J. Korean Math. Soc. 37 (2), No. 4, pp. 593{611 STABILITY AND CONSTRAINED CONTROLLABILITY OF LINEAR CONTROL SYSTEMS IN BANACH SPACES Vu Ngoc Phat, Jong Yeoul Park, and Il Hyo Jung Abstract. For linear time-varying control systems with constrained control described by both dierential and discrete-time equations in Banach spaces we give necessary and sucient conditions for exact global null-controllability. We then show that for such systems, complete stabilizability implies exact null-controllability. 1. Introduction Consider a linear time-varying control system described by dierential equations of the form (1) _x(t) =A(t)x(t)+B(t)u(t) x(t) 2 X u(t) 2 U t where X and U are innite-dimensional Banach spaces A(:) andb(:) are linear operators. From the classical control theory, it is dened that system (1) is exactly null-controllable if every point x 2 X can be controllable to the origin by some control u 2 U in nite time exponentially stabilizable (in Lyapunov sense) if there exists an operator function K(t)(:) : X! U t such that all solutions x(t x ) of the closed-loop control system _x =[A(t)+B(t)K(t)]x with the initial condition x() = x satisfy kx(t x )kme ;t kx k 8 t for some M > and >: Received December 2, 1999. 1991 Mathematics Subject Classication: 93D15, 34K2, 49M7. Key words and phrases: stability, controllability, linear operator, stabilizability, Banach space, time-varying system.
594 V. N. Phat, J. Y. Park, and I. H. Jung The problems of controllability and stabilizability of control system (1) or of its discrete analog were studied widely by many researchers in control systems theory(see, e.g. [2, 5, 9-13, 16, 19, 22]) and references therein. In particular, the relationship between controllability and stabilizability was presented in [1] for time-invariant control systems, where X U are nite-dimensional and it was shown that the exact null-controllability implies exponential stabilizability. It is obvious that all exactly null-controllable systems are exponentially stabilizable(see [4, 23]), however the exponentially stabilizable system is, in general, not exactly null-controllable. If the time-invariant system is completely stabilizable in a sense of Wonham [2], i.e., for an arbitrary > there is a matrix K such that the matrix A + BK is exponentially stable with the stability exponent then the system is exactly null-controllable. A natural question is: To what extent does complete stabilizability imply exact null-controllability for innite-dimensional control systems? In the innite-dimensional control theory [3] characterizations of controllability and stabilizability are complicated and therefore their relationships are much more complicated and require more sophisticated methods. The diculties increase to the same extent as passing from time-invariant systems to time-varying systems as well as from unconstrained control systems to systems with constrained controls. For time-invariant control systems without constrained controls in Hilbert spaces, in a stronger type of complete stabilizability, it was shown in Megan [7] and then Zabczyk [21, 23] that complete stabilizability implies exact null-controllability. In this paper, we rst give necessary and sucient conditions for exact global null-controllability of continuous and discrete-time timevarying systems, where the control u(k) is restricted to lie in some subset in a Banach space U: Secondly, and more importantly, asan extension of [7, 21], we show that for such control systems complete stabilizability implies exact global null-controllability. Unlike usual constrained controllability conditions expressed by the spectrum of the adjoint operator A (see, e.g. [1, 18]), our controllability conditions are described in terms of Schmitendorf and Barmish-type conditions [17], which can be applicable to the stability analysis. To our knowledge, the main results of this paper are also new for the case of nite-dimensional control systems. Some constrained controllability results, Corollaries
