DELAY-DIFFERENTIAL systems arise in the study

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1 438 IEEE TRANSACTIONS ON AUTOMATIC CONTROL On Feedback Stabilizability of Linear Systems With State Input Delays in Banach Spaces Said Hadd Qing-Chang Zhong, Senior Member, IEEE Abstract The feedback stabilizability of a general class of wellposed linear systems state input delays in Banach spaces is studied in this paper Using the properties of infinite dimensional linear systems, a necessary condition for the feedback stabilizability of delay systems is presented, which extends the wellknown results for finite dimensional systems to infinite dimensional ones This condition becomes sufficient as well if the semigroup of the delay-free system is immediately compact the control space is finite dimensional Moreover, under the condition that the Banach space is reflexive, a rank condition in terms of eigenvectors control operators is proposed When the delay-free state space control space are all finite dimensional, a very compact rank condition is obtained Finally, the abstract results are illustrated examples Index Terms Banach spaces, feedback stabilizability, Hautus criterion, rank condition, regular systems, time-delay systems very interesting motivates many researchers to look for conditions guaranteeing stabilizability of such systems By analyzing the existing theory in this area one can note that the results are obtained some special techniques, which do not fit into a general theory as that developed for delay-free systems Also, most of the works have been devoted to feedback stabilizability of state delay systems a finite dimensional delay-free state space eg, [4], [6], [20] or an infinite dimensional space [25] The feedback stabilizability of state-input delay systems is investigated in [19], [22], [26] [29], where the delay-free state space is finite dimensional It is shown by Olbrot [26] that the feedback stabilizability of the state-input delay system I INTRODUCTION is equivalent to the condition A Literature Review Motivation DELAY-DIFFERENTIAL systems arise in the study of many problems theoretical practical importance The behavior of these systems was studied in the seventies, especially in the book by Hale [17], where the semigroup theory was applied to reformulate delay systems as distributed parameter systems (see also Bellman Cooke [4] Halanay [16] for an introduction to this approach) The recent well-developed theory for such systems is gathered in several books eg, Bátkai Piazzera [2], Bensoussan et al [3], Hale Verduyn Lunel [18] More recently it is shown in [10], [11], [14] that delay systems form a subclass of well-posed regular infinite dimensional linear systems in the Salamon-Weiss sense [30] [33], [37] It has been recognized that the delay presence in the state /or input could induce bad performance (even instability) complicate controller design system analysis [39] This fact makes the investigation of the stability of delay systems Manuscript received June 22, 2007; revised March 14, 2008 August 07, 2008 Current version published March 11, 2009 This paper was presented in part at the 46th IEEE Conference on Decision Control, New Orleans, LA, December 2007 at the 18th International Symposium on Mathematical Theory of Networks Systems (MTNS2008), July 2008 This work was supported by the EPSRC, UK under Grant EP/C005953/1 Recommended by Associate Editor G Feng The authors are the Department of Electrical Engineering Electronics, The University of Liverpool, Liverpool L69 3GJ, UK ( said hadd@qatartamuedu; qzhong@livacuk) Digital Object Identifier /TAC (1) where is the dimension of the delay-free system We are interested in developing a more general approach to the feedback stabilizability of state-input delay systems in Banach spaces We shall introduce a general class of feedback laws which stabilize the closed-loop system This is based on the recent theory of regular linear systems [21], [31], [32], [37], the recent work on state-input delay systems in Banach spaces [11], [24], where it has been proved that systems general distributed state, input output delays can be reformulated as regular linear systems Notation Problem Statement Throughout this paper we use the following notation For a Banach space, a real number, a function, the history function of is defined as for We denote by the space of all continuous functions from to For we denote by the Lebesgue space of all -integrable functions [7], by the space of indefinite integrals of -integrable functions, by the Banach space of all linear bounded operators from a Banach space to another Banach space In this paper we consider the state-input delay system (2) /$ IEEE Authorized licensed use limited to: University of Liverpool Downloaded on March 13, 2009 at 12:45 from IEEE Xplore Restrictions apply

