Available online at www.sciencedirect.com Systems & Control Letters 52 (24) 221 232 www.elsevier.com/locate/sysconle Riesz bases and exact controllability of C -groups with one-dimensional input operators Bao-Zhu Guo a;, Gen-Qi Xu b a Academy of Mathematics and System Sciences, Academia Sinica, Beijing 18, PR China b Department of Mathematics, Tianjin University, Tianjin 372, PR China Received 26 September 23; received in revised form 18 November 23; accepted 1 December 23 Abstract This paper considers linear innite dimensional systems with C -group generators and one-dimensional admissible input operators. The exact controllability and Riesz basis generation property are discussed. The corresponding results of Jacob and Zwart (Advances in Mathematical Systems Theory, Birkhauser, Boston, MA, 2) under the assumption of algebraic simplicity for eigenvalues of the generator are generalized to the case in which the eigenvalues are allowed to be algebraically multiple but with a uniform bound on the multiplicity. c 23 Elsevier B.V. All rights reserved. Keywords: Riesz basis; Controllability; Functions of exponentials; Semigroups 1. Introduction Many systems describing vibrations of exible structures with boundary control can be put into the form of innite-dimensional systems of the following kind: ẋ(t)=ax(t)+bu(t); (1) where A: D(A) H is the generator of a C -group T (t) on the complex separable Hilbert space H and b is an admissible one-dimensional control operator, i.e., b [D(A )], the dual space of the graph space [D(A )], where A is the adjoint operator of A. t T(t s)bu(s)ds Corresponding author. Tel.: +86-162651443; fax: +86-161587343. E-mail address: bzguo@iss3.iss.ac.cn (B.-Z. Guo). denes a bounded linear operator from L 2 (;t)toh for some (and hence all) t. The input function u is assumed to be in L 2 loc (; ). Under these conditions, for any x H and u L 2 loc (; ), (1) admits a unique solution given by x(t)=t(t)x + t T(t s)bu(s)ds; t : (2) Weiss [16] showed that x( ) lies in H and is continuous. We say that system (1) isexactly controllable in time t if for any x H there exists an input function u x L 2 (;t ) such that =T(t )x + t T(t s)bu x (s)ds: (3) Exactly, the above denition is the denition of exact null-controllability. But because T( )isac -group, it coincides with exact controllability. 167-6911/$ - see front matter c 23 Elsevier B.V. All rights reserved. doi:1.116/j.sysconle.23.12.1
222 B.-Z. Guo, G.-Q. Xu / Systems & Control Letters 52 (24) 221 232 In the spirit of Jacoband Zwart [9 11], this paper continues studying the exact controllability of system (1). It is well known that system (1) cannot be exactly controllable if H is innite-dimensional and b is an element of H (see [4, Theorem 4.1.5]). However, when b is an unbounded operator, the situation is quite dierent. A typical practical example is the following Euler Bernoulli beam equation which is widely used in the control of vibrations of the exible arms: y tt (x; t)+y xxxx (x; t)=; y(;t)=y x (;t)=y xx (1;t)=; x 1; t ; y xxx (1;t)=u(t): (4) System (4) can be written as y tt (x; t)+y xxxx (x; t)+(x 1)u(t)=; y(;t)=y x (;t)=y xx (1;t)=y xxx (1;t)=; (5) where ( 1) denotes Dirac function. It is known (see, e.g., [8,13]) that system (5) is exactly controllable in any time t. Another widely used example is the following model of a NASA spacecraft control laboratory experiment (SCOLE): (x)y tt (x; t)+(ei(x)y xx (x; t)) xx =; x 1; t ; y(;t)=y x (;t)=; my tt (1;t) (EIy xx ) x (1;t)=; Jy xtt (1;t)+EI(1)y xx (1;t)=u(t): (6) It was shown in [7] that system (6) is exactly controllable in some graph space in any nite time t. The motivation of this work is using the eigenpairs of operator A to characterize the exact controllability of system (1) and vice versa. A variety of necessary and sucient conditions have been available in the literature that ensure the exact controllability of system (1), see, for instance, Ref. [1] and the references therein. Here we are concerned with the relationship between the exact controllability and Riesz basis property of the eigenvectors of the operator A. Asin[1], we assume that A generates an exponential stable C -semigroup. However, the