Stability and constrained controllability 595 3.1 and 3.2, are well-known for the nite-dimensional control systems, but the proofs are dierent and our approach can be extended to the study of the controllability and stability of some other classes of both continuous-time and discrete-time systems with constrained controls in Banach spaces. The paper is organized as follows. In Section 2, we review main notations, denitions and give some auxiliary lemmas needed later. Global null-controllability conditions for both continuous-time and discretetime control systems with constrained controls in Banach spaces are given in Section 3. In Section 4, we show that complete stabilizability implies exact global null-controllability. Discrete analog of the result is also given in this section. 2. Notations, denitions and preliminaries Let X and U be innite-dimensional Banach spaces. X denotes the topological dual space of X and <y x>denotes the value of y 2 X at x 2 X: The adjoint and the inverse operator of an operator A are denoted by A and A ;1 respectively. I denotes the identity operator. In this paper, we use the following standard notations from [3, 6]. - R; the set of all real numbers, Z + ; the set of all non-negative integers, - B 1 = fx 2 X : kx k X =1g X = X nfg - L(X Y ); the Banach space of all linear bounded operators mapping X into Y - L 1 ([ t] U); the Banach space of all U;valued strongly measurable functions u(:) 2 U such that kuk U is essentially bounded on [ t] - H (y ); the support function of a set dened by H (y )=sup<y u> u2 - M ; the polar set of a set M at 2 M dened by M = y 2 X :<y x> 1 8x 2 M : We shall consider control system (1), where x(t) 2 X u(t) 2 U is a given nonempty subset in U: Throughout this paper, we assume that A(t)andB(t) are strongly measurable and locally integrable in t and for every t A(t) 2L(X X) B(t) 2L(U X):
596 V. N. Phat, J. Y. Park, and I. H. Jung For any T > the set of admissible controls on [ T] for system (1) is dened as U T = u(:) 2 L 1 ([ T] U):u(t) 2 a.e. on [ T] : Thus, as in [6], for each u(t) 2U T and x 2 X the unique solution of (1) initiated at x is given by x(t x u)=(t )x + (t s)b(s)u(s) ds where (t s) is a nonsingular evolution operator of the linear system _x = A(t)x satisfying (s s) =I ;1 (t s) =(s t) (t s) =(t )( s) t s : Definition 2.1. The time-varying system (1) is completely stabilizable if for every > there exist a linear bounded operator function K(t)(:) : X! t and a number M > such that the solution x(t x ) of the system (1.1) _x(t) = A(t)+B(t)K(t) x(t) x() = x satises the condition kx(t x )kme ;t kx k 8 t : In other words, if K (t s) is the evolution operator of system (1.1), then the complete stabilizability is equivalent to 8 > 9 K(t)(:) :X! 9M > :k K (t )k Me ;t 8 t : Note that if the operator K and number M do not depend on then the complete stabilizability implies exponential stabilizability in usual Lyapunov sense (see [1, 23]).
Stability and constrained controllability 597 We now give analogous denition of the stabilizability for discretetime control systems of the form x(k +1)=A(k)x(k)+B(k)u(k) k 2 Z + (2) x(k) 2 X u(k) 2 U where A(k) 2 L(X X) and B(k) 2 L(U X): It is obvious that for every u 2U k = fu k =(u() u(1) ::::: u(k ; 1) 2 k g and x 2 X the discrete-time control system (2) always has a solution x(k x u) with x() = x given by x(k x u)= (k )x + k;1 X i= where (k i) is the transition operator dened by (k i +1)B(i)u(i) (k i) =A(k ; 1)A(k ; 2):::A(i) k>i (k k) =I: Definition 2.2. The discrete-time system (2) is completely stabilizable if for every q 2 ( 1) there exist a linear bounded operator function Q(k)(:) :X! k 2 Z + and a number M > such that the solution x(k x ) of the system (2.1) x(k +1)=[A(k)+B(k)Q(k)]x(k) x() = x k 2 Z + satises the condition or equivalently, kx(k x )kmq k kx k k Q(k )k Mq k 8 k 2 Z + 8 k 2 Z + where Q(k i) is the transition operator of the system (2.1). To give controllability denition, let us dene the reachable set of the control system (1) from x in time T > by and by R T (x )=fx 2 X : 9 u(t) 2U T x(t x u)=xg R(x )= [ T> R T (x ) the reachable set of the system in nite time.
