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Approximate controllability of delayed semilinear control systems

Approximate controllability of delayed semilinear control systems APPROXIMATE CONTROLLABILITY OF DELAYED SEMILINEAR CONTROL SYSTEMS LIANWEN WANG Received 22 January 2004 and in revised form 9 July 2004 We deal with the approximate controllability of control systems governed by delayed semilinear differential equations y˙ (t) = Ay(t)+ A y(t − ∆ )+ F(t, y(t), y )+(Bu)(t). Vari- 1 t ous sufficient conditions for approximate controllability have been obtained; these results usually require some complicated and limited assumptions. Results in this paper provide sufficient conditions for the approximate controllability of a class of delayed semilinear control systems under natural assumptions. 1. Introduction The main concern in this paper is the approximate controllability of the following delayed semilinear control system: y˙(t) = Ay(t)+ A y(t − ∆ )+ F t, y(t), y +(Bu)(t), t ≥ a, 1 t (1.1) y = ξ, in a real Hilbert space X with the norm ·. The meaning of all notations is listed as fol- lows: ∆ ≥ 0 is a system delay; y(·):[a− ∆ ,b] → X is the state function; ξ ∈ C([−∆ ,0];X), the Banach space of all continuous functions ψ :[−∆ ,0] → X endowed with the norm |ψ|= sup{ψ(θ) : −∆ ≤ θ ≤ 0}; A is the generator of a C semigroup T(t)inX; A is a 0 1 boundedlinearoperatorfrom X to X; F :[a,b]× X × C([−∆ ,0];X) → X is a nonlinear operator; u(·) ∈ L (a,b;U) is a control function; U is a Hilbert space; B is a bounded lin- 2 2 ear operator from L (a,b;U)to L (a,b;X). In addition, for any y ∈ C([a− ∆ ,b];X)and t ∈ [a,b], define y ∈ C([−∆ ,0];X)by y (θ) = y(t + θ)for θ ∈ [−∆ ,0]. t t Denote the state function of (1.1) corresponding to a control u(·)by y(·;u). Then y(b;u) is the state value at terminal time b. Introduce the set R (F) = y(b;u): u(·) ∈ L (a,b;U) , (1.2) which is called the reachable set of system (1.1)atterminaltime b, its closure in X is denoted R (F). Copyright © 2005 Hindawi Publishing Corporation Journal of Applied Mathematics and Stochastic Analysis 2005:1 (2005) 67–76 DOI: 10.1155/JAMSA.2005.67 68 Approximate controllability of delayed systems Definition 1.1. The system (1.1)issaidtobeapproximatelycontrollableon[a,b]if R (F) = X. The following system is called the corresponding linear system of (1.1): y(t) = Ay(t)+ A y(t − ∆ )+(Bu)(t), t ≥ a, (1.3) y = ξ. This is a special case of (1.1)with F ≡ 0. The reachable set of system (1.3)atterminal time b is denoted R (0). Similarly, system (1.3) is said to be approximately controllable on [a,b]if R (0) = X. For semilinear control systems without delays, approximate controllability has been extensively studied in the literature. We list only a few of them. Zhou [10] studied the ap- proximate controllability for a class of semilinear abstract equations. Naito [6] established the approximate controllability for semilinear control systems under the assumption that the nonlinear term is bounded. Approximate controllability for semilinear control sys- tems also can be found in Choi et al. [1], Fernandez and Zuazua [2], Li and Yong [4], Mahmudov [5], and many other papers. Most of them concentrate on finding conditions of F, A,and B such that semilinear systems are approximately controllable on [a,b]ifthe corresponding linear systems are approximately controllable on [a,b]. For semilinear delayed control systems, some papers are devoted to the approximate controllability. For example, Klamka [3] provided some approximate controllability re- sults. Naito and Park [7] dealt with approximate controllability for delayed Volterra sys- tems. In [9] Ryu et al. studied approximate controllability for delayed Volterra control systems. The purpose of this paper is to study the approximate controllability of control system (1.1). We obtain the approximate controllability of system (1.1) if the correspond- ing linear system is approximate controllable and other natural assumptions such as the local Lipschitz continuity for F and the compactness of operator W are satisfied. 