exponential stability assumption is not restrictive because both admissibility and exact controllability are invariant with respect to a scalar shift of A. In the sequel, we also use (A; b) to refer to system (1). When system (1) is exactly controllable, it has been shown in [9,1] that the spectrum of A is of a very special form. We summarize these results as follows. Theorem 1. Assume that system (1) is exactly controllable. Then (i) the spectrum of A consists of isolated eigenvalues: (A) ={ n } and inf n Re n 6 sup n Re n ; (ii) each eigenvalue has nite algebraic multiplicity and geometric multiplicity one; (iii) (A )={ n n Z} and every n is an isolated eigenvalue of A with nite algebraic multiplicity and geometric multiplicity one; (iv) both span{e( n ;A)H; } and span{e( n ;A ) H; n Z} are dense in H, where E(; ) denotes the eigen projection with respect to the spectral (v) set ; m n (7) n 1+ for any, where m n is the algebraic multiplicity of n. Let us recall that a scalar sequence of complex numbers { n n Z} is called separated if inf n m : (8) n;m Z;n m A sequence {f i } i=1 in H is called bi-orthogonal to the sequence {e i } i=1 if { 1; i= j; f i ;e j = ij = ; i j; i; j =1; 2;::: : The sequence {e i } i=1 is called a basis for H if any element x H has a unique representation x = c i e i ; (9) i=1
B.-Z. Guo, G.-Q. Xu / Systems & Control Letters 52 (24) 221 232 223 the series being convergent with respect to the norm of H. {e i } i=1 is called a Riesz basis for H if (a) span{e i } = H, and (b) there exist positive constants m and M such that for an positive integer n and any numbers c i ;i= 1; 2;:::;n, we have n m c i 2 n 2 n 6 c i e i 6 M c i 2 : i=1 i=1 i=1 In a Hilbert space, the most important bases are orthonormal. Second in importance are Riesz bases that are bases equivalent to some orthonormal basis. We refer to [18] for more details on Riesz bases. We denote by H 1 the completion of H with respect to the norm 1 = ( A) 1 for some (A): According to Weiss [16], H 1 is equivalent to [D(A )]. For any (A);R(; A) =( A) 1 has a natural extension R(; A) from H 1 to H: R(; A)x = R(; A)x for all x H. A can be extended to all of H by Ãx; y = x; A y for any x H; y D(A ): Ã is an isomorphism from H to H 1. By Proposition 3.3 of [16], for any L L(H) which commutes with A, there is an extension L on H 1 : L =( Ã)L R( ;A): In the sequel, we still use A and E(; A) to denote their extensions to H 1. Let { n } be the normalized eigenvectors of A associated with { n } and { n } the normalized eigenvectors of A associated with { n }. For the case of m n = 1, that is, each n is algebraically simple, the equivalence of the following conditions are proven in Corollary 1.1 of [1]. Theorem 2. Assume that the system (A; b) is exactly controllable and each eigenvalue of A is algebraically simple. Then the following statements are equivalent: (i) inf n m n m. (ii) inf n n ; n. (iii) { n } is a Riesz basis in H. (iv) { n } is a Riesz basis in H. A characterization of exact controllability and Riesz basis generation were obtained in Theorem 1.2 of [1] assuming that the eigenvalues are simple. Theorem 3. Assume that each eigenvalue n of A is algebraically simple and the eigenvalues are separated: inf n m n m. Then the following conditions are equivalent: (i) (A; b) is exactly controllable. (ii) For each x H, there exist functions x ( ) H 2 (C + ; H) and! x ( ) H 2 (C + ) such that x =( A) x () b! x () C + : (iii) For each x H, there exists a function! x ( ) H 2 (C + ) such that! x ( n )= x; n b; n (iv) 2 x; n b; n n Z: x H: n N (v) inf n b; n 6sup n b; n and { n } forms a Riesz basis for H. (vi) inf n b; n 6sup n b; n and { n } forms a Riesz basis for H. It is seen that in both Theorems 2 and 3, there is a basic assumption that all eigenvalues of A are algebraically simple. This, however, is not always the case in applications. The simple string equation with viscous damping is an example where A has multiple eigenvalues [3] y tt (x; t)+y xx (x; t)+y t (x; t)=; x 1; t ; y(;t)=; y x (1;t)=u(t) (1) which can be written as y tt (x; t)+y xx (x; t)+y t (x; t) (x 1)u(t)=; y(;t)=y x (;t)=: (11)
