NULL CONTROLLABILITY OF NEUTRAL EVOLUTION INTEGRODIFFERENTIAL SYSTEMS WITH INFINITE DELAY

The concept of controllability plays a major role in finite-dimensional control theory so it is natural to try to generalize this to infinite dimensions. Controllability is the property of being able to steer between two arbitrary points in the state space. For continuous time-invariant linear systems in finite-dimensional space the concepts of controllability and reachability are equivalent, and they are independent of the time. But in infinitedimensional space, the situation is more complex, and many different types of controllability and reachability have been studied in the literature. A weaker condition than exact controllability is the property of being able to steer all points exactly to the origin. This has important connections with the concept of stabilizability. Several authors have studied the null controllability of various kinds of dynamical systems [4, 5]. Neutral differential equations arise in many areas of applied mathematics and such equations have received much attention in recent years. The theory of functional differential equation with unbounded delay has been studied by several authors [6]. Almost all the work deals with the Cauchy problem


Introduction
The concept of controllability plays a major role in finite-dimensional control theory so it is natural to try to generalize this to infinite dimensions.Controllability is the property of being able to steer between two arbitrary points in the state space.For continuous time-invariant linear systems in finite-dimensional space the concepts of controllability and reachability are equivalent, and they are independent of the time.But in infinitedimensional space, the situation is more complex, and many different types of controllability and reachability have been studied in the literature.A weaker condition than exact controllability is the property of being able to steer all points exactly to the origin.This has important connections with the concept of stabilizability.Several authors have studied the null controllability of various kinds of dynamical systems [4,5].
Neutral differential equations arise in many areas of applied mathematics and such equations have received much attention in recent years.The theory of functional differential equation with unbounded delay has been studied by several authors [6].Almost all the work deals with the Cauchy problem x (t) = F t,x t , t ≥ σ, x σ = ϕ, (1.1) where x t represents the "history" of x at t, the values x(t) belong to some finite-dimensional space, and F is a function, usually continuous on appropriate spaces.Nevertheless, this class of equations does not include partial integrodifferential equations with infinite delay, which arise, for example, in the study of heat conduction in materials with memory or population dynamics for spatially distributed populations.Besides, it is well known that the behavior of the first-and second-order abstract Cauchy problem is different in many aspects.For these reasons, there has been an increasing interest in studying equations that can be described in the form where A is the infinitesimal generator of a strongly continuous semigroup of linear operators on a Banach space X.We call these equations as abstract retarded functional differential equations.
Similarly there exists an extensive theory for ordinary neutral functional differential equations, which includes qualitative behavior of classes of such equations and applications to biological and engineering processes.However, for partial neutral functional differential equations very little is known [14].Our purpose here is to study the controllability of equations that can be modelled in the form where the initial conditions x σ and F and G are appropriate functions.These equations will be called abstract neutral functional differential equations with unbounded delay.As a motivation example for this class of equations we consider the boundary value problem where the functions a 0 , a, a 1 , b, and φ satisfy appropriate conditions.These problems arise from control systems described by abstract retarded functional differential equations with a feedback control governed by a proportional integrodifferential law [1,16].On the other hand, some abstract retarded functional differential equations can be conveniently transformed into abstract neutral functional differential equations.Consider (1.2) with where C is a strongly continuous map of continuous operators from X into X.Assume that we can decompose C(s) = L(s) + N(s) where L and N are also strongly continuous K. Balachandran and A. Leelamani 3 maps of continuous operators and further the L(s) are linear.We define the operator V (t) by then (1.2) can be transformed into an abstract neutral functional differential equation which has the form (1.3) and in some cases, depending on V and N, it is easier to treat than the original equation.Motivation for neutral functional differential equations can be found in [3,5,[12][13][14].
There are several papers which have appeared on the controllability of nonlinear systems in infinite-dimensional spaces [4].Balachandran and Anandhi [2] discussed the controllability of neutral functional integrodifferential systems in abstract phase space with the help of Schauder's fixed point theorem.Recently Fu [8,9] studied the same problem in abstract phase space for neutral functional differential systems and nonlinear neutral systems with unbounded delay by utilizing the Sadovskii fixed point theorem.In this paper we will study the null controllability of neutral evolution integrodifferential system with infinite delay by utilizing the technique of Fu [9].The results are a generalization of the results established by Fu [8,9].
Consider the neutral functional integrodifferential system of the form where the state variable x(•) takes values in the Banach space X, and the control function u(•) is in L 2 (J,U), a Banach space of admissible control functions with U as a Banach space and J = [0,a] and B is a bounded linear operator from U into X.The unbounded linear operator −A(t) generates an analytic semigroup and h : appropriate functions and Ꮾ is the phase space to be specified later.
(a) U(t,s) ∈ L(X) the space of bounded linear transformation on X, whenever 0 ≤ s ≤ t ≤ a and for each x ∈ X, the mapping (t,s ) is a compact operator whenever t,s > 0. Condition (B4) ensures that the generated evolution system satisfies (d).In fact, let exp(−τA(t)) denote the analytic semigroup having infinitesimal generator A(t), we obtain the estimation exp(−τA(t)) ≤ K. Friedman [7] constructed a family of bounded linear operators {Φ(t, τ) : 0 ≤ τ ≤ t ≤ a} with Φ(t,τ) ≤ K|t − τ| α−1 that provide the following representation for U(t,s): We choose a sequence { n } which decreases to zero.If n < t − s , we define 3) The condition (B4) implies that exp(−τA(t)) is compact whenever τ >0, then each U n (t,s) is compact and we have Thus U(t,s) is the limit of a sequence of compact operators in the uniform operator topology, which implies that (d) is true.For the evolution system {U(t, s) : 0 ≤ s ≤ t ≤ a}, the following properties are well known [18]: (iii) the family of operators {U(t, s), t > s} is continuous in t in the uniform operator topology uniformly for s.We need the following fixed point theorem due to Sadovskiȋ [17].
Theorem 2.1 (Sadovskiȋ).Let P be a condensing operator on a Banach space X, that is, P is continuous and takes bounded sets into bounded sets, and let α(P(B)) ≤ α(B) for every bounded set B of X with α(B) > 0. If P(H) ⊂ H for a convex, closed, and bounded set H of X, then P has a fixed point in H (where α(•) denotes Kuratowski's measure of noncompactness).