2. Basic assumptions We start this section by introducing the fundamental solution S(t)ofthe following system: y˙ (t) = Ay(t)+ A y(t − ∆ ), t ≥ a, (2.1) y = ξ. We already know that (2.1) has a unique solution, denoted by y (t), for each ξ ∈ C([−∆ , 0];X). Hence, we can define an operator S(t)in X by y (t + a), t ≥ 0, S(t)ξ(0) = (2.2) 0, t< 0. Lianwen Wang 69 S(t) is called the fundamental solution of (2.1). It is easy to check that S(t) is the unique solution of the following operator equation: S(t) = T(t)+ T(t − s)A S(s− ∆ )ds. (2.3) Let K := max{T(t) :0 ≤ t ≤ b}.By(2.3)wehave t t (2.4) S(t) ≤ K + K A S(s− ∆ ) ds ≤ K + K A S(s) ds. 1 1 0 ∆ Gronwall’s inequality implies that S(t) ≤ K exp KA(b− ∆ ) := M,0 ≤ t ≤ b. (2.5) Throughout the paper we impose the following condition on F. (H1) F :[a,b] × X × C([−∆ ,0];X) → X is locally Lipschitz continuous in y, η uni- formly in t ∈ [a,b]; that is, for any r> 0, there is a constant L(r)suchthat F t, y ,η − F t, y ,η ≤ L(r) y − y + η − η (2.6) 1 1 2 2 1 2 1 2 for any t ∈ [a,b], y ≤ r, y ≤ r, |η |≤ r,and |η |≤ r. 1 2 1 2 With a minor modification of [8], we can prove that system (1.1)has auniquemild solution y(·;u) ∈ C([a − ∆ ,b];X) for any control u(·) ∈ L (a,b;U)under assumption (H1). This mild solution is defined as a solution of the integral equation: y(t;u) = S(t − a)ξ(0) + S(t − s) F s, y(s;u), y +(Bu)(s) ds, t ≥ a, a (2.7) y = ξ. Similarly, for any z(·) ∈ L (a,b;X), the following integral equation: x(t;z) = S(t − a)ξ(0) + S(t − s) F s,x(s;z),x + z(s) ds, t ≥ a, a (2.8) x = ξ, has a unique mild solution x(·;z). Therefore, we can define an operator W from L (a,b;X) to C([a,b];X)by (Wz)(·) = x(·;z). (2.9) Regarding the operator W, we assume that (H2) W is a compact operator. Remark 2.1. (H2) is the case if, for instance, T(t), the semigroup generated by A,isa compact semigroup. The following assumption (H3) was introduced by Naito in [6]. Define a linear oper- ator ϕ from L (a,b;X)to X by ϕp = S(b− s)p(s)ds for p(·) ∈ L (a,b;X). (2.10) a 70 Approximate controllability of delayed systems Let the kernel of the operator ϕ be N; that is, N ={p : ϕp = 0}.Then N is a closed sub- 2 2 ⊥ space of L (a,b;X). Denote its orthogonal space in L (a,b;X)by N .Let G be the projec- 2 ⊥ tion operator from L (a,b;X)into N and let R[B]bethe rangeof B. We assume that (H3) for any p(·) ∈ L (a,b;X), there is a function q(·) ∈ R[B]suchthat ϕp = ϕq. Remark 2.2. (H3) is valid for many control systems, see [6] for detailed discussion. It follows from assumption (H3) that {x + N}∩ R[B] =∅ for any x ∈ N . Therefore, the operator P from N to R[B]definedby Px = x , (2.11) ∗ ∗ where x ∈{x + N}∩ R[B]and x = min{y : y ∈{x + N}∩ R[B]},iswelldefined. It is proved in [6]that P is bounded. 3. Lemmas This section provides two lemmas that will be used to prove the main theorem. Lemma 3.1. Assume that a(t) is continuous on [a,b], b(t) is nonnegative and integrable on [a,b],and x(t) is a nonnegative continuous function satisfying the following inequality: x(t) ≤ a(t)+ b(s)x (s)ds,0 ≤ α< 1, t ∈ [a,b]. (3.1) If the equation y(t) = a(t)+ b(s)y (s)ds (3.2) has a unique solution y¯(t) on [a,b], then x(t) ≤ y¯(t), t ∈ [a,b]. (3.3) Proof. Let C[a,b] be the Banach space of all continuous functions on [a,b] endowed with the maximum norm. Define an operator E from C[a,b]to C[a,b]by (Ey)(t) = a(t)+ b(s)y (s)ds. (3.4) Construct a sequence {y } as follows: y (t) = x(t), y (t) = Ey (t), n = 0,1,.... (3.5) 0 n+1 n We have x(t) = y (t) ≤ y (t)≤··· , x= y ≤ y ≤··· . (3.6) 0 1 0 1 Note that Ey≤a +y b(s)ds, (3.7) 0 Lianwen Wang 71 then we can find a number d> 