224 B.-Z. Guo, G.-Q. Xu / Systems & Control Letters 52 (24) 221 232 If = n =2 where n =2 is some eigenvalue of the free system ( = u = ), then multiple eigenvalue appears for the uncontrolled system (u = ). In this paper, we generalize the above results to the case of multiple eigenvalues. In the next section, wegiveacharacterizationoftheresolventofaunderthe exact controllability condition. Our result implies the conclusion (iv) of Theorem 1. In Section 3, Theorems 2 and 3 are generalized to the case in which sup m n : n Note that the case sup n m n = may contradict the assumption of exact controllability. We refer to counterexamples for which A is a discrete operator (that is, the resolvent is compact) but does not satisfy the spectrum-determined growth condition (see e.g. [14]). In such a case, (A; B) is never exactly controllable for any nite-dimensional admissible input operator B, by Theorem 5.14 of [9]. 2. Some basic facts and improved (;!) representation For any, denote by S n; = { C n } the circle centered at n with radius in the complex plane. Lemma 1. Suppose that (A; b) is exactly controllable with isolated separated eigenvalues { n }. Then for any ; R(; A) is uniformly bounded in G = C S n;. Proof. Since A generates a C -group, it follows from the Hille Yosida theorem that there exist!; M such that R(; A) 6 M as Re!: Re! If the lemma is not true, then there exists a sequence {s n } with s n G such that sup n R(s n ;A) =. It follows that Re s n 6!. We need only consider the case of s n since (A; b) is exactly controllable if and only if ( A; b) is exactly controllable. Take ==2. Then {s C s s n } (A). Since the exact controllability condition is stronger than the optimizability condition, it follows from Lemma 5.12 of [9] that sup n R(s n ;A), contradiction. The result follows. Recall that an entire function f( ) is said to be of exponential type if the inequality f(z) 6 Ce L z (12) holds for some positive constants C and L and all complex values of z. The smallest of constants L is said to be the exponential type of f( ) [18]. Lemma 2. If (A; b) is exactly controllable in t then for any x H, there exist entire functions of exponential type x ( ) and! x ( ) such that x =( A) x () b! x () C; where both the exponential type of x and! x are at most t. Moreover x ( ) H 2 (C + ; H);! x ( ) H 2 (C + ). Proof. Since (A; b) is exactly controllable in t, for any x H, there exists u x L 2 (;t ) such that x = t T( s)bu x (s)ds: Dene operator B t : L 2 (;t ) H: B t u = t T( s)bu(s)ds: (13) Since b is admissible, B t is a linear bounded operator from L 2 (;t )toh. Set U = ker(b t ) : (14) Then B t is a 1 1 mapping from U to H: For any x H, there exists a unique u x U such that x = B t u x. Now for x H, dene the function û x in L 2 loc (; ) by { ux (t); 6 t 6 t ; û x = ; t t and the function x( ) inh: x(t)=t(t)x + t T(t s)bû x (s)ds: Then x( ) is continuous in H and x() = x; x(t) = ; t t. Dene entire functions x ()= t e t x(t)dt;! x ()= t e t u x (t)dt: (15) Then under the restriction u x U, both x and! x are uniquely determined by x H. It is obvious that both
B.-Z. Guo, G.-Q. Xu / Systems & Control Letters 52 (24) 221 232 225 x and! x are entire functions of exponential type at most t. We show that x D(A). Notice that x ()= t t e t T(t)x dt t + e t dt T (t s)bu x (s)ds: From C -semigroup theory, the rst term above belongs to D(A) and ( A) t e t T(t)x dt = e t x(t )+x = x: For the second term, it holds T(h) I t t lim e t dt T (t s)bu x (s)ds h + h t =e t T(t s)bu x (s)ds t t + e t dt T (t s)bu x (s)ds t b e t u x (t)dt and so x D(A) and x =( A) x () b! x ()+e t x(t ) =( A) x () b! x (): Furthermore, since x(t);u x (t) are square integrable functions, it follows from the Paley Wiener theorem that x ( ) H 2 (C + ; H);! x ( ) H 2 (C + ). Let n be an eigenvalue of A with algebraic multiplicity m n. We say that n;1 is a highest order generalized eigenvector of A if (A n ) mn n;1 =; (A n ) mn 1 n;1 : Lemma 3. Assume that (A; b) is exactly controllable. n is an eigenvalue of A with algebraic multiplicity m n and is an highest order generalized eigenvector of A corresponding to n. Choose and! as in Lemma 2 so that =( A) () b! (): Denote by the zero set of! (). Then (A) { n } and! ( ) ( k)=; {; 