Controllability of non-densely defined functional differential systems in abstract space

Applied Mathematics Letters 19 (26) 369 377 www.elsevier.com/locate/aml Controllability of non-densely defined functional differential systems in abstract space Xianlong Fu Department of Mathematics, East China Normal University, Shanghai 262, PR China Received 7 February 24; received in revised form 11 April 25; accepted 17 April 25 Abstract In this work, with the Schauder fixed point theorem applied, we establish a result concerning the controllability for a class of abstract functional differential systems where the linear part is non-densely defined and satisfies the Hille Yosida condition. As an application, an exampleisprovided to illustrate the result obtained. 25 Elsevier Ltd. All rights reserved. Keywords: Controllability; Non-densely defined; Integral solution; Schauder fixed point theorem 1. Introduction and preinaries In this work we study the controllability of semilinear functional differential systems defined nondensely. More precisely, we consider the controllability problem of the following system on a general Banach space X (with the norm. ): d dt x(t) = Ax(t) + Cu(t) + F(t, x t ), t a, (1) x = φ C([ r, ]; X), where the state variable x(.) takes values in Banach space X and the control function u(.) is given in L 2 ([, a]; U), thebanach space of admissible control functions with U abanach space. C is a bounded This work was supported by Mathematics Tianyuan Fund (No. A324624), NNSF of China (No. 13714) and Shanghai Priority Academic Discipline. E-mail address: xlfu@math.ecnu.edu.cn. 893-9659/$ - see front matter 25 Elsevier Ltd. All rights reserved. doi:1.116/j.aml.25.4.16

37 X. Fu /Applied Mathematics Letters 19 (26) 369 377 linear operator from U into X. The unbounded linear operators A are not defined densely on X, thatis, D(A) X. AndF :[, a] C([ r, ]; X) X is an appropriate function to be specified later. Let r > beaconstant; we denote by C([ r, ]; X) the space of continuous functions from [ r, ] to X with the sup-norm φ C = max s [ r,] φ(s), andforafunction x we define x t C([ r, ]; X) by x t (s) = x(t + s), s [ r, ]. The problem of controllability of linear and nonlinear systems represented by ODE in finite dimensional space has been extensively studied. Many authors have extended the controllability concept to infinite dimensional systems in Banach space with unbounded operators (see [1 8]andthereferences therein). Triggiani [5] has established sufficient conditions for controllability of linear and nonlinear systems in Banach space. Exact controllability of abstract semilinear equations has been studied by Lasiecka and Triggian [6]. Quinn and Carmichael [7] haveshownthatthe controllability problem in Banach space can be converted into a fixed point problem for a single-valued mapping. Kwun et al. [8] have investigated the controllability and approximate controllability of delay Voltera systems by using afixed point theorem. Recently Balachandran and co-workers have studied the (local) controllability of abstract semilinear functional differential systems [9] andthecontrollability of abstract integrodifferential systems [1]. In paper [11] theauthor extended the problem to neutral systems with unbounded delay. In all the work the linear operator A is always defined densely in X and satisfies the Hille Yosida condition so that it generates a C -semigroup or an analytic semigroup. However, as indicated in [12], we sometimes need to deal with non-densely defined operators. For example, when we look at a onedimensional heat equation with the Dirichlet condition on [, 1] and consider A = 2 in C([, 1]; R), 2 x in order to measure the solution in the sup-norm we take the domain D(A) ={x C 2 ([, 1]; R); x() = x(1) = }, and then it is not dense in C([, 1]; R) with the sup-norm. The example presented in Section 3 also shows the advantages of non-densely defined operators in handling some practical problems. See [12] formore examples and remarks concerning the non-densely defined operators. Up to now there have been very few papers in this direction dealing with the controllability problems for the important case where the linear parts are defined non-densely. The purpose of this work is just to investigate the controllability for the non-densely defined system (1). The result obtained here can be regarded as a continuation and an extension of those for densely defined control systems. Throughout this work we will always suppose the following hypothesis for Eq. (1): Hypothesis. (H ) The operator A : D(A) X X satisfies the Hille Yosida condition, i.e., there exist M andw R such that (w, + ) ρ(a) and sup{(λ w) n R(λ, A) n, n N,λ>w} M, where R(λ, A) = (λi A) 1. Remark 1. According to [13], if operator A satisfies the Hille Yosida condition, then A generates a non-degenerate, locally Lipschitz continuous integrated semigroup. For the theory of the integrated semigroup we refer the reader to paper [13] and[14]. Here, for the sake of brevity, we give directly the definition of integral solutions for Eq. (1)byvirtueofthis theory. Definition 1.1. Let φ C([ r, ]; X). A function x :[ r, a] X is said to be an integral solution of Eq. (1)on[ r, a] if the following conditions hold:

X. Fu / Applied Mathematics Letters 19 (26) 369 377 371 (i) x is continuous on [, a]; (ii) x(s)ds D(A) on [, a]; (iii) t φ() + A x(s)ds + [Cu(s) + F(s, x x(t) = s )]ds, t, φ(t), r t <. Let A be the part of A on D(A) defined by D(A ) ={x D(A) : Ax D(A)}, A x = Ax. Then A generates a C -semigroup {T (t)} t on D(A) (see [15]) and the integral solution in Definition 1.1 (if it exists) is given by t T x(t) = (t)φ() T (t s)b(λ)[cu(s) + F(s, x s )]ds, t, (2) φ(t), r t <, where B(λ) = λr(λ, A). Remark 2. We should point out here that, from Definition 1.1, it is not difficult to verify that if x is an integral solution of Eq. (1) on[ r, a], then for all t [, a], x(t) D(A). In particular, φ() D(A). Now we give the definition of the controllability for the non-densely defined system (1). Definition 1.2. The system (1)is said to be controllable on the interval [, a] if for every initial function φ C([ r, ]; X) with φ() D(A) and x 1 D(A),thereexists a control u L 2 ([, a]; U) such that the integralsolution x(.) of Eq. (1)satisfiesx(a) = x 1. 2. Main result To consider the controllability of system (1) weimpose the following assumptions on it. (H 1 ) F :[, a] C([ r, ]; X) X satisfies the following conditions: (i) For each t [, a], thefunction F(t,.) : C([ r, ]; X) X is continuous and for each φ C([ r, ]; X) the function F(., φ) :[, a] X is strongly measurable. (ii) For each positive number k, thereisafunction f k L 1 ([, a]) such that sup F(t,φ) f k (t) φ C k and 1 a inf f k (s)ds = γ<. k + k (H 2 ) The operator C : U X is bounded and linear. The linear operator W : L 2 ([, a]; U) D(A) defined by Wu = T (a s)b(λ)cu(s)ds

372 X. Fu /Applied Mathematics Letters 19 (26) 369 377 induces a bounded invertible operator W defined on L 2 ([, a]; U)/ ker W. (SeeAppendix for the construction of W 1.) Theorem 2.1. Suppose that the C -semigroup T (t) is compact. Let φ C([ r, ]; X) with φ() D(A). Iftheassumptions (H ) (H 2 ) are satisfied, then the system (1) is controllable on interval [, a] provided that (1 + amm C W 1 )M Mγ <1, (3) where M = sup t [,a] T (t). Proof. By means of the assumption (H 2 ),forarbitrary function x(.) we define the control ] u(t) = W [x 1 1 T (a)φ() T (a s)b(λ)f(s, x s )ds (t). Using this control we will show that the operator S defined by (Sx)(t) = T (t)φ() + T (t s)b(λ)[cu(s) + F(s, x s )]ds, t a. has a fixed point x(.). Thenfrom (2) x(.) is a integralsolution of system (1), and it is easy to verify that x(a) = (Sx)(a) = x 1. which implies that the system is controllable. Subsequently we will prove that S has a fixed point applying the Schauder fixed point theorem. Let y(.) :[ r, a] X be the function defined by { T (t)φ(), t, y(t) = φ(t), r t <. Then y = φ and the map t y t is continuous. We can assume that N = sup{ y t C : t a}. For each z C([, a]; D(A)), z() =, we denote by z the function defined by { z(t), t a, z(t) =, r t <. If x(.) satisfies (2), we can decompose it as x(t) = z(t) + y(t), t a, whichimplies that x t = z t + y t for every t a and the function z(.) satisfies z(t) = T (t s)b(λ)[cu(s) + F(s, z s + y s )]ds, t a. Let P be the operator on C([, a]; D(A)) defined by (Pz)(t) = T (t s)b(λ)[cu(s) + F(s, z s + y s )]ds. Obviously the operator S having a fixed point is equivalent to P having one, so it turns out to prove that P has a fixed point. For each positive integer k, let B k ={z C([, a]; D(A)) : z() =. z(t) k, t a}.

