function of .



We say that the well-posed system is regular (with zerofeedthrough) if the limit

exists in for the constant input .The following definitions will be used throughout this paper.Definition 1: Let and be the admissible control and ob-

servation operators issued from and , respectively.We say that the triple generates a regular system

if there exists a bounded operatorsuch that is regular on , ,

.Definition 2: Let be a regular system with input-output

operators . An operator is called an admissiblefeedback for if has uniformly bounded inverse.

Note that in the case of Hilbert spaces and one canuse transfer functions instead of input-output operators for thedefinition of admissible feedback operators [36].

We now state a very general perturbation theorem due toWeiss in Hilbert spaces [37] and due to Staffans in general Ba-nach spaces [32, Chap.7].

Theorem 3: Assume that generates a regular systemwith admissible feedback operator . Then the operator

with the sum defined in generates a -semigroup on. Moreover, for a.e. , and, for

,

(16)

III. FEEDBACK STABILIZABILITY OF SYSTEM (2)

A. Left Shift Semigroups and Main Assumptions

Here, we follow Section II-A to present two left shift semi-groups and then state the main assumptions on delay operators

and of the system (2). Define

(17)

and, similarly, . It is well known that gener-ates the left shift semigroup

(18)

for and . Similarly, wehave . The pair with

for the control function is a control systemon and , which is represented by the (strictly)unbounded admissible control operator [13]

(19)

where is the generator of the extrapolation semigroupassociated with and is the linear bounded operator

In fact, is the adjoint of the delta operator at zero. Similarly,we can have operators and .

For the control function ofwith for a.e. , the state trajectory of

is the history function of given by

Similarly, for the control system represented by ,we have

with for a.e. .According to (11), if is an admissible observation operator

of , we define

and then according to (12). Similarly, we define andfor operator . The input-output operator associated with

and will be defined according to (15), replacingwith . The input-output operator is similarly definedfor operator .

From now on we assume that the operators and in system(2) satisfy:

(A1) and the triple gen-erates a regular system , on

.(A2) and the triple gen-erates a regular system , on

.Remark 4: Here, we give an example of operators and

that satisfy the above assumptions. Let andbe functions of bounded variations such

that the total variations of and approach 0 when the intervalgoes to 0. If the operators and take the form of

for and , where andare not necessarily finite dimensional, then satisfies (A1) and

satisfies (A2); see [13].

B. Reformulation of System (2)

We recall from [11], [15] how the delay system (2) can bereformulated as a linear distributed parameter system on someappropriate Banach spaces.



Consider the Banach space en-

dowed with the norm . It is well-known

(see e.g., [2]) that the delay system (2) with is closelyrelated to the following linear operator

(20)

It is not difficult to show that if and only if

(21)

for and some constants and [9],

where is defined as

otherwise.(22)

According to the Hölder inequality, (21) implies that

(23)

for and with .

Hence, the operator generates a strongly continuous semi-group on , according to [1].

The following remark is important for the proof of the mainresults in this paper.

Remark 5: From the discussion above, we have seen that forthe estimate (23) holds with as .

If in addition the operator generates an immediately compactsemigroup on (i.e., compact for ), then generatesan eventually compact semigroup on (i.e., compact for

, with in this paper) [23].In order to consider the well-posedness of the state-input

delay systems (2), define the operators

(24)

and

(25)

It has been shown in [11, Theorem 3.1] that when satisfies thecondition (23) and , the operator generatesthe following -semigroup on ,

(26)

with given by

(27)

for and , where is the extendedoutput map associated with and . Furthermore, we have thefollowing result from [11, Proposition 3.5] and [15, Proposition3.2].

Proposition 2: Assume that satisfies (23) and satisfiesthe condition (A2). Then the operator defined in (25) is anadmissible control operator for defined in (24).