1; 2;:::;m k 1} k n: Proof. This is Proposition 12.7 of [1], we omit the details. Theorem 4. Assume that (A; b) is exactly controllable in t. Then R(; A) can be represented as F(; A) R(; A)= P() ; (16) where F( ;A) is an operator-valued entire function of exponential type and P( ) is a scalar entire function of exponential type, both with exponential type at most t. Proof. Let { n } be the spectrum of A. From Lemma 3, for any xed n; { k ;k n} is a subset of zeros of an entire function of exponential type!. So the canonical product P() of{ n } does exist [18]. Let ()= ( n) mn! () ; P() where! is determined by Lemma 3. Then is an entire function and ( n ) mn! ()=()P(): Since ( n ) mn! () is an entire function of exponential type at most t. From complex analysis, the exponential type of P() is at most t. Let F(; A)=P()R(; A): Then F( ;A) is an operator-valued function. From Lemma 1, when C S n;; R(; A 6 M() for some ; M(), and hence F(; A) 6 M() P(). This, together with the fact that F( ;A) attains its maximal value on the boundary in each closed ball of S n; by the analyticity of F( ;A), concludes that F(; A) is an entire function of exponential type at most t. In [1] it is shown that the root subspace of A and A are complete. Using Theorem 4, we obtain an alternative proof of this fact. Here we only give an outline of the proof of this fact, because the complete proof can be found in [15,17]. Let Sp(A ) denote the closed
226 B.-Z. Guo, G.-Q. Xu / Systems & Control Letters 52 (24) 221 232 subspace spanned by all generalized eigenvectors of A in H. Then from Lemmas 5 and 6 of [5] on pp. 2355 and 2296, respectively, H = (A) Sp(A ); where (A) = {x H R(; A)x is analytic in C}. Since for any x (A);R(; A)x is a H-valued entire function of and the orders of both entire functions of F and P in (16) are less than or equal to 1, therefore, from general complex analysis, the order of R(; A)x is less than or equal to 1. That is, there is a such that R(; A)x = O(e 1+ ) as : Since A generates a C semigroup and A generates an exponential stable C -semigroup, it follows that R( ;A)x is uniformly bounded in both real and imaginary axis. Notice that R( ;A)x is uniformly bounded in the left complex plane. Applying the Phragmen Lindelof s theorem to R(; A)x in each angular region { = r i r ; 6 6 =2} and { = r i r ; =2 6 6 }, we know that R( ;A)x is also uniformly bounded in the right complex plane. It then follows from Liouville s theorem that R( ;A)x is a constant vector in the whole complex plane and hence x =. Therefore, Sp(A )=H. On the other hand, notice that R( ; A )= F (; A) : P() Same arguments show that Sp(A)= H. 3. Equivalent conditions for multiple eigenvalues We introduce some notations. We always assume that A satises parts (i) (iv) of Theorem 1. For each eigenvalue n, let n;1 be a highest order generalized eigenvector of A associated with n. Then other linearly independent generalized eigenvectors can be found through n;j =(A n ) j 1 n;1 ;j=2; 3;:::;m n. Let {{ n;j } mn } be the bi-orthogonal sequence of {{ n;j } mn }. Then (A n ) n;1 =; n;j = (A n ) n;j+1 ;j=2; 3;:::;m n 1. We can always make { n;mn } uniformly bounded with respect to n. Denote b n j = b; n;j [D(A )] [D(A )] for each j and n and b n 1 b n 2 b n 1 B n = b n 3 b n 2 b n 1 ;..... b n m n b n m n 1 b n m n 2 b n m n 3 b n 1 n Z: (17) n =( n;1 ; n;2 ;:::; n;mn ) T ; n =( n;1 ; n;2 ;::: n;mn ) T ; n Z: (18) Let n(x) denote the vector n(x)=[ x; n;1 ; x; n;2 ;:::; x; n;mn ] T ; x H; n Z: (19) The vector n (x) is dened similarly, by replacing n;j with n;j in the denition of n(x). Theorem 5. Assume A satises parts of conditions (i) (iv) of Theorem 1. Then the following conditions are equivalent: (i) (A; b) is exactly controllable in t (ii) For every x H, there exist entire functions x ( ) and! x ( ) of exponential type at most t such that x =( A) x () b! x () C; where also x H 2 (C + ; H) and! x H 2 (C + ). (iii) For every x H, there exists an entire function! x ( ) H 2 (C + ) of exponential type at most t such that x ( n ) ( =! x ( n );! x( n );! x ( n ) 2! x ( n ) (m n 1)! ;:::;!(mn 1) ) T = B 1 n n(x) n Z: (2) Proof. (i) (ii) follow from Lemma 2 and Proposition 12.5 of [1].