X. Fu / Applied Mathematics Letters 19 (26) 369 377 373 Then for each k, B k is clearly a bounded closed convex set in C([, a]; D(A)). Obviously, P is well defined on B k.weclaim that there existsapositive integer k such that PB k B k.ifthisisnot true, then for each positive integer k, thereisafunction z k (.) B k,butpz k B k,thatis, Pz k (t) > k for some t(k) [, a], wheret(k) denotes t depending on k. However,onthe other hand, we have that k < (Pz k )(t) = T (t s)b(λ)cu k (s)ds T (t s)b(λ)f(s, z k,s + y s )ds = T (t s)b(λ)c W {x 1 1 T (a)φ() } T (a τ)b(λ)f(τ, z k,τ + y τ )dτ (s)ds T (t s)b(λ)f(s, z k,s + y s )ds, where u k is the corresponding control of x k, x k = z k + y.since B(λ) it holds that k < λm λ ω M F(s, z k,s + y s ) ds + (λ + ), f k+n (s)ds, { M M C W 1 x 1 +M φ() + M M F(s, z k,s + y s ) ds amm C W 1 + M M f k+n (s)ds { x 1 +M φ() +M M = M + (1 + am C W 1 )M M f k+n (s)ds } M M F(τ, z k,τ + y τ )dτ (s)ds } f k+n (τ )dτ = M + (1 + amm C W 1 )(k + N) M M f k+n (s)ds k + N for M > independent of k. Dividingboth sides by k and taking the lower it, we get (1 + amm C W 1 )M Mγ 1. This contradicts (3). Hence for some positive integer k, PB k B k. In order to apply the Schauder fixed point theorem, we need to prove that P is a compact operator. For this purpose, first we prove that P is continuous on B k.let{z n } B k with z n z in B k ;thenfor each s [, a], z n,s z s,andby (H 1 )(i), wehavethat F(s, z n,s + y s ) F(s, z s + y s ), n.

374 X. Fu /Applied Mathematics Letters 19 (26) 369 377 Since F(s, z n,s + y s ) F(s, z s + y s ) 2 f k+n (s), by the dominated convergence theorem we have Pz n Pz = sup T (t s)b(λ)c[u n (s) u(s)]ds t a T (t s)b(λ)[f(s, z n,s + y s ) F(s, z s + y s )]ds, as n +, i.e. P is continuous. Next we prove that the family {Pz : z B k } is an equicontinuous family of functions. To do this, let ɛ>small, < t 1 < t 2 ;then 1 ɛ (Pz)(t 2 ) (Pz)(t 1 ) T (t 2 s) T (t 1 s) B(λ) C u(s) ds 1 t 1 ɛ 2 t 1 1 ɛ 1 t 1 ɛ 2 t 1 T (t 2 s) T (t 1 s) B(λ) C u(s) ds T (t 2 s) B(λ) C u(s) ds T (t 2 s) T (t 1 s) B(λ) F(s, z s + y s ) ds T (t 2 s) T (t 1 s) B(λ) F(s, z s + y s ) ds T (t 2 s) B(λ) F(s, z s + y s ) ds. Noting that [ u(s) W 1 x +M φ() ] T (a τ) B(λ) F(τ, z τ + y τ ) dτ [ ] W 1 x 1 +M φ() +MM f k+n (τ )dτ and f k+n L 1,weseethat (Pz)(t 2 ) (Pz)(t 1 ) tends to zero independently of z B k as t 2 t 1 with ɛ sufficiently small, since the compactness of T (t) (t > ) implies the continuity of T (t) (t > ) in t in the uniform operator topology. Hence, P maps B k into an equicontinuous family of functions. It remains to prove that V (t) ={(Pz)(t) : z B k } is relatively compact in X.Let< t a be fixed, <ɛ<t; forz B k,wedefine