2 HADD AND ZHONG: FEEDBACK STABILIZABILITY OF LINEAR SYSTEMS 439 Here is the generator of a -semigroup on a Banach space the delay operators are linear bounded, where is a (control) Banach space The function is the history function of the function is the history function of the function The initial conditions are, which form the initial state of the system If are finite dimensional spaces then the operators can take the explicit form of Riemann-Stieltjes integrals for, where are functions of bounded variations [3, p249] In this case the system (2) is well-posed in the sense that it can be rewritten as a well-posed open-loop system [11], [13] In this paper is an arbitrary Banach space the operators are not assumed to be of the form (3) In this case, it is shown in [11] that the system (2) can be reformulated as a well-posed control system on the state space in the sense of [35] if are observation operators of regular linear systems governed by the left shift semigroup (see [13] for more details on such systems) These conditions on are always satisfied if are finite dimensional, see [13] B Overview of the Major Contributions After discussing left shift semigroups, transforming the system (2) into a well-posed open-loop system on based on [11] characterizing the structure of feedback stabilizable operators, we introduce a necessary condition for the stabilizability of the system (2), using the generalization of the Hautus criterion for the stabilizability of distributed-parameter linear systems [38, Proposition 35] This generalizes the condition (1) to the case of general Banach spaces general delay operators (see Theorem 7) This necessary condition does not require any regularity on the semigroup or on the geometry of the state control spaces However, to show the sufficiency of this condition, we need to assume that the control space is finite dimensional the semigroup is compact for, ie, immediately compact Under these conditions, the delay system (2) is feedback stabilizable if only if holds for all, which belongs to the finite unstable spectrum of the operator, where is defined as Furthermore, a rank condition, derived from the above condition, is given in terms of eigenvectors control operators in the case a reflexive Banach space This extends the result in [25] where a rank condition was given for systems state delays only When the delay-free state space control space are finite dimensional, a compact (3) rank condition equivalent to that of Olbrot [26] is obtained We illustrate our abstract results examples involving a single delay, multiple delays, distributed delays elliptic operators in bounded domains of the Euclidean space C Organization of the Paper The organization of the paper is as follows In Section II we recall some preliminaries about well-posed regular infinite dimensional systems In Section III after we discuss left shift semigroups, state the main assumptions reformulate the delay system (2) to an infinite dimensional open loop system, we present the definition of feedback stabilizability characterize the structure of feedback stabilizable operators Section IV is devoted to state prove the main results on feedback stabilizability of the delay system (2) in Section V we present some examples Conclusions are made in Section VI, followed by two appendices, in which the frequently-cited known results are gathered II PRELIMINARIES: WELL-POSED AND REGULAR INFINITE DIMENSIONAL LINEAR SYSTEMS Here, we briefly recall the framework of infinite dimensional well-posed regular linear systems in the Salamon-Weiss sense from [31], [32], [34] [36], [33], [37] A Basic Concepts Let be the generator of a -semigroup on a Banach space We denote by the resolvent set of, ie, the set of all such that is invertible The spectrum of is by definition We define the resolvent operator of as The domain endowed the graph norm,isa Banach space We also define the norm for some The completion of respect to the norm is a Banach space denoted by, which is called the extrapolation space associated Moreover, the continuous injection holds The semigroup can be naturally extended to a strongly continuous semigroup on, of which the generator is the extension of from to ; see [8] for more details The pair is called a control system on, if is a -semigroup on, is a linear bounded operator satisfying, for, See [35], [36] for more details on the maps By the representation theorem due to Weiss [35, Theorem 39], there exists a unique operator, called an admissible control operator for (or ), such that for any, where the integral exists in We say that is represented by the control operator Each control system (4) (5) Authorized licensed use limited to: University of Liverpool Downloaded on March 13, 2009 at 12:45 from IEEE Xplore Restrictions apply