B.-Z. Guo, G.-Q. Xu / Systems & Control Letters 52 (24) 221 232 227 (i) (iii). Let! x ( ) be the function dened in (15). It is found directly that! x (k) ()= t e s ( s) k u x (s)ds: For each n, note that E( n ;A)b = m n b; n;j n;j and hence E( n ;A)x = t = k=1 T( s)e( n ;A)bu x (s)ds t ns ( s)k 1 e (k 1)! (A n) k 1 E( n ;A)bu x (s)ds = b; n;j k+1!(k 1) x ( n ) n;j (k 1)! k=1 j=k [ j = k=1 On the other hand, E( n ;A)x = x; n;j n;j : b n! (k 1) j k+1 ] x ( n ) n;j : (k 1)! Comparing these two expressions, we obtain B n x ( n )= n(x) n Z: (21) By the exact controllability assumption, b n 1.So Bn 1 exists, proving the conclusion. (iii) (i). Suppose such an entire function of exponential type! x ( ) does exist. Then since! x () is square integrable along the imaginary axis, by the Paley Wiener theorem [18, Theorem 18, p. 11] there exists a u x (s) L 2 (;t ) such that! x ()= t e s u x (s)ds: (22) For this function u x, we compute B t u x + x as follows: B t u x + x; n;j = t T( s)e( n ;A)b; n;j u x (s)ds + x; n;j = k=1 t ns ( s)k 1 e (k 1)! u x(s)ds E( n ;A)b; (A n ) k 1 n;j + x; n;j i = b n! x (k 1) ( n ) j k+1 + x; n;j (k 1)! =: k=1 Since { n;j j =1; 2;:::;m n } is complete in H, the above implies that B t u x = x. The proof is complete. Remark 1. From the last paragraph of the proof of the Theorem 5, we see that if! x ( ) is the function of exponential type at most t in the (;!) representation (ii) of Theorem 5, then the Paley Wiener theorem ensures that! x and u x are related by (22). And the function u x is just the control function which drives x into zero at time t. Moreover, the Plancherel s theorem shows that! x (i) 2 d =2 t u x (t) 2 dt: (23) Besides, from Theorem 17 and its Remark in [18] (see pp. 96 98), for any separated sequence { n } satisfying sup n Re n, there exists a constant C such that! x ( n ) 2 6 C! x (i) 2 d t =2C u x (t) 2 dt: (24) Remark 2. Suppose (A; b) is exactly controllable in t and take x = in (2), that is B t u x =. Then we have x ( n ) = for all n Z. By(22), we see that t t j e nt u x (t)dt = 6 j 6 m n 1 ;n Z: That is, any element u x in the kernel of operator B t in L 2 (;t ) is orthogonal to functions t j e nt ; 6 j 6 m n 1; n Z. Suppose that we have ordered eigenvalues { n } of A so that {Im n } forms a nonincreasing sequence in
228 B.-Z. Guo, G.-Q. Xu / Systems & Control Letters 52 (24) 221 232 C with respect to n Z. We make the following assumptions: inf n Re n 6 sup Re n 6 h n for some h and all n Z; inf n m n m ; sup m n : (25) In the sequel, we need the basis property of the family of exponential functions {g n;j (t)} in L 2 (;t ) dened as in following: = {G n (t)} = {[g n;1 (t);g n;2 (t);:::;g n;mn (t)] T } ; g n;j (t)= ( t)j 1 (j 1)! e nt : (26) The basis property of such exponential functions in L 2 space has been studied extensively by former Soviet mathematicians (Levin, Pavlov, Nikolskii and many others). Here we refer to recent work by Avdonin and Ivanov [2]. Denition 1. Let k ;k=1; 2;:::;m, be arbitrary complex numbers (not necessarily dierent). The generalized divided dierence (GDD) of order zero of the function e t corresponding to the point 1 is dened as [ 1 ](t)=e 1t. GDD of the order n 1;n6 m of e t corresponding to { k ;k=1; 2;:::;n} is dened by [ 1 ; 2 ;:::; n ](t) = [ 1; 2;:::; n 1](t) [ 2; 3;:::; n](t) 1 n ; 1 n ; @ @ [; 2;:::; n 1 ](t) =1 ; 1 = n : Note that if i = ; i =1; 2;:::;n, then [ 1 ; 2 ;:::; i ](t)=t i 1 e t ; 1 6 i 6 n: (27) It should be pointed out that in Denition 1 of GDD in [2], all n are written i n. Here we remove i. The reason is that in [2], the basis property of exponential functions {( it) j 1 =(j 1)! e in } is considered, where { n } are located in the strip paralleling to the real axis. But here we need the basis property of functions {g n;j (t)} which can be written as {g n;j (t) = ( t) j 1 =(j 1)! e i(in)t }.Soi n here plays the role of n in [2]. The following proposition is key to the proof of our main results of this paper. Proposition 1. Under assumption (25), there exists a t (and hence for all t t ) such that forms a Riesz basis for span in L 2 (;t ). In particular, if {F n (t)} = {(f n;1 (t);f n;2 (t);:::;f n;mn (t)) T } is the bi-orthogonal sequence of in span, then there are constants C 1 and C 2 so that