According to (5), the control maps associated with the controloperator are given by

for and . According to [11, Proposition3.2], this is equivalent to

(28)

for and , where is the extendedinput-output map of the regular system . Now, the open loopsystem

(29)

is well-posed according to Proposition 2. Combining (26) and(28), one can see that the state trajectory of the system (29), forthe initial state , is given by

according to (7). This is the same as the state trajectory of thesystem (2), see [11, Proposition 3.3]. This gives a natural con-nection between the systems (2) and (29). Letand be sufficiently large. According to [13], the Laplace trans-form of in is equal to and that of in

is exactly . Hence, by taking the Laplace transform onboth sides of (28), we have

(30)

C. The Structure of Feedback Stabilizable Operators

According to the reformulation of the delay system above,Proposition 2 and [38, Definition 2] (see also Appendix B), weintroduce the following definition.

Definition 6: Let the conditions (A1) and (A2) be satisfied.We say that the state-input delay system (2) is feedback stabiliz-able if there exists an operator such that:

i) The triple generates a regular system on, , .

ii) The identity operator is an admissible feed-back operator for .



iii) The semigroup generated by the operatorendowed with the domain

is exponentially stable.In this case we say that the operator stabilizes the delay

system (2). The following result shows an explicit structure ofsuch operators .

Proposition 3: Let the conditions (A1) and (A2) be satisfied.If stabilizes the delay system (2) then it isof the form

with , , and with thetriple generating a regular linear system withas an admissible feedback operator.

Proof: Assume that stabilizes thesystem (2), then at least we have . Decom-pose into with and

. Then, for any and ,we have

for . This means .In order to characterize the operators , decom-

pose into with

From [1], the operator is the generator of the -semigroup

(31)

on for , where the operators are defined in (22).Thus, the fact that implies that .According to the invariance of admissibility of observation op-erators [12, Theorem 3.1], which is reproduced in Appendix Afor the reader’s convenience, there is . Thisallows us to concentrate on operators . Using (31),one can easily show that an operator if and only ifit is of the form

(32)

with and . This shows that theoperators stabilizing the open loop system haveat least the following form

with , and .Now prove the second part. Let be the regular system

generated by the triple . According to [11, The-orem 5.1], at least should be an observation operator of theregular system generated by thetriple on and such thatis an admissible feedback operator. Hence, . Thisends the proof.

IV. MAIN RESULTS

A. A Necessary Condition

For each , we redefine

(33)

for the system (2). From [2, p.56], we have

(34)

with defined in (20). In view of this equivalence, iscalled the characteristic operator. The result below gives a nec-essary condition for the feedback stabilizability of the system(2).

Theorem 7: Assume the conditions (A1) and (A2) are satis-fied. The feedback stabilizability of the delay system (2) impliesthe existence of such that

(35)

for any with .Proof: Denote

According to the generalized Hautus criterion [38, Prop. 3.5],which is reproduced in Appendix B for the reader’s conve-nience, the feedback stabilizability of system (29), and henceof (2), implies that there exists such that

for any with . This indicates, for anyand hence for any , there existand such that, for ,

(36)

Let with and let be fixed1.According to (30), we have

1This is because � does not always belong to �� .



Now, by (36), we obtain

(37)

It is clear that

which implies that

(38)

and, as a result

��

�

��

��

�

��

� ��

�

��

��

�

��

��

�

��

��

��

��

�

��

�

(39)

Here, the identity was used. Note that

(40)

By combining (37), (39) and (40), we have

By using the expression of in (20), we obtain

(41)

and

The last identity implies , or

Similarly, we have

As we are only interested in any , substitute andwith , into (41), then

Hence, the condition (35) holds. This completes the proof.

B. A Necessary and Sufficient Condition

Theorem 8: Assume the conditions (A1) and (A2) are satis-fied and the control space is finite dimensional. Moreover, as-sume that is compact for . The delay system (2) isfeedback stabilizable if and only if

(42)

holds for all with .Proof: The necessity has been proved in Theorem 7. Only

the sufficiency will be shown here. At first, we will transformthe system (2) into an equivalent system different from (29).