3 440 IEEE TRANSACTIONS ON AUTOMATIC CONTROL the representing control operator is completely determined by an abstract differential equation of the form It is well-posed in the sense that it has a unique strong solution, called the state trajectory initial state, given by (6) (7) The right h side goes to 0 when, according to Lemma 34 in [8, p73] Hence, for any Note that the definition of does not require that However, if this is the case, for all ae, (see [34, Theorem 45]) For, denote its convolution the semigroup as is called an admissible obser- An operator vation operator for (or )if (8) Then, we have the following result [9, Proposition 33] Proposition 1: Assume Define holds for any for some constants For simplicity, denote (9) Then For, the linear system (10) is well-posed in the sense that the observation function can be extended to a function in In fact, from (8), the map (11) can be extended to a linear bounded operator from to, called the extended output map Then we can set for any almost every If we define (12) for, 0 for, then is a family of bounded linear operators from to We say that is an observation system on,, the observation equation in (10) satisfies for almost every (ae) all Note that is extended in the abstract way In order to obtain a pointwise representation in terms of the observation operator, consider the Yosida extension[34] of respect to defined as Let be endowed the graph norm For, (13) for a constant, which is independent of approaches 0 as B Well-Posed Regular Systems Here, we focus on a control system represented by an observation system represented by We say that the linear system (14) the initial state,iswell-posed on,, if there exists a family of bounded linear operators from to, satisfying (15) for, For more details on operators we refer to [36], [37] In this case we also say that the quadruple is a well-posed system on,, Let be the operator of truncation to, that is for zero otherwise The operators are compatible in the sense that for zero otherwise This property provides a unique operator, called the extended input-output map, which satisfies for We recall from [36, Theorem 36] that there exist a unique bounded analytic function such that if Here are the Laplace transform of, respectively Here, is called the transfer function of Authorized licensed use limited to: University of Liverpool Downloaded on March 13, 2009 at 12:45 from IEEE Xplore Restrictions apply

4 HADD AND ZHONG: FEEDBACK STABILIZABILITY OF LINEAR SYSTEMS 441 We say that the well-posed system feedthrough) if the limit is regular ( zero for the control function is a control system on, which is represented by the (strictly) unbounded admissible control operator [13] (19) exists in for the constant input The following definitions will be used throughout this paper Definition 1: Let be the admissible control observation operators issued from, respectively We say that the triple generates a regular system if there exists a bounded operator such that is regular on,, Definition 2: Let be a regular system input-output operators An operator is called an admissible feedback for if has uniformly bounded inverse Note that in the case of Hilbert spaces one can use transfer functions instead of input-output operators for the definition of admissible feedback operators [36] We now state a very general perturbation theorem due to Weiss in Hilbert spaces [37] due to Staffans in general Banach spaces [32, Chap7] Theorem 3: Assume that generates a regular system admissible feedback operator Then the operator where is the generator of the extrapolation semigroup associated is the linear bounded operator In fact, is the adjoint of the delta operator at zero Similarly, we can have operators For the control function of for ae, the state trajectory of is the history function of given by Similarly, for the control system represented by, we have for ae According to (11), if is an admissible observation operator of, we define the sum defined in generates a -semigroup on Moreover, for ae,, for, III FEEDBACK STABILIZABILITY OF SYSTEM (2) (16) A Left Shift Semigroups Main Assumptions Here, we follow Section II-A to present two left shift semigroups then state the main assumptions on delay operators of the system (2) Define (17), similarly, It is well known that generates the left shift semigroup (18) for Similarly, we have The pair then according to (12) Similarly, we define for operator The input-output operator associated will be defined according to (15), replacing The input-output operator is similarly defined for operator From now on we assume that the operators in system (2) satisfy: (A1) the triple generates a regular system, on (A2) the triple generates a regular system, on Remark 4: Here, we give an example of operators that satisfy the above assumptions Let be functions of bounded variations such that the total variations of approach 0 when the interval goes to 0 If the operators take the form of for, where are not necessarily finite dimensional, then satisfies (A1) satisfies (A2); see [13] B Reformulation of System (2) We recall from [11], [15] how the delay system (2) can be reformulated as a linear distributed parameter system on some appropriate Banach spaces Authorized licensed use limited to: University of Liverpool Downloaded on March 13, 2009 at 12:45 from IEEE Xplore Restrictions apply