for any u span ; u= U n T G n (t); U n 2, it holds 2 C 1 U n 2 6 Un T G n (t) L 2 (;t ) 6 C 2 U n 2 ; (28) where U n =( u; f n;1 ;:::; u; f n;mn ) T. Proof. Let =inf n m n m. Denote by D n (R) a disk with center n and radius R =2. Let nj = n ; 1 6 j 6 m n ;n Z; = { nj 1 6 j 6 m n ;n Z}. We use the same notations as Avdonin and Ivanov [2]: n + (r) = sup #{Im [x; x + r)}; x R D + n + (r) () = lim : r r For any x R, suppose there are M number of balls with radius R=2, which covers the compact region (x) ={ Re 6 h; Im [x; x +1]} of C. Note that M is independent of x by unit shift. Then there are at most km number of inside (x);k= sup n m n. Hence for any r, we have n + (r) = sup #{Im [x; x + r)} x R 6 sup #{Im [x; x +([r] + 1))} x R 6 ([r]+1)km; where [r] denotes the maximal integer not exceeding r. Therefore, D + () 6 km. Note that there are m n number of nj in the disk D n (R). Make GDD in D n (R) of following: {[ n1 ](t); [ n1 ; n2 ](t);:::;[ n1 ; n2 ;:::; nmn ](t)}:
B.-Z. Guo, G.-Q. Xu / Systems & Control Letters 52 (24) 221 232 229 Then it follows from Theorem 3 of [2] that for any t 2D + (), the family {[ n1 ](t); [ n1 ; n2 ](t);:::; [ n1 ; n2 ;:::; nmn ](t);n Z} forms a Riesz basis in the closed subspace spanned by itself in L 2 (;t ). The result then follows from (27) and the assumption that sup n m n. Remark 3. Assume that t is as in Proposition 1 which makes (A; b) exactly controllable in t. Then we have the explicit representation of U dened by (14), U = span. Indeed, on the one hand, span U follows from Remark 2. On the other hand, if u x L 2 (;t ) satises t tj 1 e nt u x (t)dt =, i.e.,! (j 1) x ( n ) = for all 1 6 j 6 m n ;, then it follows from (2) that n(x) =for all n Z. Since { n } is dense in H, this implies that x =. That is, B t u x =oru x Ker(B t ). Therefore, U span. Remark 4. Assume that t is as in Proposition 1 which makes (A; b) exactly controllable in t. Then for any x H, we dene (motivated from (21)) u x (t)= (B 1 n n(x)) T F n (t): By (2) and (24), B 1 n n(x) 2 and hence u x L 2 (;t ). Dene t! x ()= e t u x (t)dt: Then (2) is satised. As we mentioned in Remark 1 that such a u x (t) is nothing but the control which drives x into zero at time t. However, in this form u x takes the feedback form with respect to x. Remark 5. Assume that t is as in Proposition 1 which makes (A; b) exactly controllable in t. Let B t be dened by (13) and B t f n;j = x n;j. Then by letting! x =! x n; j in (2), we have (; ;:::;1 j ; ;:::;) T = B 1 n (x n;j ); where 1 j denotes the element 1 at jth position. Hence (B t f n;1 ;B t f n;2 ;:::;B t f n;mn ) T =B t F n (t)=b T n n ; n Z: (29) Lemma 4. Assume that (A; b) is exactly controllable in t and condition (25) is satised. Then inf b; n;1 n n;1 6 inf B n 6 sup B n : n Proof. The rst inequality comes from Jacoband Zwart [9]. The second one is trivial. For the third inequality, we rst show that there exists a M, such that for any n, it holds (A n ) k E( n ;A)b k! 6 M k ; n Z: (3) Indeed, by Lemma 1 and assumption (25), there exists a such that E( n ;A)= 1 R(; A)d 2i n = is uniformly bounded with respect to n Z. Since b is admissible, B t u = t T( s)bu(s)ds is bounded from L 2 (;t )toh. Sois E( n ;A)B t u = t = T( s)e( n ;A)bu(s)ds t n ns ( s)j 1 e (j 1)! u(s)ds (A n ) j 1 E( n ;A)b: Set k = sup n m n. Since {1; ( s); ( s) 2 ; ( s) 3 ;:::;( s) k 1 } is linearly independent in L 2 (;t ), there exists its bi-orthogonal sequence {u 1 ;u 2 ;u 3 ;:::;u k } such that t ( t) j 1 u i (t)dt { 1; i= j; = ij = ; i j; 1 6 i; j 6 k: Now, we choose function u n;j (s)=e ns u j (s). It has u n;j 2 = t u n;j (s) 2 ds t = e ns u j (s) 2 ds 6 e 2th u j 2 :