0suchthat Ey≤y for y≥ d. (3.8) If y ≤ d holds for any integer n = 0,1,...,then y  is bounded. Otherwise, it follows n n from (3.6)thatasufficiently large integer N exists such that y ≤···≤ y ≤ d, y >d for n>N. (3.9) 0 N n Thus max y : n ≥ 0 ≤ max d, y := m. (3.10) n N+1 Consequently, 0 ≤ x(t) = y (t) ≤ y (t)≤···≤ y (t)≤···≤ m. (3.11) 0 1 n Therefore, lim y (y) = y¯(t). (3.12) n→∞ Note that y¯(t) is the unique solution of (3.2). The conclusion of Lemma 3.1 follows from (3.11). Lemma 3.2. Assume that (H1) is fulfilled. Furthermore, for any y ∈ X and η∈C([−∆ ,0];X) α α F(t, y,η) ≤ M 1+y +|η| ,0 ≤ α< 1, t ∈ [a,b]. (3.13) Then the mild solution x(t;z) of (2.8) has the estimate x ≤ H z , (3.14) where H(r) is an increasing function and H(r) = O(r) as r →∞. Proof. Recall that M = max S(t) :0 ≤ t ≤ b . (3.15) It follows from x(s)≤|x | and (2.8)that t t α α x(t) ≤ M |ξ| + M M 1+ x(s) + x ds + M z(s) ds 1 1 s 1 a a (3.16) ≤ M |ξ| + M M(b− a)+ M b− az+2M M x ds. 1 1 1 1 s For any θ ∈ [−∆ ,0], we have t+θ x(t + θ) ≤ M |ξ| + M M(b− a)+ M b− az+2M M x ds. (3.17) 1 1 1 1 s a 72 Approximate controllability of delayed systems Hence x ≤ M |ξ| + M M(b− a)+ M b− az+2M M x ds. (3.18) t 1 1 1 1 s Note that for any two constants V and V , the following equation 1 2 y(t) = V + V y (s)ds (3.19) 1 2 has a unique solution 1/(1−α) 1−α y(t) = (1− α)V t + V . (3.20) Applying Lemma 3.1 to (3.18), we obtain 1/(1−α) 1−α x ≤ 2(1− α)MM (b− a)+ M |ξ| + M b− az + MM (b− a) t 1 1 1 1 := H z . (3.21) Clearly, the function H(r) satisfies all requirements of Lemma 3.2 and the proof of the lemma is complete. 4. Approximate controllability The following theorem is the main result of this paper. Theorem 4.1. Assume that linear system (1.3) is approximately controllable on [a,b].If (H1), (H2), (H3), and (3.13) are fulfilled, then system (1.1) is approximately controllable on [a,b]. Proof. Note that system (1.3) is approximately controllable on [a,b]bythe assumption, then R (0) = X. To prove the approximate controllability of (1.1); that is, R (F) = X,itis b b sufficient to show that R (0) ⊂ R (F). (4.1) b b b b That means for any  > 0and x ∈ R (0), there exists a ν ∈ R (F)suchthat ν− x  < . b b By the definition of reachable set R (0)ofsystem(1.3), there is a control u(·) ∈ L (a,b;U)suchthat x = S(b− a)ξ(0) + S(b− s)(Bu)(s)ds. (4.2) ⊥ ⊥ ⊥ Let z¯ = Bu, z = Gz¯ .Then z ∈ N . Define an operator J from N to N by 0 0 0 0 Jv = z − GΓPv, v ∈ N , (4.3) 0 Lianwen Wang 73 2 2 where Γ is the operator from L (a,b;X)to L (a,b;X)definedby (Γz)(t):= F t,(Wz)(t),(Wz) = F t,x(t;z),x . (4.4) t t ⊥ 2 2 ⊥ For any v ∈ N ,wehave Pv ∈ L (a,b;X), ΓPv ∈ L (a,b;X), and GΓPv ∈ N . Therefore, J is well defined. Since W is compact by assumption (H2), for any bounded sequence z (·) ∈ L (a,b;X); that is,z ≤ r for some r > 0, there is a subsequence z (·)of z (·)suchthat(Wz )(·) n 1 1 n n n k k converges to x¯(·)in C([a,b];X)as k →∞.So, Wz is bounded in C([a,b];X); that is, Wz ≤ r for some constant r > 0. (H1) implies that a constant L(r) > 0 exists such n 2 2 that ¯ ¯ F t, Wz (t), Wz − F t,x(t),x n n t k k (4.5) ≤ L(r) Wz (t)− x¯(t) + Wz − x¯ , n n t k k t where r = max(r ,r ). Hence, we have 1 2 Γz −F ·,x¯(·),x¯ n · = F ·, Wz (·), Wz − F ·,x¯(·),x¯ n n · k k · (4.6) ≤ L (r)(b− a) sup Wz (t)− x¯(t) a≤t≤b +sup Wz − x¯ −→ 0 n t k t a≤t≤b as k →∞. Therefore, Γ is compact and J is compact as well. From Lemma 3.2,for any z(·) ∈ L (a,b;X), we have α α F t,x(t;z),x ≤ M 1+ x(t;z) + x ≤ M 1+2H z . (4.7) t t Note that H(r)isincreasingand P is a bounded operator, then z − GΓPv ≤ z +GΓPv 0 0 (4.8) ≤ z + M b− a+2M b− aH Pv . Taking into account √ √ z + M b− a+2M b− aH Pv lim = 0, (4.9) v→∞ v then z − GΓPv lim = 0. (4.10) v→∞ v Therefore, we can find a sufficiently large number r¯ such that z − GΓPv ≤ r¯ for v≤ r. ¯ (4.11) 0 74 Approximate controllability of delayed