For the system (2), take and, respectively, and introduce the Banach

space

and the operator

which generates the -semigroup

on . Since is compact for and is compact,is compact for . Define

where is assumed to be smooth. Then the system (2) is con-verted into

(43)

Here the new control operator is defined by. Denote

and define



with

Then, the operator

is the generator of the following semigroup on :

Note that on . In particular, the assumptions(A1) and (A2) show that . Hence generates astrongly continuous semigroup on , according to [2, p.67], [9].Define

which is bounded, then the delay system (43), and hence (2), isequivalent to the open loop system

(44)

where the state is defined as .We have now converted the feedback stabilizability of thesystem (43) or system (2) to that of the above system, whereis linear bounded.

Now let us characterize the spectrum of . According to [2,p.56], we have

with

Hence, by (34) we obtain

By assumption that is compact for generatesan eventually compact semigroup on , according to Remark 5.As a result, the following unstable set is finite:

Take any and let . The condition (42)implies the existence of and such that

, which yields

(45)

Furthermore, take any and such that anddefine

(46)then

Substitute (46), after rearrangement, into (45), then

Putting the last two identities together, we have

where . This is equivalent to

which means that the system (44), or system (2), is feedbackstabilizable, according to [5, Theorem 1]. This ends the proof.

Remark 9: Denote by the adjoint space of . Then theadjoint of the characteristic operator defined in (33) is given by

(47)

where is the complex conjugate of . Using Theorem 7, [25,Prop.5.1] and the proof of Theorem 8 one can see that

(48)where

A simple computation shows that

and

(49)

Since

(50)

we have

(51)



Combining (49), (51) and (48), the following holds under theconditions of Theorem 8:

(52)

C. A Rank Condition for a Special Case in Banach Spaces

In addition to the assumptions in Theorem 8 we assume thatis a reflexive Banach space. As the control space is

finite dimensional, denote as

with is linear and bounded for all.

For , we define

with for and .As , we have

(53)

for and . We say that sat-isfies (A2) if the triple generates a regular linearsystem on , and . Naturally, satisfies (A2)if every satisfies (A2) for all .

We recall that if is compact for , then generatesan eventually compact semigroup on . In this case, the setof unstable eigenvalues can be denoted by

(54)

which is finite with finite algebraic multiplicity. Because of (34),for defined in (47) with , wecan set the dimension of as

(55)

and the basis of by .Throughout the following we denote the duality pairing be-

tween and by and the orthogonal space of a set inby

When is reflexive and , the adjoint space ofis the product space with satisfying

.Theorem 10: Assume that (i) and with

satisfy the conditions (A1) and (A2), respectively, (ii) the Ba-nach space is reflexive and the control space is finite dimen-sional and (iii) is compact for . The system (2) isfeedback stabilizable if and only if

(56)

with

......

...

(57)

for , given in (54).Proof: We only need to prove that the condition (52) is

equivalent to (56) under the extra conditions. Since

we have

...

and, as a result,

(58)

Now, assume that the condition (56) does not hold. This meansthat there exists such that

, i.e.,

...

...(59)

Now since is a basis of , we have

On the other hand, by using (58) and (59) one can obtain. This is a contradiction, which

shows that the equivalence (52) implies the rank conditions(56). The converse can be obtained in a direct way using (58)and the basis . This ends the proof.



D. A Rank Condition for Delay Systems WithFinite-Dimensional State (When Delay-Free) and ControlSpaces

Consider the delay system

(60)

where and are real matrices, andare matrices, and is the delay.

Here, and. Let the control function

for . We have

with

...

