APPROXIMATE CONTROLLABILITY OF DELAYED SEMILINEAR CONTROL SYSTEMS

We deal with the approximate controllability of control systems governed by delayed semilinear differential equations ẏ(t)=Ay(t) +A1y(t−∆) +F(t, y(t), yt) + (Bu)(t). Various 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.


Introduction
The main concern in this paper is the approximate controllability of the following delayed semilinear control system: ẏ(t) = Ay(t) + A 1 y(t − ∆) + F t, y(t), y t + (Bu)(t), t ≥ a, y a = ξ, (1.1) in a real Hilbert space X with the norm • .The meaning of all notations is listed as follows: ∆ ≥ 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 0 semigroup T(t) in X; A 1 is a bounded linear operator from X to X; F : The following system is called the corresponding linear system of (1.1): This is a special case of (1.1) with F ≡ 0. The reachable set of system (1.
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 approximate 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 systems 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] if the 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 results.Naito and Park [7] dealt with approximate controllability for delayed Volterra systems.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 corresponding 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.

Basic assumptions
We start this section by introducing the fundamental solution S(t) of the following system: (2.1) We already know that (2.1) has a unique solution, denoted by y ξ (t), for each ξ ∈ C([−∆, 0];X).Hence, we can define an operator S(t) in X by 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: 3) we have Gronwall's inequality implies that Throughout the paper we impose the following condition on F.
With a minor modification of [8], we can prove that system (1.1) has a unique mild solution y( a,b;U) under assumption (H1).This mild solution is defined as a solution of the integral equation: Similarly, for any z(•) ∈ L 2 (a,b;X), the following integral equation: has a unique mild solution x(•;z).Therefore, we can define an operator (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, is a compact semigroup.
The following assumption (H3) was introduced by Naito in [6].Define a linear operator ϕ from L 2 (a,b;X) to X by (2.10) Let the kernel of the operator ϕ be N; that is, N = {p : ϕp = 0}.Then N is a closed subspace of L 2 (a,b;X).Denote its orthogonal space in L 2 (a,b;X) by N ⊥ .Let G be the projection operator from L 2 (a,b;X) into N ⊥ and let R[B] be the range of B. We assume that (H3) for any p(•) ∈ L 2 (a,b;X), there is a function q(•) ∈ R[B] such that 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] defined by where It is proved in [6] that P is bounded.

Lemmas
This section provides two lemmas that will be used to prove the main theorem.Construct a sequence {y n } as follows: We have Consequently, Proof.Recall that x s α ds. (3.16) For any θ ∈ [−∆,0], we have x s α ds.
(3.17) Hence x s α ds. (3.18) Note that for any two constants V 1 and V 2 , the following equation has a unique solution . (3.20) Applying Lemma 3.1 to (3.18), we obtain Clearly, the function H(r) satisfies all requirements of Lemma 3.2 and the proof of the lemma is complete.

Approximate controllability
The following theorem is the main result of this paper.Proof.Note that system (1.3) is approximately controllable on [a,b] by the assumption, then R b (0) = X.To prove the approximate controllability of (1.1); that is, That means for any > 0 and Lianwen Wang 73 where Γ is the operator from L 2 (a,b;X) to L 2 (a,b;X) defined by (Γz)(t) := F t,(Wz)(t),(Wz) t = F t,x(t;z),x t .( For any v ∈ N ⊥ , we have Pv ∈ L 2 (a,b;X), ΓPv ∈ L 2 (a,b;X), and GΓPv ∈ N ⊥ .Therefore, J is well defined.Since W is compact by assumption (H2), for any bounded sequence z n (•) ∈ L 2 (a,b;X); that is, z n ≤ r 1 for some r 1 > 0, there is a subsequence where r = max(r 1 ,r 2 ).Hence, we have as k → ∞.Therefore, Γ is compact and J is compact as well.From Lemma 3.2, for any z(•) ∈ L 2 (a,b;X), we have Note that H(r) is increasing and P is a bounded operator, then Taking into account lim Therefore, we can find a sufficiently large number r such that z 0 − GΓPv ≤ r for v ≤ r. (4.11)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) such that On account of

Example
Let X = L 2 (0,π) and e n (x) = sin(nx) for n = 1,.... Then {e n : n = 1,2,...} is an orthogonal base for X.Define A : X → X by Ay = y with domain (5. 2) It is well known that A is the infinitesimal generator of an analytic group T(t), t ≥ 0, in X and is given by e −n 2 t y,e n e n , y ∈ X. Define a mapping B from U to X as follows: u n e n . (5.6) Consider the following delayed semilinear heat equation:

. 1 )
If the equation y(t) = a(t) + t a b(s)y α (s)ds (3.2) has a unique solution ȳ(t) on [a,b], then x(t) ≤ ȳ(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) +
− s)(Pv * )(s)ds = b a S(b − s)v * (s)ds.(4.14)Note that G is the projection operator from L 2 (0,T;X) into N ⊥ , then we have b a S(b − s)Gp(s)ds = b a S(b − s)p(s)ds for p(•) ∈ L 2 (a,b;X), b a S(b − s)(Bu)(s)ds = b a S(b − s) F s,x s;Pv * ,x s + v * (s) ds = b a S(b − s) F s,x s;Pv * ,x s ) + Pv * (s) ds.
S(b − a)ξ(0) + b a S(b − s) F s,x s;Pv * ,x s + Pv * (s) ds = x b;Pv * .(4.16) Observe that Pv * ∈ R[B], then there is a sequence u n (•) ∈ L 2 (a,b;U) such that Bu n → Pv * as n → ∞.W is continuous due to its compactness, then WBu n −→ WPv * in C [a,b];X .(4.17)This implies x b;Bu n −→ x b;Pv * = x b (4.18) as n → ∞.Since x(b;Bu n ) = y(b;u n ) ∈ R T (F), we obtain x b ∈ R T (F) and complete the proof of the theorem.

(5. 3 )
T(t) is compact because it is an analytic semigroup.Define an infinite dimensional space U by U = u :