23 B.-Z. Guo, G.-Q. Xu / Systems & Control Letters 52 (24) 221 232 Hence sup u n;j 6 e th max u j : n;j 16j6k Under this group of functions (A n ) j 1 (j 1)! E( n ;A)b = E( n ;A)B t u n;j 6 e th E( n ;A) B t u j 1 6 j 6 m n ; n Z: Next, notice that B n 2 6 m n b; n;j 2 : We need only show that b; n;j 2 is uniformly bounded with respect to n Z. Since (A n ) mn j n;m n = n;j ; we have b; n;j 2 6 b; (A n ) mn j n;m n 2 6 (A ) mn j E( n ;A)b; n;mn 2 6 m n! n;mn 2 m n (A n ) mn j E( n ;A)b (m n j)! 6 e 2th m n! n;mn 2 E( n ;A) 2 B t 2 u j 2 which is uniformly bounded since we have chosen sup n n;mn to be uniformly bounded which is mentioned in the beginning of this section. The proof is complete. 2 Remark 6. Since sup n n;mn is uniformly bounded, by assumption (25) and Lemma 1 (A n ) k E( n ;A)= 1 ( n ) k R(; A)d 2i n = is uniformly bounded with respect to n and 1 6 k 6 m n, where is properly chosen so that n is the unique eigenvalue of A inside the disk { n 6 }. Therefore, E (; A)(A n ) mn j n;m n =(A n ) mn j n;m n = n;j is also uniformly bounded with respect to n and 1 6 k 6 m n. We rst generalize Theorem 2 into the case of multiple eigenvalues. Theorem 6. Assume that (A; b) is exactly controllable and the multiplicities of the eigenvalues of A have a nite upper bound: sup m n. Then the following conditions are equivalent: (i) inf n m n m. (ii) { n;j j =1; 2; 3;:::;m n } forms a Riesz basis for H. That is, there are constants C 1 ;C 2 so that for any x = (x)t n n, we have C 1 n(x) 2 2 6 n(x) T n 6 C 2 n(x) 2 : (iii) { n;j j =1; 2; 3;:::;m n } forms a Riesz basis for H. Proof. The equivalent between (ii) and (iii) follows from general basis theory (see e.g. [18, p. 37]). (i) (ii). Take t as in Proposition 1 which makes (A; b) exactly controllable in t. Then B t is bounded invertible from U to H. By formulae (4.12) of Kato [12] onp.28, Bn 1 6 B n mn 1 det B n ; (31) where is independent of n. From Lemma 4, there exists M such that Bn 1 6 M; B n 6 M for all n Z. Furthermore, by Remark 5, n =(Bn 1 ) T B t F n (t)
B.-Z. Guo, G.-Q. Xu / Systems & Control Letters 52 (24) 221 232 231 for all n Z. Since (Bn 1 ) T B t is uniformly bounded with respect to n; { n } forms a Riesz basis for H. (ii) (i). Take t as in Proposition 1 which makes (A; b) exactly controllable in t. Then B t is bounded invertible from U to H. From Remark 5, we know that B t (Bn 1 ) T F n (t) forms a Riesz basis for span in L 2 (;t ). Since from (31), B t (Bn 1 ) T is uniformly bounded with respect to n; F n (t) forms a Riesz basis for span in L 2 (;t ). Therefore {e nt } forms a Riesz basis for the closed subspace spanned by itself in L 2 (;t ). Thus, { n } is separated by the necessary condition of Riesz basis for the functions of exponentials (see e.g., [1, Theorem II.4.22]). Our nal result of following generalizes Theorem 3 to the case of multiple eigenvalues. Theorem 7. Assume that (25) is satised. Then the following conditions are equivalent: (i) (A; b) is exactly controllable. (ii) inf n b; n;1 = n;1 6sup n B n and Bn 1 n(x) 2 x H; n=1 where B 1 n n(x) denotes the Euclidean norm of R mn. (iii) inf n b; n;1 = n;1 6 sup n B n and { n;j j =1; 2; 3;:::;m n } forms a Reisz basis for H. (iv) inf n b; n;1 = n;1 6 sup n B n and { n;j j =1; 2; 3;:::;m n } forms a Reisz basis for H. Proof. The equivalence between (iii) and (iv) is ensured by the general Riesz basis theory (see e.g. [18, p. 37]). (i) (ii). The rst part follows from Lemma 4. The second part follows from (2) of Theorem 5 and (24). (ii) (i). Let t be as in Proposition 1. For any x H, dene control u x and! x as in Remark 4, we see that (2) is satised. The result then follows from Theorem 5. (i) (iii). The rst part follows from Lemma 4. The second part follows from Theorem 6. (iii) (i). Take t as in Proposition 1. Then {G n (t)} forms a Riesz basis for span in L 2 (;t ) and so does {F n (t)} in span. Since {} is a Riesz basis in H, ithas n(x) 2 : This, together with (31), gives Bn 1 n(x) 2 : For any x H, dene function u x (t)= (Bn 1 n(x)) T F n (t) and B t as before with respect to t and above dened u x. A direct computation shows that E( n ;A)B t u x = ( n(x)) T n : Therefore, x = ( n(x)) T n = E( n ;A)B t u x = B t u x : That is, (A; b) is exactly controllable in t. The proof is complete. To end the paper, we give an example which does not satisfy the assumption of Jacoband Zwart [1]. Certainly, system (1) can serve such an example which is signicant in practice. However, the uncontrolled system of (1) has only one eigenvalue which is of multiple 2. For the case where only nite number of eigenvalues are multiple, the proof of these article can be simplied signicantly by the method of Guo [6]. Our following example gives the case where each eigenvalue is of multiple 2. Example 1. Let n = n; n N. Consider the following system in usual H = 2 space: ẋ 1n (t)=i n x 1n (t)+u(t); ẋ 2n (t)=i n x 2n (t)+x 1n (t)+u(t); n N: (32) Dene A =(A n ); A n (x 1 ;x 2 )=(i n x 1 ;i n x 2 + x 1 ) (x 1 ;x 2 ) C 2 ; b=(b n ); b n =(1; 1): It is easily seen that (A)= p (A)={i n n N}; b [D(A )] :