ɛ (P ɛ z)(t) = X. Fu / Applied Mathematics Letters 19 (26) 369 377 375 T (t s)b(λ)[cu(s) + F(s, z s + y s )]ds ɛ = T (ɛ) T (t ɛ s)b(λ)[cu(s) + F(s, z s + y s )]ds. Using the estimation on u(s) as above and by the compactness of T (t)(t > ), weobtain that V ɛ (t) ={(P ɛ z)(t) : z B k } is relative compact in X for every ɛ, <ɛ<t.moreover,for every z B k, we have that (Pz)(t) (P ɛ z)(t) t ɛ T (t s)b(λ)[cu(s) + F(s, z s + y s )] ds { [ M M C W 1 x 1 +M φ() t ɛ ] } + M M f k+n (τ )dτ + f k+n (s) ds. Therefore there are relative compact sets arbitrarily close to the set V (t) ={(Pz)(t) : z B k };hence the set V (t) is also relative compact in X. Thus, by the Arzela Ascoli theorem P is a compact operator and by Schauder s fixed point theorem there exists afixedpoint z(.) for P on B k.ifwedefinex(t) = z(t) + y(t), r t a, itiseasy to see that x(.) is a integral solution of (1) satisfying x = φ,x(a) = x 1 which shows that system (1) is controllable. The proof is completed. 3. An example As an application of Theorem 2.1, weconsider the following system: 2 z(t, x) = z(t, x) + Cu(t) + f (t, z(t r, x)), t a, x π t x2 u(t, ) = u(t,π)= u(θ, x) = φ(θ,x), r θ, x π. To write system (4) intheform of (1), we choose X = C([,π]) and consider the operator A defined by Af = f with the domain D(A) ={f (.) X : f X, f () = f (π) = }. We have D(A) ={f (.) X : f () = f (π) = } X, and ρ(a) (, + ), (4) (λi A) 1 1 λ, for λ>. This implies that A satisfies the Hille Yosida condition on X. It is well known that A generates a compact C -semigroup {T (t)} t on D(A) such that T (t) e t for t.

376 X. Fu /Applied Mathematics Letters 19 (26) 369 377 In addition, we set that, for t a and φ C([ r, ]; X), F(t,φ)(x) = f (t,φ( r)(x)). A case where the system (4) can be handled by using the classical semigroup theory is that when the function f is assumed to satisfy f (t, ) =, for all t a. In this case, the function F takes its values in the space D(A) and the operator A generates a strongly continuous semigroup on D(A). However,here the integrated semigroup theory allows the range of F to be X without the condition (5). Now it is easy to adapt our previous result to obtain the controllability of system (4). We assume that: (i) For the function f :[, a] R R the following three conditions are satisfied: (1) For each t [, a], f (t,.)is continuous. (2) For each z X, f (., z) is measurable. (3) There are positive functions h 1, h 2 L 1 ([, a]) such that f (t, z) h 1 (t) z +h 2 (t), for all (t, z) [, a] C([, a]; X). Clearly, these conditions ensure that F yields condition (H 1 ) with γ = h 1 (.) L 1. (ii) C : U X is a bounded linear operator. (iii) The linear operator W : U X defined by Wu = T (t s)b(λ)cu(s)ds satisfies the condition (H 2 ).Thus, all the conditions of Theorem 2.1 are verified. Therefore, from Theorem 2.1, foranyinitial function φ with φ(, ) = φ(,π) =, the system (4) iscontrollable on [, a] provided that (1 + a C W 1 )γ < 1(clearly here M = M = 1). Appendix Construction of W 1 (see [7]). Let Y = L2 ([, a]; U). ker W Since ker W is closed, Y is abanach space under the norm [u] Y = inf u L u [u] 2 = inf u +ũ L 2, W ũ= where [u] denotes the equivalence class of u. Define W ; Y D(A) by W [u] =Wu, u [u]. Then W is one-to-one and W [u] X W [u] Y. (5)