598 V. N. Phat, J. Y. Park, and I. H. Jung Definition 2.3. A point x 2 X is said to be null-controllable by the system (1) in time T > if 2 R T (x ): The system (1) is globally null-controllable if every point x 2 X is null-controllable in some nite time T > i.e., 2 R(x ) for all x 2 X: Similar denition is applied for global null-controllability of discrete-time system (2). We need the following lemma for later use. Lemma 2.1. Assume that is a convex, compact subset in U: Then 8 y 2 X : sup u2u T <y L T u>= where L T u = R T (T s)b(s)u(s) ds: Z T H B (s) (T s)y ds Proof. We shall prove the lemma based on the same arguments used in the proof of Theorem 5.3.8 in [1]. By the assumption that is a convex compact set, it is easy to see that the admissible control set U T is convex, weakly compact in L 1 ([ T] U) (e.g., [1, 3]). For every xed y the linear function u 7;! F (y u) =< y L T u > is weakly continuous and it attains its supremum on the convex weakly compact set U T at some u (t) 2U T : sup u2u T F (y u)=f (y u ): Let t 2 ( T)=I T and for every v 2, we dene and I n T = fs 2 R : t ; 1 n s t + 1 n g\ I T u (t) if t 2 I T n IT n u(t) = v if t 2 IT n: For the admissible control u(t) 2U T, we have and consequently, Z I n T F (y u ) F (y u) <B (s) (T s)y u (s) > ds Z I n T <B (s) (T s)y v > ds:
Stability and constrained controllability 599 Denoting (IT n) the Lebesgue length measure of In T, we have 1 (I n T ) ZI n T <B (s) (T s)y u (s) > ds 1 (I n T ) ZI n T <B (s) (T s)y v > ds: Letting n!1,we get, for all v 2 the estimate <B (t) (T t)y u (t) >< B (t) (T t)y v > and hence <B (t) (T t)y u (t) > H (B (t) (T t)y ): Since u (t) 2 for every t 2 ( T), we also get <B (t) (T t)y u (t) > H (B (t) (T t)y ): Therefore (3) <B (t) (T t)y u (t) >= H (B (t) (T t)y ): Integrating both sides of the equation (3) over [ T], we obtain F (y u )= which proves the lemma. Z T H (B (t) (T t)y ) dt We rst give a necessary and sucient condition for the controllability of a point x 2 X to an arbitrary convex set in a xed time by the system (1). For this, we consider the following operator equation (4) x =x + T u u 2U U where 2L(X X) T 2L(U X) x 2 X is a xed element. As in [8], the system (4) is controllable with respect to (x U M) if there is a u 2U such that x + T u 2 M:
6 V. N. Phat, J. Y. Park, and I. H. Jung Lemma 2.2. ([8]) Assume that U is a convex, weakly compact subset in U: The system (4) is controllable with respect (x U M) if and only if 8y 2 M :sup<y T u>< y x > ;1: u2u Remark 2.1. Note that if M = fg then M = X and if we take =(T ) T = L T U = U T then from Lemmas 2.1 and 2.2 it follows that a point x is nullcontrollable at time T > by system (1), where is a convex, compact subset in U if and only if y 2 X : J(T x y ) where J(T x y )= Z T H (B (s) (T s)y ) ds; <x (T )y > +1: The similar result is obtained for discrete-time control systems (2), where we dene G(k x y )= k;1 X i= H (B (i) (k i+1)y )+ <x (k )y > k 2 Z + : Lemma 2.3. Assume that is a convex, compact subset in U: A point x 2 X is null-controllable at step K > by the discrete-time control system (2) if and only if 8y 2 X : G(K x y ) : Proof. Assume that x is null-controllable at step K > i.e., 2 R K (x ) where R K (x )= n K;1 X (K )x + i= o (K i +1)B(i)u(i) :u(i) 2 i = 1 ::: K ; 1 :
Therefore, for all y 2 X, we get Stability and constrained controllability 61 sup <y x>< y >= : x2r K (x ) By the denition of R K (x ) we then have G(K x y ) for all y 2 X : Conversely, we assume that G(K x y ) for all y 2 X : By the assumption that is a convex compact set, it is easy to see that the reachable set R K (x ) is also convex and compact. If a point x is not null-controllable at step K > then =2 R K (x ): By the separation theorem of convex sets in Banach space [14], there exist y 2 X and some > such that <y x><; 8x 2 R K(x ): Therefore, we have sup <y x>< x2r K (x ) which contradicts the assumption and then completes the proof. 3. Global null-controllability Consider the time-varying control system (1), where, throughout this section we assume that is a convex compact subset in U: We are now in position to prove the following theorem of the global nullcontrollability of system (1). (5) Theorem 3.1. If 8 c> 9 T > : Z T H (B (s) (T s)y ) ds ck (T )y k 8y 2 X then the system (1) is globally null-controllable. system (1) is globally null-controllable, then Conversely, if the (6) 8y 2 X : Z 1 H (B (s) ( s)y ) ds =+1:
62 V. N. Phat, J. Y. Park, and I. H. Jung Proof. Assume that the condition (5) holds but the system (1) is not globally null-controllable. Then we can nd a point x which can not be null-controllable in any time t>: By Lemma 2.2 and Remark 2.1, for every t>, there is y t 2 X such that J(t x y t )=1; <x (t )y t > + H (B (s) (t s)y t ) ds < : Therefore H (B (s) (t s)y t ) ds << x (t )y t >j<x (t )y t > j kx kk (t )y t k which contradicts the condition (5). Conversely, we assume that the system (1) is globally null-controllable, but the condition (6) is not satised, i.e., (7) 9a > 9y 2 X : For every t, we dene Z 1 H (B (s) ( s)y ) ds a<+1: y t = ;1 (t )y : Since (t ) is nonsingular, we get y t x 2 X, the following relation 2 X : Let us consider for any J(t x y t )=1; <x (t )y t > + H (B (s) (t s)y t ) ds: Since we have (t )y t = y (t s) = ( s) (t ) J(t x y t )=1; <x y > + H (B (s) ( s)y ) ds:
From (7), it follows that Stability and constrained controllability 63 J(t x y t ) 1; <x y > + a: Therefore, for any xed > we take x = ;( + a +1)x 1 where x 1 2 X is chosen by the Hahn-Banach theorem [14], due to y 6= such that <y x 1 >= 1: We then obtain J(t x y t ) ;< which implies, by Lemma 2.2, the point x can not be null-controllable in any time. This contradicts the global null-controllability of the system. Remark 3.1. Note that if X U are nite-dimensional, we can check that the conditions (5) and (6) are equivalent and then for nitedimensional systems we derive the following global null-controllability criterion. Corollary 3.1. [1] Let dim X < +1 dim U < +1: The system (1) is globally null-controllable if and only if (8) 8y 2 B 1 : Z 1 H B (s) ( s)y ds =+1: It is worth to note that the condition (8) can be formulated in terms of the solution (t) of the adjoint equation (9) _ (t) =;A (t) (t) t : Indeed, since ;1 (t ) is the evolution operator of the adjoint system (9) and ( s) () = (s) we can take the solution of (9) in the form (t) = ;1 (t )y with the initial condition () = y 6= : We then obtain the null-controllability criterion in terms of nontrivial solutions of the adjoint system (9) presented in [1, 17] as follows.
64 V. N. Phat, J. Y. Park, and I. H. Jung Corollary 3.2. [17] Assume that dim X < +1 and dim U < +1: The system (1) is globally null-controllable if and only if Z 1 H B (s) (s) ds =+1 for all nontrivial solutions (t) of system (9). Based on Lemma 2.3 and using the same arguments that used in the proof of Theorem 3.1, the following discrete analog of Theorem 3.1, which gives a sucient condition for the global null-controllability of discrete-time control system (2), is consequently derived. Theorem 3.2. The discrete-time control system (2) is globally null controllable if 8 c> 9 K> 8y 2 X : K;1 X i= H B (i) (K i +1)y ck (K )y k: 4. Complete stabilizability implies null-controllability Consider system (1), where is a nonempty subset in U: In [4] it was shown that if the system (1), where A B are constants and =U X U are Hilbert spaces, is null-controllable then the system is exponentially stabilizable and the converse is, in general, not true. In a stronger type of complete stabilizability, i.e., if for any > there exist a linear bounded operator K : X! U and a number M > such that the generator S K (t) of the system _x = (A + BK)x satises the condition ks K (t)k Me ;t 8t it was proved in [7, 23] that the complete stabilizability implies exact null controllability. In this section we extend these results to the timevarying systems (1), (2) with constrained controls in Banach space U: Let us rst remark that if A is a constant operator, the evolution operator (t ) = S(t) is, as in [14, 22], the generator of the innitesimal operator A and has the following important property: 9N > 9 2 R : ks(t)k Ne jtj 8 t 2 R:
Stability and constrained controllability 65 In time-varying case, it is, in general, not true and therefore, we shall assume, throughout this section, that for every t s the evolution operator (t s) generated by A(t) is a linear operator acting in X and satises the following condition. (A) 9N > 9 2 R : k(t s)k Ne jt;sj for all t s : It is obvious that if A(t) is uniformly bounded for all t then (t s) satises condition (A). Theorem 4.1. Assume the condition (A). If the system (1), where X U are Banach spaces, is a convex compact subset of U is completely stabilizable, then the system is globally null-controllable. Proof. Let N > 2 R be given numbers dened by the condition (A) such that For all y : ky k =1 we get k ( t)k = k( t)k Ne t t : 1=k ( t) (t )y kk ( t)kk (t )y k and hence (1) 1 k (t )y k k ( t)k Ne t 8t : Taking any > maxf g by the complete stabilizability of system (1), there exist a linear bounded operator function K(t) :X! t and M > such that k K (t )k Me ;t t where K (t s) is the evolution operator of system (1.1). For all y : ky k =1we get the estimate (11) k K(t )y kk K(t )k = k K (t )k Me ;t 8 t : For the operator K(t) we consider the following linear dierential system _x(t) =A(t)x(t)+B(t)K(t)x(t) t x() = x :
