Control Problems for Semilinear Neutral Differential Equations in Hilbert Spaces

We construct some results on the regularity of solutions and the approximate controllability for neutral functional differential equations with unbounded principal operators in Hilbert spaces. In order to establish the controllability of the neutral equations, we first consider the existence and regularity of solutions of the neutral control system by using fractional power of operators and the local Lipschitz continuity of nonlinear term. Our purpose is to obtain the existence of solutions and the approximate controllability for neutral functional differential control systems without using many of the strong restrictions considered in the previous literature. Finally we give a simple example to which our main result can be applied.


Introduction
Let and be real Hilbert spaces such that is a dense subspace in . Let be a Banach space of control variables. In this paper, we are concerned with the global existence of solution and the approximate controllability for the following abstract neutral functional differential system in a Hilbert space : where is an operator associated with a sesquilinear form on × satisfying Gårding's inequality, is a nonlinear mapping of [0, ]× into satisfying the local Lipschitz continuity, : 2 (0, ; ) → 2 (0, ; ) and : 2 (0, ; ) → 2 (0, ; ) are appropriate bounded linear mapping.
This kind of equations arises in population dynamics, in heat conduction in material with memory and in control systems with hereditary feedback control governed by an integrodifferential law.
Recently, the existence of solutions for mild solutions for neutral differential equations with state-dependence delay has been studied in the literature in [1,2]. As for partial neutral integrodifferential equations, we refer to [3][4][5][6]. The controllability for neutral equations has been studied by many authors, for example, local controllability of neutral functional differential systems with unbounded delay in [7], neutral evolution integrodifferential systems with state dependent delay in [8,9], impulsive neutral functional evolution integrodifferential systems with infinite delay in [10], and second order neutral impulsive integrodifferential systems in [11,12]. Although there are few papers treating the regularity and controllability for the systems with local Lipschitz continuity, we can just find a recent article by Wang [13] in case of semilinear systems. Similar considerations of semilinear systems have been dealt with in many references [14][15][16][17].
In this paper, we propose a different approach from the earlier works (briefly introduced in [1][2][3][4][5][6] about the mild solutions of neutral differential equations. Our approach is that results of the linear cases of Di Blasio et al. [18] and semilinear cases of [19] on the 2 -regularity remain valid under the above formulation of the neutral differential equation (1). For the basics of our study, the existence of local 2 The Scientific World Journal solutions of (1) is established in 2 (0, ; ) ∩ 1,2 (0, ; * ) → ([0, ]; ) for some > 0 by using fractional power of operators and Sadvoskii's fixed point theorem. Thereafter, by showing some variations of constant formula of solutions, we will obtain the global existence of solutions of (1) and the norm estimate of a solution of (1) on the solution space. Consequently, in view of the properties of the nonlinear term, we can take advantage of the fact that the solution mapping ∈ 2 (0, ; ) → is Lipschitz continuous, which is applicable for control problems and the optimal control problem of systems governed by nonlinear properties.
The second purpose of this paper is to study the approximate controllability for the neutral equation (1) based on the regularity for (1); namely, the reachable set of trajectories is a dense subset of . This kind of equations arises naturally in biology, physics, control engineering problem, and so forth.
The paper is organized as follows. In Section 2, we introduce some notations. In Section 3, the regularity results of general linear evolution equations besides fractional power of operators and some relations of operator spaces are stated. In Section 4, we will obtain the regularity for neutral functional differential equation (1) with nonlinear terms satisfying local Lipschitz continuity. The approach used here is similar to that developed in [13,19] on the general semilinear evolution equations, which is an important role to extend the theory of practical nonlinear partial differential equations. Thereafter, we investigate the approximate controllability for the problem (1) in Section 5. Our purpose in this paper is to obtain the existence of solutions and the approximate controllability for neutral functional differential control systems without using many of the strong restrictions considered in the previous literature.
Finally, we give a simple example to which our main result can be applied.