systems ⊥ ⊥ This means that J maps the bounded closed set D(r¯)={v : v≤ r¯,v ∈ N } of N into itself. Consequently, a fixed point of operator J exists due to the Schauder fixed point theorem; that is, there is a v ∈ D(r¯)suchthat ∗ ∗ ∗ Jv = z − GΓPv = v . (4.12) On account of ∗ ∗ Pv ∈ v + N ∩ R[B], (4.13) we have b b ∗ ∗ S(b− s)(Pv )(s)ds = S(b− s)v (s)ds. (4.14) a a 2 ⊥ Note that G is the projection operator from L (0,T;X)into N ,thenwehave b b S(b− s)Gp(s)ds = S(b− s)p(s)ds for p(·) ∈ L (a,b;X), a a b b ∗ ∗ (4.15) S(b− s)(Bu)(s)ds = S(b− s) F s,x s;Pv ,x + v (s) ds a a ∗ ∗ = S(b− s) F s,x s;Pv ,x )+ Pv (s) ds. Finally, b ∗ ∗ ∗ x = S(b− a)ξ(0) + S(b− s) F s,x s;Pv ,x + Pv (s) ds = x b;Pv . (4.16) ∗ 2 Observe that Pv ∈ R[B], then there is a sequence u (·) ∈ L (a,b;U)suchthat Bu → n n Pv as n→∞. W is continuous due to its compactness, then WBu −→ WPv in C [a,b];X . (4.17) This implies ∗ b x b;Bu −→ x b;Pv = x (4.18) as n→∞.Since x(b;Bu ) = y(b;u ) ∈ R (F), we obtain x ∈ R (F) and complete the n n T T proofofthe theorem. Remark 4.2. If A = 0, (H3) implies the approximate controllability of (1.3)on[a,b] (see [6]). Therefore, Naito’s result in [6]isaspecialcaseof Theorem 4.1 when A = 0, ∆ = 0, and F(t,x(t),x ) = F(x(t)). In particular, we improve Naito’s result by weakening the uniform Lipschitz continuity and the uniform boundedness imposed on the nonlinear term. Lianwen Wang 75 5. Example Let X = L (0,π)and e (x) = sin(nx)for n = 1,.... Then {e : n = 1,2,...} is an orthogonal n n base for X.Define A : X → X by Ay = y with domain D(A) = y ∈ X : y and y are absolutely continuous, y ∈ X, y(0) = y(π) = 0 . (5.1) Then Ay =− n y,e e , y ∈ D(A). (5.2) n n n=1 It is well known that A is the infinitesimal generator of an analytic group T(t), t ≥ 0, in X and is given by −n t T(t)y = e y,e e , y ∈ X. (5.3) n n n=1 T(t) is compact because it is an analytic semigroup. Define an infinite dimensional space U by ∞ ∞ U = u : u = u e , u < ∞ (5.4) n n n=2 n=2 with the norm defined by 1/2 u = u . (5.5) n=2 Define a mapping B from U to X as follows: Bu = 2u e + u e . (5.6) 2 1 n n n=2 Consider the following delayed semilinear heat equation: ∂y(t,x) ∂ y(t,x) = + y(t − ∆ ,x)+ F y(t,x), y(t− ∆ ,x) ∂t ∂x + Bu(t,x), 0 <t <b,0 <x <π, (5.7) y(t,0) = y(t,π) = 0, 0 ≤ t ≤ b, y(t,x) = ξ(x), −∆ ≤ t ≤ 0, 0 ≤ x ≤ π. Then system (5.7) can be written to the abstract form (1.1). (H2) holds because T(t)is a compact semigroup. Following the same arguments as in [6]wecan provethat(H3)is valid and that the corresponding linear system is approximately controllable on [0,b]. By Theorem 4.1,system(5.7)isapproximatelycontrollableon[0,b]if F is locally Lipschitz continuous and condition (3.13) is satisfied. 76 Approximate controllability of delayed systems References [1] J. R. Choi, Y. C. Kwun, and Y. K. Sung, Approximate controllability for nonlinear integrodifferen- tial equations, J. Korea Soc. Math. Educ. Ser. B Pure Appl. Math. 2 (1995), no. 2, 173–181. [2] L.A.Fernandez ´ and E. Zuazua, Approximate controllability for the semilinear heat equation involving gradient terms, J. Optim. Theory Appl. 101 (1999), no. 2, 307–328. [3] J. Klamka, Controllability of Dynamical Systems, Mathematics and its Applications (East Euro- pean Series), vol. 48, Kluwer Academic Publishers Group, Dordrecht, 1991. [4] X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkhauser, Boston, Massachusettes, USA, 1995. [5] N.I.Mahmudov, Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces,SIAMJ.Control Optim. 42 (2003), no. 5, 1604–1622. [6] K. Naito, Controllability of semilinear control systems dominated by the linear part,SIAMJ.Con- trol Optim. 25 (1987), no. 3, 715–722. [7] K. Naito and J. Y. Park, Approximate controllability for trajectories of a delay Volterra control system,J.Optim.TheoryAppl. 61 (1989), no. 2, 271–279. [8] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,Ap- plied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. [9] J.W.Ryu,J.Y.Park, andY.C.Kwun, Approximate controllability of delay Volterra control system, Bull. Korean Math. Soc. 30 (1993), no. 2, 277–284. [10] H. X. Zhou, Approximate controllability for a class of semilinear abstract equations,SIAMJ.Con- trol Optim. 21 (1983), no. 4, 551–565. 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Approximate controllability of delayed semilinear control systems