66 V. N. Phat, J. Y. Park, and I. H. Jung The solution of this system is dened either by x(t x )=(t )x + (t s)b(s)k(s)x(s) ds or by Therefore, we get x(t x )= K (t )x : K (t )x =(t )x + For every y 2 X, we also get < K(t )y x > =< (t )y x > + Since K(s)x(s) 2, we have <B (s) (t s)y K(s)x(s) > ds and hence for every y 2 X we also have < K(t )y x >< (t )y x > + or equivalently, (t s)b(s)k(s)x(s) ds: <B (s) (t s)y K(s)x(s) > ds: < (t )y ;x >< K(t )y ;x > + H (B (s) (t s)y ) ds H (B (s) (t s)y ) ds Therefore, since x is an arbitrary, we obtain the relation (12) < (t )y x>< K(t )y x>+ H (B (s) (t s)y ) ds: H (B (s) (t s)y ) ds
Stability and constrained controllability 67 for all x 2 X y 2 X : We now assume to the contrary that the system (1) is not globally null-controllable. By Theorem 3.1, there is a number c> and for any sequence t k! +1 and y k 2 X such that (13) k H (B (s) (t k s)y k) ds < ck (t k )y kk: From the above strict inequality it follows that y k and (13) gives 6=: Combining (12) < (t k )y k x><< K(t k )y k x>+ck (t k )y kk for all x 2 X: Since y k 6= the above estimate holds for all y k 2 B 1 : On the other hand, since k (t k )y kk6= we then get where Consequently, < y k x><<f K(t k ) x>+c F K (t k )= K (t k )y k k (t k )y k k y k = (t k )y k k (t k )y k k 2 B 1 : < y k ; F K (t k ) x><c for all x 2 X: Let us take a number a > such that ac = < 1: We get < y k ; F K (t k ) y >< for all y = ax 2 X: The above estimate holds for all y 2 X we then get (14) ky k ; F K (t k )k: Let us set W K (k) =ky k ; F K (t k )k: As remarked above, ky kk =1 then from (14) it follows that (15) 1 ;kf K (t k )k = ky kk;kf K (t k )kw K (k) :
68 V. N. Phat, J. Y. Park, and I. H. Jung On the other hand, since y k account we nally obtain 2 B 1 taking (1), (11) and (15) into 1 ; kf K (t k )kce (;+)t k where C = NM: Since > as chosen before, letting k! 1 the right-hand side of the above inequality tends to while the left-hand side is 1 ; > which leads to a contradiction. The theorem is proved. Remark 4.1. Note that Theorem 4.1 is an extension of a result of [2] for nite-dimensional systems without constrained controls and of [7, 21] for unconstrained control systems in a Hilbert space, where one requires 2 R: We now consider the discrete-time control system (2). Let us make the following assumption : (B) 9N > 9p >: k (k i)k Np jk;ij for all k i : Note that the condition (B) holds if the operator A(k) is uniformly bounded for all k 2 Z + : Theorem 4.2. Assume the condition (B). If the discrete-time system (2), where X and U are Banach spaces and is a convex compact subset of U is completely stabilizable, then the system is globally nullcontrollable. Proof. As in the proof of Theorem 4.1, for numbers N > p > given by the assumption (B), for every k 2 Z + and for all y 2 X : ky k =1 we get (16) 1 k (k )y k k ( k)k = k ( k)k Np k : Let us take a number q > such that q < minf1 1=pg: By the denition of the complete stabilizability of system (2), there exist a linear operator Q(k) : X! k 2 Z + and a number M > such that for every k 2 Z + and for all y 2 X : ky k =1 we have (17) k Q(k )y kk Q(k )k = k Q(k )k Mq k
Stability and constrained controllability 69 where Q(k i) is the transition operator of system (2.1). We now assume to the contrary that the system (2) is not globally null-controllable. By Theorem 3.2, there is a number c > such that for any sequence t k!1and y k 2 X such that t k ;1 X i= H (B (i) (t k i+1)y k) <ck (t k )y kk: By the same arguments that used in the proof of Theorem 4.1, we will arrive at the following estimate (18) 1 ;kf Q (t k )k where <1 and F Q (t k )= Q (t k )y k k (t k )y k k y k 2 B 1 : Taking into (16), (17) and (18) into account, we obtain 1 ; MN(qp) k : Since < 1 and < q < 1=p as choosen before, letting k! 1 we again arrived at a contradiction, which completes the proof. 5. Conclusions Global null-controllability of linear time-varying control systems with constrained controls in Banach spaces was studied. New necessary and sucient conditions for global null-controllability of the systems were established. We showed that complete stabilizability implies exact global null-controllability for both continuous and discrete-time systems with restrained controls. The obtained results can be considered as extensions of the results obtained earlier for linear timeinvariant control systems without constrained controls in nite dimensional spaces.