232 B.-Z. Guo, G.-Q. Xu / Systems & Control Letters 52 (24) 221 232 Each eigenvalue of A is of algebraic multiple 2 and corresponds two linearly independent generalized eigenvectors n;1 =(;:::;;e n1 ; ;:::;); e n1 =(1; ); n;2 =(;:::;;e n2 ; ;:::;); e n2 =(; 1) n Z: { n ; n1 n N} forms an orthonormal basis for H. The C -semigroup generated by A is e At =(e Ant ): Hence e At b =(e int (1;t+ 1)): It is found directly from Fourier series that 2 2 e A(2 s) bu(s)ds 6 C 2 u(s) 2 ds u L 2 (; 2) for some constant C. This shows that b is admissible with respect to the C -semigroup e At. Now since n;1 = n;1 ; n;2 = n;2 n N it follows that B n = I 2 2 n N: Therefore, all conditions (iii) of Theorem 7 are satis- ed. It follows then that system (32) is exactly controllable. Acknowledgements This work was started during a visit of Bao-Zhu Guo at the University of Twente, the Netherlands, supported by the Netherlands Organization for Scientic Research (NWO). Bao-Zhu Guo is grateful to Hans Zwart for showing him his works in this eld as well as for many valuable discussions. The special thanks go to anonymous referees for their valuable comments and suggestions for the revision of this article. The support of the National Natural Science Foundation of China is gratefully acknowledged. References [1] S.A. Avdonin, S.A. Ivanov, Families of Exponentials, The Method of Moments in Controllability Problems for Distributed Parameter Systems, Cambridge University Press, Cambridge, 1995. [2] S.A. Avdonin, S.A. Ivanov, Riesz bases of exponentials and divided dierences, translation in St. Petersburg Math. J. 13 (3) (22) 339 351. [3] S. Cox, E. Zuazua, The rate at which energy decays in a damped string, Comm. Partial Dierential Equations 19 (1994) 213 244. [4] R.F. Curtain, H. Zwart, An Introduction to Innite-Dimensional Linear Systems Theory, Texts in Applied Mathematics, Vol. 21, Springer, New York, 1995. [5] N. Dunford, J.T. Schwartz, Linear Operators, Part III, Wiley-Interscience, New York, 1971. [6] B.Z. Guo, Riesz basis approach to the stabilization of a exible beam with a tip mass, SIAM J. Control Optim. 39 (21) 1736 1747. [7] B.Z. Guo, On boundary control of a hybrid system with variable coecients, J. Optim. Theory Appl. 114 (2) (22) 373 395. [8] B.Z. Guo, Y.H. Luo, Controllability and stability of a second order hyperbolic system with collocated sensor/actuator, Systems Control Lett. 46 (1) (22) 45 65. [9] B. Jacob, H. Zwart, Equivalent conditions for stabilizability of innite-dimensional systems with admissible control operators, SIAM J. Control Optim. 37 (1999) 1419 1455. [1] B. Jacob, H. Zwart, Exact controllability of C -groups with one-dimensional input operators, in: F. Colonius, et al. (Eds.), Advances in Mathematical Systems Theory, Systems Control Found. Appl, Birkhauser, Boston, MA, 2, pp. 221 242. [11] B. Jacob, H. Zwart, Exact observability of diagonal systems with a nite-dimensional output operator, Systems Control Lett. 43 (21) 11 19. [12] T. Kato, A Short Introduction to Perturbation Theory for Linear Operators, Springer, New York, 1982. [13] J.E. Lagnese, Recent progress in exact boundary controllability and uniform stabilizability of thin beams and plates, in: Distributed Parameter Control Systems: New Trends and Applications, Minneapolis, MN, 1989, pp. 61 111, Lecture Notes in Pure and Applied Mathematics, Vol. 128, Dekker, New York, 1991. [14] Z.H. Luo, B.Z. Guo, O. Morgul, Stability and Stabilization of Innite Dimensional Systems with Applications, Springer, London, 1999. [15] S.M. Verduyn Lunel, The closure of the generalized eigenspace of a class of innitesimal generators, Proc. Roy. Soc. Edinburgh 117A (1991) 171 192. [16] G. Weiss, Admissibility of unbounded control operators, SIAM J. Control Optim. 27 (1989) 527 545. [17] G.Q. Xu, B.Z. Guo, Riesz basis property of evolution equations in Hilbert spaces and application to a coupled string equation, SIAM J. Control Optim. 42 (3) (23) 966 984. [18] R.M. Young, An Introduction to Nonharmonic Fourier Series, Academic Press, London, 198.