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Abstract

APPROXIMATE CONTROLLABILITY OF DELAYED SEMILINEAR CONTROL SYSTEMS LIANWEN WANG Received 22 January 2004 and in revised form 9 July 2004 We deal with the approximate controllability of control systems governed by delayed semilinear differential equations y˙ (t) = Ay(t)+ A y(t − ∆ )+ F(t, y(t), y )+(Bu)(t). Vari- 1 t ous sufficient conditions for approximate controllability have been obtained; these results usually require some complicated and limited assumptions. Results in this paper provide sufficient conditions for the approximate controllability of a class of delayed semilinear control systems under natural assumptions. 1. Introduction The main concern in this paper is the approximate controllability of the following delayed semilinear control system: y˙(t) = Ay(t)+ A y(t − ∆ )+ F t, y(t), y +(Bu)(t), t ≥ a, 1 t (1.1) y = ξ, in a real Hilbert space X with the norm ·. The meaning of all notations is listed as fol- lows: ∆ ≥ 0 is a system delay; y(·):[a− ∆ ,b] → X is the state function; ξ ∈ C([−∆ ,0];X), the Banach space of all continuous functions ψ :[−∆ ,0] → X endowed with the norm |ψ|= sup{ψ(θ) : −∆ ≤ θ ≤ 0}; A is the generator of a C semigroup T(t)inX; A is a 0 1 boundedlinearoperatorfrom X to X; F :[a,b]× X × C([−∆ ,0];X) → X is a nonlinear operator; u(·) ∈ L (a,b;U) is a control function; U is a Hilbert space; B is a bounded lin- 2 2 ear operator from L (a,b;U)to L (a,b;X). In addition, for any y ∈ C([a− ∆ ,b];X)and t ∈ [a,b], define y ∈ C([−∆ ,0];X)by y (θ) = y(t + θ)for θ ∈ [−∆ ,0]. t t Denote the state function of (1.1) corresponding to a control u(·)by y(·;u). Then y(b;u) is the state value at terminal time b. Introduce the set R (F) = y(b;u): u(·) ∈ L (a,b;U) , (1.2) which is called the reachable set of system (1.1)atterminaltime b, its closure in X is denoted R (F). Copyright © 2005 Hindawi Publishing Corporation Journal of Applied Mathematics and Stochastic Analysis 2005:1 (2005) 67–76 DOI: 10.1155/JAMSA.2005.67 68 Approximate controllability of delayed systems Definition 1.1. The system (1.1)issaidtobeapproximatelycontrollableon[a,b]if R (F) = X. The following system is called the corresponding linear system of (1.1): y(t) = Ay(t)+ A y(t − ∆ )+(Bu)(t), t ≥ a, (1.3) y = ξ. This is a special case of (1.1)with F ≡ 0. The reachable set of system (1.3)atterminal time b is denoted R (0). Similarly, system (1.3) is said to be approximately controllable on [a,b]if R (0) = X. For semilinear control systems without delays, approximate controllability has been extensively studied in the literature. We list only a few of them. Zhou [10] studied the ap- proximate controllability for a class of semilinear abstract equations. Naito [6] established the approximate controllability for semilinear control systems under the assumption that the nonlinear term is bounded. Approximate controllability for semilinear control sys- tems also can be found in Choi et al. [1], Fernandez and Zuazua [2], Li and Yong [4], Mahmudov [5], and many other papers. Most of them concentrate on finding conditions of F, A,and B such that semilinear systems are approximately controllable on [a,b]ifthe corresponding linear systems are approximately controllable on [a,b]. For semilinear delayed control systems, some papers are devoted to the