X. Fu / Applied Mathematics Letters 19 (26) 369 377 377 We claim that V = Range W is a Banach space with the norm v V = W 1 v Y. This norm is equivalent to the graph norm on D( W 1 ) = Range W. W is bounded and since D( W) = Y is closed, W 1 is closed, and so the above norm makes Range W = V abanachspace. Moreover, Wu V = W 1 Wu Y = W 1 W [u] = [u] = inf u u, u [u] so W L(L 2 ([, a]; U), V ). SinceL 2 ([, a]; U) is reflexive and kerw is weakly closed, the infimum in the definition of the norm on Y is attained. Foranyv V,wecan therefore choose a control u L 2 ([, a]; U) such that u = W 1 v. References [1] E.N. Chuckwu, S.M. Lenhart, Controllability questions for nonlinear systems in abstract space, J. Optim. Theory Appl. 68 (1991) 437 462. [2] K. Naito, Controllability of semilinear control systems dominated by linear part, SIAM J. Control Optim. 25 (1987) 715 722. [3] K. Naito, J.Y. Park, Approximate controllability for trajectoics of a delay Voltera control system, J. Optim. Theory Appl. 61 (1989) 271 279. [4] S. Nakagri, R. Yamamoto, Controllability and observability of linear retarded systems in Banach space, Internat. J. Controll. 49 (1989) 1489 154. [5] R. Triggiani, Controllability and observability in Banach space with bounded operators, SIAM J. Control. 13 (1975) 462 491. [6] L. Lasiecka, R. Triggiani, Exact controllability of semilinear abstract systems with application to waves and plates boundary control problems, Appl. Math. Optim. 23 (1991) 19 154. [7] M.D. Quinn, N. Carmichael, An approach to nonlinear control problems using fixed point methods, degree theory and pseudo-inverses, Numer. Funct. Anal. Optim. 7 (1984 1985) 197 219. [8] Y.C. Kwun, J.Y. Park, J.W. Ryu, Approximate controllability and controllability for delay Voltera systems, Bull. Korean Math. Soc. 28 (1991) 131 145. [9] K. Balachandran, R. Sakthivel, Controllability of Sobolev-type semilinear integro-differential systems in Banach Spaces, Appl. Math. Lett. 12 (1999) 63 71. [1] K. Balachandran, P. Balasubramaniam, J.P. Dauer, Local null controllability of nonlinear functional differential systems in Banach space, J. Optim. Theory Appl. 88 (1996) 61 75. [11] X. Fu, Controllability of neutral functional differential systems in abstract space, Appl. Math. Comput. 141 (23) 281 296. [12] G. Da Prato, E. Sinestrari, Differential operators with non-dense domain, Ann. Scuola Norm. Sup. Pisa Sci. 14 (1987) 285 344. [13] H. Kellermann, M. Hieber, Integrated semigroup, J. Funct. Anal. 84 (1989) 16 18. [14] H. Thiems, Integrated semigroup and integral solutions to abstract Cauchy problems, J. Math. Anal. Appl. 152 (199) 416 447. [15] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.