61 V. N. Phat, J. Y. Park, and I. H. Jung 6. Acknowledgements The paper was written during the rst author's stay at the Department of Mathematics, Pusan National University, Pusan, Republic of Korea. Research is supported by the Korea Scientic Engineering Foundation (KOSEF). The authors would like to thank Prof. J. Zabczyk for his valuable comments and discussions. References [1] N. U. Ahmed, Elements of Finite-dimensional Systems and Control Theory Pitman SPAM, Longman Sci. Tech. Publ. 37 (199). [2] O. Carja, On constrained controllability of linear systems in Banach spaces, J. Optim. Theory Appl. 56 (1988), 215{225. [3] R. F. Curtain and A. J. Pritchard, Innite-dimensional Linear Systems Theory, Lecture Notes in Contr. Inform. Sci., Springer-Verlag 8 (1981). [4] R. Datko, Uniform asymptotic stability of evolutionary process in a Banach space, SIAM J. Math. Anal. 3 (1972), 428{445. [5] V. Komornik, Exact Controllability and Stabilization, John Wiley & Sons, New York, 1994. [6] S. G. Krein, Linear Dierential Equation in Banach Spaces, Nauka, Moscow (in Russian), 1967. [7] G. Megan, On the stabilizability and controllability of linear dissipative systems in Hilbert space, S.E.F., Universitate din Timisoara, 3 (1975). [8] S. K. Mitter, Theory of inequalities and the controllability of linear systems In: "Math. Control Theory", Academic Press, 1967, pp. 23{212. [9] S. Nikitin, Global Controllability and Stabilization of Nonlinear Systems, World Scientic, Singapore, 1994. [1] V. N. Phat, Constrained Control Problems of Discrete Processes, World Scientic, Singapore, 1996. [11], Weak asymptotic stabilizability of discrete-time systems given by setvalued operators, J. Math. Anal. Appl. 22 (1996), 363{378. [12] V. N. Phat and J. Y. Park, Further generations of Farkas' theorem and applications in optimal control, J. Math. Anal. Appl. 216 (1997), 23{39. [13] A. Pritchard and J. Zabczyk, Stability and stabilizability of innite-dimensional systems, SIAM Review 23 (1981), 25{52. [14] S. Rolewics, Functional Analysis and Control Theory,Polish Sci. Publ., Warszawa, 1987. [15] Robert val Til P., Constrained controllability of discrete-time systems, Ph.D thesis, Northwestern University, Evanston IL, 1985. [16] T. Schanbacher, Aspect of positivity in control theory, SIAM J. Contr. Optim. 27 (1989), 457{475.
Stability and constrained controllability 611 [17] W. Schmitendorf and B. Barmish, Null-controllability of linear systems with contrained controls, SIAM J. Contr. Optim., 18 (198), 327{345. [18] N. K. Son, A unied approach to constrained controllability for the heat equations and the retarded equations, J. Math. Anal. Appl. 15 (199), 1{19. [19] Y. J. Sun and J. G. Hsieh, Robust stabilization for a class of uncertain systems with time-varying delays, J. Math. Anal. Appl. 98 (1998), 161{173. [2] W. M. Wonham, On pole assignment in multi-input controllable linear systems, IEEE Trans. Automat. Control 12 (1967), 66{665. [21] J. Zabczyk, Complete stabilizability implies exact controllability, Seminarul Ecuati Functionale 38 (1977), 1{1. [22], Some comments on stabilizability, Appl. Math. Optim. 19 (1988), 1{9. [23], Mathematical Control Theory: An Introduction, Birkhauser, 1992. Vu Ngoc Phat Institute of Mathematics P.O. BOX 631, BO HO, Hanoi, Vietnam E-mail: vnphat@hanimath.ac.vn Jong Yeoul Park Department of Mathematics Pusan National University Pusan, 69-735, Korea E-mail: jyepark@hyowon.pusan.ac.kr Il Hyo Jung Department of Mathematics Pusan National University Pusan, 69-735, Korea E-mail: ilhjung@hyowon.pusan.ac.kr