5 442 IEEE TRANSACTIONS ON AUTOMATIC CONTROL Consider the Banach space endowed the norm It is well-known (see eg, [2]) that the delay system (2) is closely related to the following linear operator (20) It is not difficult to show that if only if (21) given by (27) for, where is the extended output map associated Furthermore, we have the following result from [11, Proposition 35] [15, Proposition 32] Proposition 2: Assume that satisfies (23) satisfies the condition (A2) Then the operator defined in (25) is an admissible control operator for defined in (24) According to (5), the control maps associated the control operator are given by for some constants [9], where is defined as otherwise According to the Hölder inequality, (21) implies that (22) (23) for Hence, the operator generates a strongly continuous semigroup on, according to [1] The following remark is important for the proof of the main results in this paper Remark 5: From the discussion above, we have seen that for the estimate (23) holds as If in addition the operator generates an immediately compact semigroup on (ie, compact for ), then generates an eventually compact semigroup on (ie, compact for, in this paper) [23] In order to consider the well-posedness of the state-input delay systems (2), define the operators for According to [11, Proposition 32], this is equivalent to (28) for, where is the extended input-output map of the regular system Now, the open loop system (29) is well-posed according to Proposition 2 Combining (26) (28), one can see that the state trajectory of the system (29), for the initial state, is given by according to (7) This is the same as the state trajectory of the system (2), see [11, Proposition 33] This gives a natural connection between the systems (2) (29) Let be sufficiently large According to [13], the Laplace transform of in is equal to that of in is exactly Hence, by taking the Laplace transform on both sides of (28), we have (24) (25) It has been shown in [11, Theorem 31] that when satisfies the condition (23), the operator generates the following -semigroup on, (26) (30) C The Structure of Feedback Stabilizable Operators According to the reformulation of the delay system above, Proposition 2 [38, Definition 2] (see also Appendix B), we introduce the following definition Definition 6: Let the conditions (A1) (A2) be satisfied We say that the state-input delay system (2) is feedback stabilizable if there exists an operator such that: i) The triple generates a regular system on,, ii) The identity operator is an admissible feedback operator for Authorized licensed use limited to: University of Liverpool Downloaded on March 13, 2009 at 12:45 from IEEE Xplore Restrictions apply

6 HADD AND ZHONG: FEEDBACK STABILIZABILITY OF LINEAR SYSTEMS 443 iii) The semigroup generated by the operator endowed 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 of such operators Proposition 3: Let the conditions (A1) (A2) be satisfied If stabilizes the delay system (2) then it is of the form,, the triple generating a regular linear system as an admissible feedback operator Proof: Assume that stabilizes the system (2), then at least we have Decompose into Then, for any, we have, Now prove the second part Let be the regular system generated by the triple According to [11, Theorem 51], at least should be an observation operator of the regular system generated by the triple on such that is an admissible feedback operator Hence, This ends the proof IV MAIN RESULTS A A Necessary Condition For each, we redefine for the system (2) From [2, p56], we have (33) (34) defined in (20) In view of this equivalence, is called the characteristic operator The result below gives a necessary condition for the feedback stabilizability of the system (2) Theorem 7: Assume the conditions (A1) (A2) are satisfied The feedback stabilizability of the delay system (2) implies the existence of such that (35), decom- for This means In order to characterize the operators pose into for any Proof: Denote From [1], the operator is the generator of the -semigroup According to the generalized Hautus criterion [38, Prop 35], which is reproduced in Appendix B for the reader s convenience, the feedback stabilizability of system (29), hence of (2), implies that there exists such that (31) on for, where the operators are defined in (22) Thus, the fact that implies that According to the invariance of admissibility of observation operators [12, Theorem 31], which is reproduced in Appendix A for the reader s convenience, there is This allows us to concentrate on operators Using (31), one can easily show that an operator if only if it is of the form for any This indicates, for any hence for any, there exist such that, for, (36) Let let be fixed 1 According to (30), we have (32) This shows that the operators stabilizing the open loop system have at least the following form 1 This is because does not always belong to (A ) Authorized licensed use limited to: University of Liverpool Downloaded on March 13, 2009 at 12:45 from IEEE Xplore Restrictions apply