K. Balachandran and A. Leelamani 5
We assume that the delay x t : (−∞,0] → X defined by x t (θ) = x(t + θ) belongs to some abstract phase space Ꮾ, which will be a linear space of functions mapping (−∞,0] into X endowed with the seminorm • Ꮾ and satisfying the following axiom [10,11,15]. (A1) If x : (−∞,a) → X, a > 0 , is continuous on [0, a] and x 0 ∈ Ꮾ, then for every t ∈ [0,a) the following conditions hold: Here The space Ꮾ is complete.We assume the following assumptions on the system (1.8).
(H1) g : J × Ꮾ × X → X is a continuous function, and there exist constants L,L 1 > 0 such that the function A(t)g satisfies the Lipschitz condition

, and the inequality
is continuous and there exist constants L 2 ,L 3 > 0 such that the function A(t)h satisfies the Lipschitz condition for 0 ≤ t 1 , t 2 ≤ a, ψ 1 ,ψ 2 ∈ Ꮾ, and the inequality (2.10) (H5) The linear operator W from L 2 (J;U) into X is defined by which induces an invertible operator W defined on L 2 (J;U)/ ker W, and there exists a constant M 3 > 0 such that B W −1 ≤ M 3 .The system (1.8) is said to be (local) null controllable on the interval [0, a] if for every initial function φ ∈ (Ω ⊂)Ꮾ there exists a control u ∈ L 2 ([0,a];U) such that the mild solution x(•) of (1.8) satisfies x(a) = 0.
Proof.Using the assumption (H5), for an arbitrary function x(•) define the control q s,τ,x τ dτ ds (t). (3.1) It will be shown that when using this control the operator S defined by which implies that the system is null controllable.Next we will prove that the operator S has a fixed point.Let y(•) : (−∞,a) → X be the function defined by then y 0 = φ and the map t → y t is continuous.Take N = sup{ y t Ꮾ : 0 ≤ t ≤ a}.For each z ∈ C([0,a];X), z(0) = 0, we denote by z the function defined by If x(•) satisfies the mild solution of (1.8), we can decompose it as x(t) = z(t) + y(t), 0 ≤ t ≤ a, which implies that x t = zt + y t for every 0 ≤ t ≤ a, and the function z(•) satisfies z(t) = −U(t,0)g(0, φ,0)   U(t,s) f s, zk,s + y s , s 0 q s,τ, zk,τ + y τ dτ ds ≤ M 1 g(0,φ,0) + g t, zk,t + y t , t 0 h t,s, zk,s + y s ds    (3.11)where k * = (1 + δ)(kK a + N), there holds (3.12) Dividing both sides by k and taking the lower limit, we get for 0 ≤ t ≤ a, respectively.We will show that P 1 is a contraction mapping and P 2 is a compact operator.
To prove that P 1 is a contraction, we take z 1 ,z 2 ∈ B k , then for each t ∈ [0,a] and by (A1)(iii) and (2.6), (2.13), we have and so P 1 satisfies contraction condition with L * < 1.
To prove that P 2 is compact, first we prove that P 2 is continuous on B k .Let {z n } ⊆ B k with z n → z in B k , then for each s ∈ [0,a], zn,s → zs and by (H4)(i), we have f s, zn,s + y s , s 0 q s,τ, zn,τ + y τ dτ −→ f s, zs + y s , s 0 q s,τ, zτ + y τ dτ as n −→ ∞.
Next we prove that the family {P 2 z : z ∈ B k } is an equicontinuous family of functions.To do this, let > 0 be small, 0 < t 1 < t 2 , then and μ k * (τ) ∈ L 1 , we see that (P 2 z)(t 1 ) − (P 2 z)(t 2 ) tends to zero independent of z ∈ B k as (t 2 − t 1 ) → 0 with sufficiently small, since the compactness of U(t,s) implies the continuity of U(t,s) in the uniform operator topology.Hence, P 2 maps B k into an equicontinuous family of functions.It remains to prove that Using the estimation of u(s) and by the compactness of U(t,s) we prove that V (t) = {(P 2, z)(t) : z ∈ B k } is relatively compact in X for every , 0 < < t.Moreover, for every z ∈ B k , we have (3.23) Therefore there are relatively compact sets arbitrarily close to the set Thus by the Arzela-Ascoli theorem P 2 is a compact operator.These arguments show that P = P 1 + P 2 is a condensing mapping on B k , and by the Sadovskiȋ fixed point theorem [17] there exists a fixed point z(•) for P on B k .If we define x(t) = z(t) + y(t), −∞ < t ≤ a, then it is easy to see that x(•) is a mild solution of (1.8) satisfying x 0 = φ, x(a) = 0. Hence the proof.Now we discuss the local null controllability of the system (1.8).For this purpose, we impose the weaker assumptions on the system (1.8) as follows.
(H7) g : J × Ω × X → X is a continuous function, and there exists a constant L > 0 such that the function A(t)g satisfies the Lipschitz condition: ) ) for all 0 ≤ t ≤ a. Define the set then S(ρ) is also a nonempty bounded, closed, and convex subset of C([0,a];X) and P well defined on S(ρ).We will show that P maps S(ρ) into S(ρ).In fact, let z ∈ S(ρ); we have The remaining part of the proof is similar to that of Theorem 3.1 and hence it is omitted.