We note that the operator and satisfy the conditions (A1)and (A2), see [13, Remark 2]. Since is a finite dimensionalmatrix, is compact for . In this case,

with. The dimension of is for

and the basis of is .Denote the matrix formed by the basis as

According to Theorem 10, we have the following:Corollary 1: The system (60) is feedback stabilizable if and

only if

(61)

As this condition is obtained from (42), which is an extensionof (1), it should be equivalent to the condition (1) proposed in[26]. This is indeed true. In fact, we have

(62)

Assume that for each ,

(63)

which implies that has full column rank

and

On the other hand, without loss of generality (otherwise carryingout some elementary operations), assume that

which gives the following having rank :

Now with the partition of com-patible with , we have (62) as follows:

Since (61) holds, has full column rank . Multiply

the above with from the left, then

Since has full column rank , the leftmost big matrixin the above identity has full column rank as well. This meansthat has full column rank. In otherwords, the condition (63) holds.

V. EXAMPLES

We start from giving two examples which illustrate the resultof Corollary 1. In the third example we treat the case of sys-tems with multiple delays. We end this section by an exampleof distributed delay systems governed by an uniformly ellipticdifferential operator of second order on a bounded domain ofthe Euclidean space.

Example 1: Consider the system (60) with

Then, and . Hence

which implies that . Now (61) is re-duced to

Hence, the delay system is feedback stabilizable if and only if. In other words, the system

is not stabilizable if



Example 2: Consider the system (60) with

This was studied in [26] (with a typo in ). Here

Now we have

Thus, is stabilizable but is not stabilizable.Example 3: Consider the state-input delay system

(64)

Here the operator generates an immediately compact semi-group on a reflexive Banach space , the family

, and for each ,

where is linear and continuous for any. As a result, . We assume that

and the control takes values in .Here,

for and , whichsatisfy the conditions (A1) and (A2), respectively, by [13, The-orem 3, Remark 2]. In this example,

with is not in-vertible and . The dimension of is for

and the basis of is .According to Theorem 10, the system (64) is feedback stabiliz-able if and only if, for , the rank of the followingmatrix is :

��

��

......

...

��

Example 4: Let be a bounded open set of withboundary and outer unit normal . We consider the el-liptic operator

where withand for some and all

, withfor , and is essentially bounded in , i.e.,

. We now consider the controlled partial differentialequation with state and input delays

(65)

Here we assume that is a bounded variation function on andfor all it satisfies

for some constant . Similarly, we assume that is thebounded variations function of for each , andsatisfies

for any and a constant . On the other hand,we assume that the initial state and the historyfunctions and for

, the control function. We now introduce the following realization

operator

It is known that generates an immediately compact semigroupon . Furthermore the spectrum

, where is isolated with finite multiplicity forany . We now define the linear operators



...

for and . By [13, The-orem 3] the operators and satisfy the conditions (A1) and(A2). In this example,

with is not in-vertible and . The dimension of is for

and the basis of is .According to Theorem 10, the system (65) is feedback stabiliz-able if and only if, for each , see the equationshown at the bottom of the page.

VI. CONCLUSION

In this paper, we have presented an analytic approach to ad-dress the feedback stabilizability of state-input delay systemsin Banach spaces. Based on the transformation of the delaysystem into a delay-free open loop system, we have introduced alarge class of feedback operators that stabilize the delay system.Making use of the recent results on infinite dimensional well-posed and regular linear systems, we have extended the wellknown necessary condition for the feedback stabilizability ofstate-input delay systems when the delay-free state space is fi-nite dimensional to the infinite dimensional case. To make it suf-ficient we have introduced extra conditions that the initial delay-free system is governed by an immediately compact semigroupand that the control space is finite dimensional. This shows thatthe state-delay equation is governed by an immediately com-pact semigroup on a product space, which allows us to use thewell-known results on feedback stabilizability of distributed-pa-rameter linear systems. Moreover, when the state space is re-flexive, we have presented a rank condition for the feedbackstabilizability of the state-input delay system in terms of controloperators and eigenvectors. Some examples are given to showthe applications.