7 444 IEEE TRANSACTIONS ON AUTOMATIC CONTROL As we are only interested in any, substitute, into (41), then Now, by (36), we obtain Hence, the condition (35) holds This completes the proof B A Necessary Sufficient Condition It is clear that (37) Theorem 8: Assume the conditions (A1) (A2) are satisfied the control space is finite dimensional Moreover, assume that is compact for The delay system (2) is feedback stabilizable if only if (42) which implies that (38) holds for all Proof: The necessity has been proved in Theorem 7 Only the sufficiency will be shown here At first, we will transform the system (2) into an equivalent system different from (29) For the system (2), take, respectively, introduce the Banach space, as a result 0(A ) = 0A (0A = ) z ' z Be u +R(; A ) ' 0 +e u z Be u +R(;A ) ' 0 +e u +R(;A ) Be u 0 0 B( +e u) 0 : the operator which generates the -semigroup 0 +(0)e u (39) Here, the identity was used Note that on Since is compact for is compact, is compact for Define (40) By combining (37), (39) (40), we have where is assumed to be smooth Then the system (2) is converted into By using the expression of in (20), we obtain (41) Here the new control operator Denote (43) is defined by The last identity implies,or define Similarly, we have Authorized licensed use limited to: University of Liverpool Downloaded on March 13, 2009 at 12:45 from IEEE Xplore Restrictions apply

8 HADD AND ZHONG: FEEDBACK STABILIZABILITY OF LINEAR SYSTEMS 445 Furthermore, take any such that define Then, the operator then (46) is the generator of the following semigroup on : Substitute (46), after rearrangement, into (45), then Note that on In particular, the assumptions (A1) (A2) show that Hence generates a strongly continuous semigroup on, according to [2, p67], [9] Define Putting the last two identities together, we have which is bounded, then the delay system (43), hence (2), is equivalent to the open loop system where This is equivalent to (44) where the state is defined as We have now converted the feedback stabilizability of the system (43) or system (2) to that of the above system, where is linear bounded Now let us characterize the spectrum of According to [2, p56], we have which means that the system (44), or system (2), is feedback stabilizable, according to [5, Theorem 1] This ends the proof Remark 9: Denote by the adjoint space of Then the adjoint of the characteristic operator defined in (33) is given by (47) where is the complex conjugate of Using Theorem 7, [25, Prop51] the proof of Theorem 8 one can see that where (48) Hence, by (34) we obtain A simple computation shows that By assumption that is compact for generates an eventually compact semigroup on, according to Remark 5 As a result, the following unstable set is finite: Take any let The condition (42) implies the existence of such that, which yields Since we have (49) (50) (45) (51) Authorized licensed use limited to: University of Liverpool Downloaded on March 13, 2009 at 12:45 from IEEE Xplore Restrictions apply

9 446 IEEE TRANSACTIONS ON AUTOMATIC CONTROL Combining (49), (51) (48), the following holds under the conditions of Theorem 8: (56) (52) C A Rank Condition for a Special Case in Banach Spaces In addition to the assumptions in Theorem 8 we assume that is a reflexive Banach space As the control space is finite dimensional, denote as (57) For is linear bounded for all, we define for, given in (54) Proof: We only need to prove that the condition (52) is equivalent to (56) under the extra conditions Since for As,wehave (53) for We say that satisfies (A2) if the triple generates a regular linear system on, Naturally, satisfies (A2) if every satisfies (A2) for all We recall that if is compact for, then generates an eventually compact semigroup on In this case, the set of unstable eigenvalues can be denoted by (54) which is finite finite algebraic multiplicity Because of (34), for defined in (47), we can set the dimension of as (55) the basis of by Throughout the following we denote the duality pairing between by the orthogonal space of a set in by we have, as a result, (58) Now, assume that the condition (56) does not hold This means that there exists such that, ie, (59) Now since is a basis of,wehave When is reflexive, the adjoint space of is the product space satisfying Theorem 10: Assume that (i) satisfy the conditions (A1) (A2), respectively, (ii) the Banach space is reflexive the control space is finite dimensional (iii) is compact for The system (2) is feedback stabilizable if only if On the other h, by using (58) (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) the basis This ends the proof Authorized licensed use limited to: University of Liverpool Downloaded on March 13, 2009 at 12:45 from IEEE Xplore Restrictions apply