Example
Consider the following partial functional differential system: where a(t,x) is a continuous function and is uniformly Holder continuous in t.Let X = L 2 ([0,π]) and let A(t) be defined by with the domain Then A(t) generates an evolution system U(t,s) satisfying assumptions (B1)-(B4).

Z 1 (
z)(x) = π 0 b(y,x)z(y)dy.(4.10)From (i) it is clear that Z 1 is a bounded linear operator on Ꮾ.Furthermore, by (4.2) and (iii) Z 1 (z) ∈ D(A) and A(t)Z 1 (z) ≤ N 1 .In fact, from the definition of Z 1 and (ii) it follows that Z 1 (z),z n = x t , x t , Then x(•) is a mild solution of system (1.8), and it is easy to verify that t 0 U(t,s)A(s)g s,x s , s 0 h s,τ,x τ dτ ds + t 0 U(t,s) Bu(s) + f s,x s , s 0 q s,τ,x τ dτ ds, 0≤ t ≤ a, (3.2) K. Balachandran and A. Leelamani 7 has a fixed point x(•).
) then B k , for each k, is a bounded, closed, convex set in C([0,a];X).Since by (2.6) the following relation holds: zτ + y τ dτ ≤ ML 1 1 + aM 2 L 3 kK a + N + 1 , (3.9) zτ + y τ )dτ) is integrable on [0,a], so P is well defined on B k .We claim that there exists a positive number k such that PB k ⊆ B k .If it is not true, then for each positive number k, there is a functionz k ∈ B k but Pz k / ∈ B k , that is, Pz k (t) > k for some t ∈ [0,a].However, on the other hand, we havek ≤ Pz k (t) aM 1 L 1 1 + aM 2 L 3 kK a + N + 1 , ≤ Now define the operators P 1 , P 2 on B k byP 1 z (t) = −U(t,0)g(0, φ,0) + g t, zt + y t , .13)This contradicts(2.14).Hence PB k ⊆ B k , for some positive number k.
z1,t + y t , .24) (H8) Let Ω ⊂ Ꮾ be an open set and let the function f :J × Ω × X → X satisfy the following conditions.(i)Foreach t ∈ [0,a], the function f (t,•,•) : Ω × X → X is continuous and for each (φ,x) ∈ Ω × X the function f (•,φ,x) : J → X is strongly measurable.(ii)For each positive number k, there is a positive function μ k ∈ L 1 ([0,a]) such thatProof.We prove this theorem by using again the Sadovskiȋ fixed point theorem.Let y(•), u(•), S, P, P 1 , P 2 be as in the proof of Theorem 3.1.It is enough to prove that P has a fixed point, which implies that S has a fixed point.×B r (φ) × X.As y 0 = φ, we choose 0 < b 2 < b 1 such that y t − φ Ꮾ ≤ r/2 for all 0 ≤ t ≤ b 2 .Let ρ = r/2K b2 and > 0, then from the continuity of the functions g, K(t) and t → y t , the compactness of the set {g(t, y t , y t , (t) ≤ U(t,0) A −1 (t) L t + y t − φ Ꮾ +