APPENDIX AADMISSIBILITY OF OBSERVATION OPERATORS

FOR PERTURBED SEMIGROUPS

Here, we recall some results on admissibility of observationoperators for perturbed semigroups. Let , be two Banachspaces, be a -semigroup on with generator

, and be a real number.An operator is called a Miyadera-Voigt

perturbation for if

(66)

for all and some constants . It isknown that if satisfies the estimate (66) then the operator

generates a -semigroup on , see [8, p.196]. Now if we usethe notation (9) then by Hölder inequality it is clear that every

satisfies (66), and then is a generator on .The following theorem, taken from [12], shows the invarianceof admissibility of observation operators.

Theorem 11: Let then generates a-semigroup on and

(67)

APPENDIX BGENERALIZED HAUTUS CRITERION

Here, we recall from [38] the generalized Hautus criterion.Let be the generator of a strongly continuous semi-

group on a Banach space , be another Banachspace, and . The open-loop system de-fined by and is denoted by . Let be theYosida extension of . The operator

is endowed with its natural domain defined by.

The following definition is taken from [38, Def.2.1].Definition 12: is feedback stabilizable if there exists

such thati) generates a regular system on , , ;

ii) the identity operator is an admissible feed-back operator for ;

......

......

......



iii) generates an exponentially stable semigroupon .

In this case, we say that stabilizes .We denote by the subspace of

which consists of all vectors of the form ,where and .

The following proposition, taken from [38, Prop.3.5] (com-bined with [38, Remark 1]), is an extension of the Hautus crite-rion for stabilizability of infinite-dimensional systems.

Proposition 4: If is feedback stabilizable, then thereexists a such that

for all with .

ACKNOWLEDGMENT

The authors greatly appreciate the reviewers’ comments,which have helped improve the quality of the paper.

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Said Hadd received the M.Sc degree in pure math-ematics and the Ph.D. degree in mathematics fromUniversity Cadi Ayyad, Morocco, in 1999 and 2005,respectively.

He held a postdoctoral position at INRIA,Grenoble, France, and also had several researchstays at the Functional Analysis Group (AGFA),Department of Mathematics, The University ofTübingen, Germany. From July 2006 to August 2008he was a Research Associate with the Department ofElectrical Engineering and Electronics, University

of Liverpool, Liverpool, U.K. His current research interests include functionalanalysis of operator semigroups, control theory of general infinite-dimensionalsystems, and time-delay systems.



Qing-Chang Zhong (M’04–SM’04) received theM.Sc. degree in electrical engineering from HunanUniversity, Hunan, China, in 1997, the Ph.D. degreein control theory and engineering from ShanghaiJiao Tong University, China, in 1999, and the Ph.D.degree in control and power engineering fromImperial College London, London, U.K., in 2004.

He started working in the area of control engi-neering after graduating from the Hunan Instituteof Engineering, Hunan, China, in 1990. He wasa Postdoctoral Research Fellow at the Faculty of

Mechanical Engineering, Technion-Israel Institute of Technology, Haifa,Israel, from 2000 to 2001, and then a Research Associate at Imperial CollegeLondon from 2001 to 2003. He took up a Senior Lectureship at the Schoolof Electronics, University of Glamorgan, U.K., in January 2004 and wassubsequently promoted to Reader in May 2005. He joined the Departmentof Electrical Engineering and Electronics, The University of Liverpool, Liv-erpool, U.K., in August 2005 as a Senior Lecturer. He is currently leadingan EPSRC-funded Network for New Academics in Control Engineering(New-ACE, www.newace.org.uk), which has attracted more than 60 membersfrom academia and industry. He is the author of the monograph Robust Controlof Time-delay Systems (New York: Springer-Verlag, Ltd., 2006). His currentresearch interests cover control theory (including H-infinity control, time-delaysystems and infinite-dimensional systems) and control engineering (includingprocess control, power electronics, renewable energy, embedded control, rapidcontrol prototyping and hardware-in-the-loop, engine control, hybrid vehiclesand control using delay elements such as input-shaping technique and repetitivecontrol).

Dr. Zhong received the Best Doctoral Thesis Prize from Imperial CollegeLondon.


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