10 HADD AND ZHONG: FEEDBACK STABILIZABILITY OF LINEAR SYSTEMS 447 D A Rank Condition for Delay Systems With Finite-Dimensional State (When Delay-Free) Control Spaces Consider the delay system On the other h, out loss of generality (otherwise carrying out some elementary operations), assume that (60) where are real matrices, are matrices, is the delay Here, Let the control function for Wehave com- which gives the following having rank : Now the partition of patible, we have (62) as follows: Since (61) holds, has full column rank Multiply the above from the left, then We note that the operator satisfy the conditions (A1) (A2), see [13, Remark 2] Since is a finite dimensional matrix, is compact for In this case, Denote the The dimension of is for the basis of is matrix formed by the basis as According to Theorem 10, we have the following: Corollary 1: The system (60) is feedback stabilizable if only if (61) As this condition is obtained from (42), which is an extension of (1), it should be equivalent to the condition (1) proposed in [26] This is indeed true In fact, we have Assume that for each, (62) Since has full column rank, the leftmost big matrix in the above identity has full column rank as well This means that has full column rank In other words, the condition (63) holds V EXAMPLES We start from giving two examples which illustrate the result of Corollary 1 In the third example we treat the case of systems multiple delays We end this section by an example of distributed delay systems governed by an uniformly elliptic differential operator of second order on a bounded domain of the Euclidean space Example 1: Consider the system (60) Then, Hence Now (61) is re- which implies that duced to (63) which implies that has full column rank Hence, the delay system is feedback stabilizable if only if In other words, the system is not stabilizable if Authorized licensed use limited to: University of Liverpool Downloaded on March 13, 2009 at 12:45 from IEEE Xplore Restrictions apply

11 448 IEEE TRANSACTIONS ON AUTOMATIC CONTROL Example 2: Consider the system (60) Example 4: Let be a bounded open set of boundary outer unit normal We consider the elliptic operator This was studied in [26] ( a typo in Now we have ) Here where for some all, for, is essentially bounded in, ie, We now consider the controlled partial differential equation state input delays Thus, is stabilizable but is not stabilizable Example 3: Consider the state-input delay system (64) Here the operator generates an immediately compact semigroup on a reflexive Banach space, the family, for each, (65) Here we assume that is a bounded variation function on for all it satisfies where Here, is linear continuous for any As a result, We assume that the control takes values in for some constant Similarly, we assume that is the bounded variations function of for each, satisfies for, which satisfy the conditions (A1) (A2), respectively, by [13, Theorem 3, Remark 2] In this example, for any a constant On the other h, we assume that the initial state the history functions for, the control function We now introduce the following realization operator is not invertible The dimension of is for the basis of is According to Theorem 10, the system (64) is feedback stabilizable if only if, for, the rank of the following matrix is : hb (e );' i hb (e );' i hb (e );' i hb (e );' i hb (e );' i hb (e );' i It is known that generates an immediately compact semigroup on Furthermore the spectrum, where is isolated finite multiplicity for any We now define the linear operators hb (e );' i hb (e );' i hb (e );' i Authorized licensed use limited to: University of Liverpool Downloaded on March 13, 2009 at 12:45 from IEEE Xplore Restrictions apply

12 HADD AND ZHONG: FEEDBACK STABILIZABILITY OF LINEAR SYSTEMS 449 for By [13, Theorem 3] the operators satisfy the conditions (A1) (A2) In this example, is not invertible The dimension of is for the basis of is According to Theorem 10, the system (65) is feedback stabilizable if only if, for each, see the equation shown at the bottom of the page VI CONCLUSION In this paper, we have presented an analytic approach to address the feedback stabilizability of state-input delay systems in Banach spaces Based on the transformation of the delay system into a delay-free open loop system, we have introduced a large class of feedback operators that stabilize the delay system Making use of the recent results on infinite dimensional wellposed regular linear systems, we have extended the well known necessary condition for the feedback stabilizability of state-input delay systems when the delay-free state space is finite dimensional to the infinite dimensional case To make it sufficient we have introduced extra conditions that the initial delayfree system is governed by an immediately compact semigroup that the control space is finite dimensional This shows that the state-delay equation is governed by an immediately compact semigroup on a product space, which allows us to use the well-known results on feedback stabilizability of distributed-parameter linear systems Moreover, when the state space is reflexive, we have presented a rank condition for the feedback stabilizability of the state-input delay system in terms of control operators eigenvectors Some examples are given to show the applications APPENDIX A ADMISSIBILITY OF OBSERVATION OPERATORS FOR PERTURBED SEMIGROUPS Here, we recall some results on admissibility of observation operators for perturbed semigroups Let, be two Banach spaces, be a -semigroup on generator, be a real number An operator is called a Miyadera-Voigt perturbation for if (66) for all some constants Itis known that if satisfies the estimate (66) then the operator generates a -semigroup on, see [8, p196] Now if we use the notation (9) then by Hölder inequality it is clear that every satisfies (66), then is a generator on The following theorem, taken from [12], shows the invariance of admissibility of observation operators Theorem 11: Let then generates a -semigroup on APPENDIX B GENERALIZED HAUTUS CRITERION (67) Here, we recall from [38] the generalized Hautus criterion Let be the generator of a strongly continuous semigroup on a Banach space, be another Banach space, The open-loop system defined by is denoted by Let be the Yosida extension of The operator is endowed its natural domain defined by The following definition is taken from [38, Def21] Definition 12: is feedback stabilizable if there exists such that i) generates a regular system on,, ; ii) the identity operator is an admissible feedback operator for ; Authorized licensed use limited to: University of Liverpool Downloaded on March 13, 2009 at 12:45 from IEEE Xplore Restrictions apply