Notations
Let Ω be a region in an -dimensional Euclidean space R and closure Ω.
(Ω) is the set of all -times continuously differential functions on Ω. 0 (Ω) will denote the subspace of (Ω) consisting of these functions which have compact support in Ω.
Let be a closed linear operator in a Banach space. Then

Regularity for Linear Equations
If is identified with its dual space we may write ⊂ ⊂ * densely and the corresponding injections are continuous. The norm on , , and * will be denoted by ‖ ⋅ ‖, | ⋅ | and ‖ ⋅ ‖ * , respectively. The duality pairing between the element V 1 of * and the element V 2 of is denoted by (V 1 , V 2 ), which is the ordinary inner product in if V 1 , V 2 ∈ .
For ∈ * we denote ( ,V) by the value (V) of at V ∈ . The norm of as element of * is given by Therefore, we assume that has a stronger topology than and, for brevity, we may consider Let (⋅, ⋅) be a bounded sesquilinear form defined in × and satisfying Gårding's inequality: Let be the operator associated with this sesquilinear form: The Scientific World Journal 3 Then is a bounded linear operator from to * by the Lax-Milgram theorem. The realization of in which is the restriction of to is also denoted by . From the following inequalities where is the graph norm of ( ), it follows that there exists a constant 0 > 0 such that Thus we have the following sequence: where each space is dense in the next one and continuous injection.

of [20]).
It is also well known that generates an analytic semigroup ( ) in both and * . The following lemma is from Lemma 3.6.2 of [21].

Lemma 2. Let ( ) be the semigroup generated by − . Then there exists a constant such that
For all > 0 and every ∈ or * there exists a constant > 0 such that the following inequalities hold: By virtue of (6), we have that 0 ∈ ( ) and the closed half plane { : Re ≥ 0} is contained in the resolvent set of . In this case, there exists a neighborhood of 0 such that Hence, we can choose that the path Γ runs in the resolvent set of from ∞ to ∞ − , < < , avoiding the negative axis. For each > 0, we put where − is chosen to be for > 0. By assumption, − is a bounded operator. So we can assume that there is a constant For each ≥ 0, we define an operator as follows: The subspace ( ) is dense in and the expression defines a norm on ( ).

Lemma 3. (a)
is a closed operator with its domain dense.
(c) For any > 0, there exists a positive constant such that the following inequalities hold for all > 0: Proof. From [21, Lemma 3.6.2] it follows that there exists a positive constant such that the following inequalities hold for all > 0 and every ∈ or * : which implies (21) by properties of fractional power of . For more details about the above lemma, we refer to [21,22].
Proof. By (27) we have Since it follows that From (11), (30), and (32) it holds that So, the proof is completed.

Semilinear Differential Equations
Consider the following abstract neutral functional differential system: Then we will show that the initial value problem (34) has a solution by solving the integral equation: Now we give the basic assumptions on the system (34).
Moreover, there is a constant 3 independent of 0 and the forcing term such that The Scientific World Journal 5 One of the main useful tools in the proof of existence theorems for functional equations is the following Sadvoskii's fixed point theorem.
Lemma 7 (see [23]). Suppose that Σ is a closed convex subset of a Banach space . Assume that 1 and 2 are mappings from Σ to such that the following conditions are satisfied: 1 is a completely continuous mapping, (iii) 2 is a contraction mapping.
Then the operator 1 + 2 has a fixed point in Σ.
From now on we establish a variation of constant formula (41) of solution of (34). Let be a solution of (34) and 0 ∈ . Then we have that from (47)-(52) it follows that ‖ ‖ 2 (0, 1 ; ) Taking into account (44) there exists a constant 3 such that which obtain the inequality (41). Since the conditions (43) and (44) are independent of initial value and by (25)  Proof. Let 0 < 1. Then instead of condition (44), we can choose 1 such that For every 1 and 2 ∈ Σ, we have Similar to (49) and (52), we have The Scientific World Journal 7 So by virtue of condition (64) the contraction mapping principle gives that the solution of (34) exists uniquely in [0, 1 ].

(67)
Moreover, there exists a constant 3 such that where 3 is a constant depending on .
The following inequality is refered to as the Young inequality.
Lemma 10 (Young inequality). Let > 0, > 0, and 1/ + 1/ = 1, where 1 ≤ < ∞, and 1 < < ∞. Then for every > 0 one has From the following result, we obtain that the solution mapping is continuous, which is useful for physical applications of the given equation.

Approximate Controllability
In this section, we show that the controllability of the corresponding linear equation is extended to the nonlinear differential equation. Let be a Banach space of control variables. Here is a linear bounded operator from 2 (0, ; ) to 2 (0, ; ), which is called a controller. For ∈ 2 (0, ; ) we set  We define the linear operator̂from 2 (0, ; ) to bŷ for ∈ 2 (0, ; ). We need the following hypothesis.
There exists a function 1 : R + → R such that hold for | | ≤ and | | ≤ .