approximate controllability. For example, Klamka [3] provided some approximate controllability re- sults. Naito and Park [7] dealt with approximate controllability for delayed Volterra sys- tems. In [9] Ryu et al. studied approximate controllability for delayed Volterra control systems. The purpose of this paper is to study the approximate controllability of control system (1.1). We obtain the approximate controllability of system (1.1) if the correspond- ing linear system is approximate controllable and other natural assumptions such as the local Lipschitz continuity for F and the compactness of operator W are satisfied. 2. Basic assumptions We start this section by introducing the fundamental solution S(t)ofthe following system: y˙ (t) = Ay(t)+ A y(t − ∆ ), t ≥ a, (2.1) y = ξ. We already know that (2.1) has a unique solution, denoted by y (t), for each ξ ∈ C([−∆ , 0];X). Hence, we can define an operator S(t)in X by y (t + a), t ≥ 0, S(t)ξ(0) = (2.2) 0, t< 0. Lianwen Wang 69 S(t) is called the fundamental solution of (2.1). It is easy to check that S(t) is the unique solution of the following operator equation: S(t) = T(t)+ T(t − s)A S(s− ∆ )ds. (2.3) Let K := max{T(t) :0 ≤ t ≤ b}.By(2.3)wehave t t (2.4) S(t) ≤ K + K A S(s− ∆ ) ds ≤ K + K A S(s) ds. 1 1 0 ∆ Gronwall’s inequality implies that S(t) ≤ K exp KA(b− ∆ ) := M,0 ≤ t ≤ b. (2.5) Throughout the paper we impose the following condition on F. (H1) F :[a,b] × X × C([−∆ ,0];X) → X is locally Lipschitz continuous in y, η uni- formly in t ∈ [a,b]; that is, for any r> 0, there is a constant L(r)suchthat F t, y ,η − F t, y ,η ≤ L(r) y − y + η − η (2.6) 1 1 2 2 1 2 1 2 for any t ∈ [a,b], y ≤ r, y ≤ r, |η |≤ r,and |η |≤ r. 1 2 1 2 With a minor modification of [8], we can prove that system (1.1)has auniquemild solution y(·;u) ∈ C([a − ∆ ,b];X) for any control u(·) ∈ L (a,b;U)under assumption (H1). This mild solution is defined as a solution of the integral equation: y(t;u) = S(t − a)ξ(0) + S(t − s) F s, y(s;u), y +(Bu)(s) ds, t ≥ a, a (2.7) y = ξ. Similarly, for any z(·) ∈ L (a,b;X), the following integral equation: x(t;z) = S(t − a)ξ(0) + S(t − s) F s,x(s;z),x + z(s) ds, t ≥ a, a (2.8) x = ξ, has a unique mild solution x(·;z). Therefore, we can define an operator W from L (a,b;X) to C([a,b];X)by (Wz)(·) = x(·;z). (2.9) Regarding the operator W, we assume that (H2) W is a compact operator. Remark 2.1. (H2) is the case if, for instance, T(t), the semigroup generated by A,isa compact semigroup. The following assumption (H3) was introduced by Naito in [6]. Define a linear oper- ator ϕ from L (a,b;X)to X by ϕp = S(b− s)p(s)ds for p(·) ∈ L (a,b;X). (2.10) a 70 Approximate controllability of delayed systems Let the kernel of the operator ϕ be N; that is, N ={p : ϕp = 0}.Then N is a closed sub- 2 2 ⊥ space of L (a,b;X). Denote its orthogonal space in L (a,b;X)by N .Let G be the projec- 2 ⊥ tion operator from L (a,b;X)into N and let R[B]bethe rangeof B. We assume that (H3) for any p(·) ∈ L (a,b;X), there is a function q(·) ∈ R[B]suchthat ϕp = ϕq. Remark 2.2. (H3) is valid for many control systems, see [6] for detailed discussion. It follows from assumption (H3) that {x + N}∩ R[B] =∅ for any x ∈ N . Therefore, the operator P from N to R[B]definedby Px = x , (2.11) ∗ ∗ where x ∈{x + N}∩ R[B]and x = min{y : y ∈{x + N}∩ R[B]},iswelldefined. It is proved in [6]that P is bounded. 