13 450 IEEE TRANSACTIONS ON AUTOMATIC CONTROL iii) generates an exponentially stable semigroup on In this case, we say that stabilizes We denote by the subspace of which consists of all vectors of the form, where The following proposition, taken from [38, Prop35] (combined [38, Remark 1]), is an extension of the Hautus criterion for stabilizability of infinite-dimensional systems Proposition 4: If is feedback stabilizable, then there exists a such that for all ACKNOWLEDGMENT The authors greatly appreciate the reviewers comments, which have helped improve the quality of the paper REFERENCES [1] A Bátkai S Piazzera, Semigroup linear partial differential equations delay, J Math Anal Appl, vol 264, no 1, pp 1 20, 2001 [2] A Bátkai S Piazzera, Semigroups for Delay Equations Wellesley, MA: A K Peters, Ltd, 2005, vol 10 [3] A Bensoussan, G Da Prato, M C Delfour, S K Mitter, Representation Control of Infinite-Dimensional Systems Boston, MA: Birkhäuser, 2007 [4] R Bellman K L Cooke, 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Associate the Department of Electrical Engineering Electronics, University of Liverpool, Liverpool, UK His current research interests include functional analysis of operator semigroups, control theory of general infinite-dimensional systems, time-delay systems Authorized licensed use limited to: University of Liverpool Downloaded on March 13, 2009 at 12:45 from IEEE Xplore Restrictions apply

14 HADD AND ZHONG: FEEDBACK STABILIZABILITY OF LINEAR SYSTEMS 451 Qing-Chang Zhong (M 04 SM 04) received the MSc degree in electrical engineering from Hunan University, Hunan, China, in 1997, the PhD degree in control theory engineering from Shanghai Jiao Tong University, China, in 1999, the PhD degree in control power engineering from Imperial College London, London, UK, in 2004 He started working in the area of control engineering after graduating from the Hunan Institute of Engineering, Hunan, China, in 1990 He was a Postdoctoral Research Fellow at the Faculty of Mechanical Engineering, Technion-Israel Institute of Technology, Haifa, Israel, from 2000 to 2001, then a Research Associate at Imperial College London from 2001 to 2003 He took up a Senior Lectureship at the School of Electronics, University of Glamorgan, UK, in January 2004 was subsequently promoted to Reader in May 2005 He joined the Department of Electrical Engineering Electronics, The University of Liverpool, Liverpool, UK, in August 2005 as a Senior Lecturer He is currently leading an EPSRC-funded Network for New Academics in Control Engineering (New-ACE, wwwnewaceorguk), which has attracted more than 60 members from academia industry He is the author of the monograph Robust Control of Time-delay Systems (New York: Springer-Verlag, Ltd, 2006) His current research interests cover control theory (including H-infinity control, time-delay systems infinite-dimensional systems) control engineering (including process control, power electronics, renewable energy, embedded control, rapid control prototyping hardware-in-the-loop, engine control, hybrid vehicles control using delay elements such as input-shaping technique repetitive control) Dr Zhong received the Best Doctoral Thesis Prize from Imperial College London Authorized licensed use limited to: University of Liverpool Downloaded on March 13, 2009 at 12:45 from IEEE Xplore Restrictions apply