Approximate Controllability of Neutral Functional Differential System with Unbounded Delay

We consider a class of control systems governed by the neutral functional differential equation with unbounded delay and study the approximate controllability of the system. An example is given to illustrate the result.


Introduction.
Let Ꮾ be an abstract phase space.Consider the following nonlinear control equation: where F,G : [0,T ] × Ꮾ → X are continuous functions, A is the infinitesimal generator of an analytic semigroup S(•) of bounded linear operators on a Banach space X, the state function x(t), 0 ≤ t ≤ T , takes values in X, and the control function v(•) is given in L 2 (0,T : V ), which is a Banach space of admissible control functions, with V as a Banach space.Also, B is a bounded linear operator from L 2 (0,T : V ) into L 2 (0,T : X).
The theory of functional differential equations with unbounded delay has been studied by many authors.Hale and Kato [1] have established the local existence and continuation of solutions for retarded equations with infinite delay with initial values in an abstract phase space.Henríquez [2] proved the existence of solutions and the periodic solutions of a class of partial functional differential equations.Recently, Hernández and Henríquez [3] have studied the existence problem for partial neutral functional differential equations with initial values in phase space.
In this paper, we study the approximate controllability of system (1.1) by using the results of Hernández and Henríquez [3].Similar results on controllability and approximate controllability of linear and nonlinear control systems have been studied in [5,6,8].
To study the nonlinear system (1.1), we assume that the histories x t : (−∞, 0] → X, x t (θ) := x(t +θ), belong to some abstract phase space Ꮾ, that is, a phase space defined axiomatically.Here, Ꮾ is a linear space of functions mapping (−∞, 0] into X endowed with a seminorm • Ꮾ and Ꮾ satisfies the following axioms (see [1]): (A 1 ) If x : (−∞,σ +a) → X, a > 0, is continuous on [σ , σ +a), σ is fixed, and x σ ∈ Ꮾ, then for every t ∈ [σ , σ + a) the following conditions hold: ), K is continuous and M is locally bounded, and H, K, and M are independent of x(•). (A 2 ) For the function x(•) in (A 1 ), x t is a Ꮾ-valued continuous function on [σ , σ +a). (A 3 ) The space Ꮾ is complete.Denote by Ꮾ the quotient Banach space Ꮾ/ • Ꮾ and if ϕ ∈ Ꮾ, we write φ for the coset determined by ϕ.Examples of phase space satisfying the above axioms can be found in [3,4].

Preliminaries.
Let the norm of the space X be denoted by • and for the other spaces we use • ∞ , and so on.
We assume the following hypotheses: (H 1 ) −A is the infinitesimal generator of an analytic semigroup S(•) of bounded linear operator on X, where the semigroup S(t) is uniformly bounded, S(t) ≤ M for some constant M ≥ 1 and for every t ≥ 0, and 0 ∈ ρ(A).
(H 2 ) There exist constants β ∈ (0, 1) and L 1 ≥ 0, such that the function F : [0,T ]×Ꮾ → X is X β -valued and satisfies the Lipschitz condition for every 0 ≤ s, t ≤ T , and ψ 1 ,ψ 2 ∈ Ꮾ, and for every 0 ≤ s ≤ T , and Under the above hypotheses it is well known [3] that for each u ∈ L 2 (0,T : X) there exists a unique mild solution The solution mapping W from L 2 (0,T : X) to C(0,T : X) can be defined by (2.6) We also define the continuous linear operator Φ from L 2 (0,T : X) to X by (2.7) Definition 2.1.Let the reachable set of the system (1.1) at time T be where x t (Bv) is a mild solution which satisfies (2.5) with u = Bv.
Definition 2.2.The system (1.1) is said to be approximate controllable on the interval (2.9) where x t (Bv) is a solution of (1.1) associated with the nonlinear term G and control Bv at the time t.
To simplify our task we consider the linear case of F .We introduce the following assumptions.
Proof.Since the domain D(A) of the operator A is dense in X (see [7]), it is sufficient to prove that D(A) ⊂ K T (0), that is, for any given > 0 and ξ ∈ D(A) there exists a To prove the approximate controllability of system (1.1), we need the following lemma.