3. Lemmas This section provides two lemmas that will be used to prove the main theorem. Lemma 3.1. Assume that a(t) is continuous on [a,b], b(t) is nonnegative and integrable on [a,b],and x(t) is a nonnegative continuous function satisfying the following inequality: x(t) ≤ a(t)+ b(s)x (s)ds,0 ≤ α< 1, t ∈ [a,b]. (3.1) If the equation y(t) = a(t)+ b(s)y (s)ds (3.2) has a unique solution y¯(t) on [a,b], then x(t) ≤ y¯(t), t ∈ [a,b]. (3.3) Proof. Let C[a,b] be the Banach space of all continuous functions on [a,b] endowed with the maximum norm. Define an operator E from C[a,b]to C[a,b]by (Ey)(t) = a(t)+ b(s)y (s)ds. (3.4) Construct a sequence {y } as follows: y (t) = x(t), y (t) = Ey (t), n = 0,1,.... (3.5) 0 n+1 n We have x(t) = y (t) ≤ y (t)≤··· , x= y ≤ y ≤··· . (3.6) 0 1 0 1 Note that Ey≤a +y b(s)ds, (3.7) 0 Lianwen Wang 71 then we can find a number d> 0suchthat Ey≤y for y≥ d. (3.8) If y ≤ d holds for any integer n = 0,1,...,then y  is bounded. Otherwise, it follows n n from (3.6)thatasufficiently large integer N exists such that y ≤···≤ y ≤ d, y >d for n>N. (3.9) 0 N n Thus max y : n ≥ 0 ≤ max d, y := m. (3.10) n N+1 Consequently, 0 ≤ x(t) = y (t) ≤ y (t)≤···≤ y (t)≤···≤ m. (3.11) 0 1 n Therefore, lim y (y) = y¯(t). (3.12) n→∞ Note that y¯(t) is the unique solution of (3.2). The conclusion of Lemma 3.1 follows from (3.11). Lemma 3.2. Assume that (H1) is fulfilled. Furthermore, for any y ∈ X and η∈C([−∆ ,0];X) α α F(t, y,η) ≤ M 1+y +|η| ,0 ≤ α< 1, t ∈ [a,b]. (3.13) Then the mild solution x(t;z) of (2.8) has the estimate x ≤ H z , (3.14) where H(r) is an increasing function and H(r) = O(r) as r →∞. Proof. Recall that M = max S(t) :0 ≤ t ≤ b . (3.15) It follows from x(s)≤|x | and (2.8)that t t α α x(t) ≤ M |ξ| + M M 1+ x(s) + x ds + M z(s) ds 1 1 s 1 a a (3.16) ≤ M |ξ| + M M(b− a)+ M b− az+2M M x ds. 1 1 1 1 s For any θ ∈ [−∆ ,0], we have t+θ x(t + θ) ≤ M |ξ| + M M(b− a)+ M b− az+2M M x ds. (3.17) 1 1 1 1 s a 72 Approximate controllability of delayed systems Hence x ≤ M |ξ| + M M(b− a)+ M b− az+2M M x ds. (3.18) t 1 1 1 1 s Note that for any two constants V and V , the following equation 1 2 y(t) = V + V y (s)ds (3.19) 1 2 has a unique solution 1/(1−α) 1−α y(t) = (1− α)V t + V . (3.20) Applying Lemma 3.1 to (3.18), we obtain 1/(1−α) 1−α x ≤ 2(1− α)MM (b− a)+ M |ξ| + M b− az + MM (b− a) t 1 1 1 1 := H z . (3.21) Clearly, the function H(r) satisfies all requirements of Lemma 3.2 and the proof of the lemma is complete. 4. Approximate controllability The following theorem is the main result of this paper. Theorem 4.1. Assume that linear system (1.3) is approximately controllable on [a,b].If (H1), (H2), (H3), and (3.13) are fulfilled, then system (1.1) is approximately controllable on [a,b]. Proof. Note that system (1.3) is approximately controllable on [a,b]bythe assumption, then R (0) = X. To prove the approximate controllability of (1.1); that is, R (F) = X,itis b b sufficient to show that R (0) ⊂ R (F). (4.1) b b b b That means for any  > 0and x ∈ R (0), there exists a ν ∈ R (F)suchthat ν− x  < . b b By the definition of reachable set R (0)ofsystem(1.3), there is a control u(·) ∈ L (a,b;U)suchthat x = S(b− a)ξ(0) + S(b− s)(Bu)(s)ds. (4.2) ⊥ ⊥ ⊥ Let z¯ = Bu, z = Gz¯ .Then z ∈ N . Define an operator J from N to N by 0 0 0 0 Jv = z − GΓPv, v ∈ N , (4.3) 0 Lianwen Wang 73 2 2 where Γ is the operator from L (a,b;X)to L (a,b;X)definedby (Γz)(t):= F t,(Wz)(t),(Wz) = F t,x(t;z),x . (4.4) t t ⊥ 2 2 ⊥ For any v ∈ N ,wehave Pv ∈ L (a,b;X), ΓPv ∈ L (a,b;X), and GΓPv ∈ N . Therefore, J is well defined. Since W is compact by assumption (H2), for any bounded sequence z (·) ∈ L (a,b;X); that is,z ≤ r for some r > 0, there is a subsequence z (·)of z (·)suchthat(Wz )(·) n 1 1 n n n k k converges to x¯(·)in C([a,b];X)as k →∞.So, Wz is bounded in C([a,b];X); that is, Wz ≤ r for some constant r > 0. (H1) implies that a constant L(r) > 0 exists such n 2 2 that ¯ ¯ F t, Wz (t), Wz − F t,x(t),x n n t k k (4.5) ≤ L(r) Wz (t)− x¯(t) + Wz − x¯ , n n t k k t where r = max(r ,r ). Hence, we have 1 2 Γz −F ·,x¯(·),x¯ n · = F ·, Wz (·), Wz − F ·,x¯(·),x¯ n n · k k · (4.6) ≤ L (r)(b− a) sup Wz (t)− x¯(t) a≤t≤b +sup Wz − x¯ −→ 0 n t k t a≤t≤b as k →∞. Therefore, Γ is compact and J is compact as well. From Lemma 3.2,for any z(·) ∈ L (a,b;X), we have α α F t,x(t;z),x ≤ M 1+ x(t;z) + x ≤ M 1+2H z . (4.7) t t Note that H(r)isincreasingand P is a bounded operator, then z − GΓPv ≤ z +GΓPv 0 0 (4.8) ≤ z + M b− a+2M b− aH Pv . Taking into account √ √ z + M b− a+2M b− aH Pv lim = 0, (4.9) v→∞ v then z − GΓPv lim = 0. (4.10) v→∞ v Therefore, we can find a sufficiently large number r¯ such that z − GΓPv ≤ r¯ for v≤ r. ¯ (4.11) 0 74 Approximate controllability of delayed systems ⊥ ⊥ This means that J maps the bounded closed set D(r¯)={v : v≤ r¯,v ∈ N } of N into itself. Consequently, a fixed point of operator J exists due to the Schauder fixed point theorem; that is, there is a v ∈ D(r¯)suchthat ∗ ∗ ∗ Jv = z − GΓPv = v . (4.12) On account of ∗ ∗ Pv ∈ v + N ∩ R[B], (4.13) we have b b ∗ ∗ S(b− s)(Pv )(s)ds = S(b− s)v (s)ds. (4.14) a a 2 ⊥ Note that G is the projection operator from L (0,T;X)into N ,thenwehave b b S(b− s)Gp(s)ds = S(b− s)p(s)ds for p(·) ∈ L (a,b;X), a a b b ∗ ∗ (4.15) S(b− s)(Bu)(s)ds = S(b− s) F s,x s;Pv ,x + v (s) ds a a ∗ ∗ = S(b− s) F s,x s;Pv ,x )+ Pv (s) ds. Finally, b ∗ ∗ ∗ x = S(b− a)ξ(0) + S(b− s) F s,x s;Pv ,x + Pv (s) ds = x b;Pv . (4.16) ∗ 2 Observe that Pv ∈ R[B], then there is a sequence u (·) ∈ L (a,b;U)suchthat Bu → n n Pv as n→∞. W is continuous due to its compactness, then WBu −→ WPv in C [a,b];X . (4.17) This implies ∗ b x b;Bu −→ x b;Pv = x (4.18) as n→∞.Since x(b;Bu ) = y(b;u ) ∈ R (F), we obtain x ∈ R (F) and complete the n n T T proofofthe theorem. Remark 4.2. If A = 0, (H3) implies the approximate controllability of (1.3)on[a,b] (see [6]). Therefore, Naito’s result in [6]isaspecialcaseof Theorem 4.1 when A = 0, ∆ = 0, and F(t,x(t),x ) = F(x(t)). In particular, we improve Naito’s result by weakening the uniform Lipschitz continuity and the uniform boundedness imposed on the nonlinear term. Lianwen Wang 75 5. Example Let X = L (0,π)and e (x) = sin(nx)for n = 1,.... Then {e : n = 1,2,...} is an orthogonal n n base for X.Define A : X → X by Ay = y with domain D(A) = y ∈ X : y and y are absolutely continuous, y ∈ X, y(0) = y(π) = 0 . (5.1) Then Ay =− n y,e e , y ∈ D(A). (5.2) n n n=1 It is well known that A is the infinitesimal generator of an analytic group T(t), t ≥ 0, in X and is given by −n t T(t)y = e y,e e , y ∈ X. (5.3) n n n=1 T(t) is compact because it is an analytic semigroup. Define an infinite dimensional space U by ∞ ∞ U = u : u = u e , u < ∞ (5.4) n n n=2 n=2 with the norm defined by 1/2 u = u . (5.5) n=2 Define a mapping B from U to X as follows: Bu = 2u e + u e . (5.6) 2 1 n n n=2 Consider the following delayed semilinear heat equation: ∂y(t,x) ∂ y(t,x) = + y(t − ∆ ,x)+ F y(t,x), y(t− ∆ ,x) ∂t ∂x + Bu(t,x), 0 <t <b,0 <x <π, (5.7) y(t,0) = y(t,π) = 0, 0 ≤ t ≤ b, y(t,x) = ξ(x), −∆ ≤ t ≤ 0, 0 ≤ x ≤ π. Then system (5.7) can be written to the abstract form (1.1). (H2) holds because T(t)is a compact semigroup. Following the same arguments as in [6]wecan provethat(H3)is valid and that the corresponding linear system is approximately controllable on [0,b]. By Theorem 4.1,system(5.7)isapproximatelycontrollableon[0,b]if F is locally Lipschitz continuous and condition (3.13) is satisfied. 76 Approximate controllability of delayed systems References [1] J. R. Choi, Y. C. Kwun, and Y. K. Sung, Approximate controllability for nonlinear integrodifferen- tial equations, J. Korea Soc. Math. Educ. Ser. B Pure Appl. Math. 2 (1995), no. 2, 173–181. [2] L.A.Fernandez ´ and E. Zuazua, Approximate controllability for the semilinear heat equation involving gradient terms, J. Optim. Theory Appl. 101 (1999), no. 2, 307–328. [3] J. Klamka, Controllability of Dynamical Systems, Mathematics and its Applications (East Euro- pean Series), vol. 48, Kluwer Academic Publishers Group, Dordrecht, 1991. [4] X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkhauser, Boston, Massachusettes, USA, 1995. 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