Control Problems for Semilinear Neutral Differential Equations in Hilbert Spaces

Jeong, Jin-Mun; Cho, Seong Ho

doi:https://doi.org/10.1155/2014/431978

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Volume 2014 | Article ID 431978 | https://doi.org/10.1155/2014/431978

Control Problems for Semilinear Neutral Differential Equations in Hilbert Spaces

Jin-Mun Jeong¹and Seong Ho Cho¹

Academic Editor: F. Başar, P.-C. Lee

Received06 Aug 2013

Accepted04 Nov 2013

Published23 Feb 2014

Abstract

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.

1. 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 , into satisfying the local Lipschitz continuity, ,;,; and ,;,; 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 [3–6]. 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 [14–17].

In this paper, we propose a different approach from the earlier works (briefly introduced in [1–6] 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 -regularity remain valid under the above formulation of the neutral differential equation (1). For the basics of our study, the existence of local solutions of (1) is established in ,;,;,; for some 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 ,; 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.

2. Notations

Let be a region in an -dimensional Euclidean space and closure . is the set of all -times continuously differential functions on . will denote the subspace of consisting of these functions which have compact support in . is the set of all functions whose derivative up to degree in distribution sense belong to . As usual, the norm is then given by where . In particular, with the norm . is the closure of in . For we denote and . Let , . stands for the dual space of whose norm is denoted by .

If is a Banach space and , ,; is the collection of all strongly measurable functions from , to , the th powers of norms are integrable, ,; will denote the set of all -times continuously differentiable functions from , to . If and are two Banach spaces, , is the collection of all bounded linear operators from to , and , is simply written as . For an interpolation couple of Banach spaces and , and , denote the real and complex interpolation spaces between and , respectively.

Let be a closed linear operator in a Banach space. Then denotes the domain of (A) and the range of ; denotes the resolvent set of , the spectrum of , and the point spectrum of ; the kernel or null space of is denoted by .

3. 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 of and the element of is denoted by ,, which is the ordinary inner product in if .

For we denote , by the value of at . 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: 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 such that Thus we have the following sequence: where each space is dense in the next one and continuous injection.

Lemma 1. With the notations (11), (12), one has where , denotes the real interpolation space between and (Section of [20]).

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

Lemma 2. Let be the semigroup generated by . Then there exists a constant such that For all and every or there exists a constant such that the following inequalities hold:

By virtue of (6), we have that and the closed half plane is contained in the resolvent set of . In this case, there exists a neighborhood of such that Hence, we can choose that the path runs in the resolvent set of from to , , avoiding the negative axis. For each , we put where is chosen to be for . By assumption, is a bounded operator. So we can assume that there is a constant such that For each , 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.
(b) If , then .
(c) For any , there exists a positive constant such that the following inequalities hold for all :

Proof. From [21, Lemma ] it follows that there exists a positive constant such that the following inequalities hold for all and every or : which implies (21) by properties of fractional power of . For more details about the above lemma, we refer to [21, 22].

Let the solution spaces and of strong solutions be defined by Here, we note that by using interpolation theory, we have Thus, there exists a constant such that

First of all, consider the following linear system:

By virtue of Theorem 3.3 of [6] (or Theorem 3.1 of [3, 21]), we have the following result on the corresponding linear equation of (26).

Lemma 4. Suppose that the assumptions for the principal operator stated above are satisfied. Then the following properties hold: (1)for (see Lemma 1) and ,;, , there exists a unique solution of (26) belonging to ,; and satisfying where is a constant depending on ;(2)let and ,;, ; then there exists a unique solution of (26) belonging to ,; and satisfying where is a constant depending on .

Lemma 5. For every ,;, let for . Then there exists a constant such that

Proof. By (27) we have Since it follows that From (11), (30), and (32) it holds that So, the proof is completed.

4. 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).

Assumption B. Let ,;,; be a bounded linear mapping such that there exist constants , , and a continuous nondecreasing function , with such that

Assumption F. is a nonlinear mapping of into satisfying the following.(i)There exists a function such that hold for and .(ii)The inequality holds for every , and .Let us rewrite , for each ,;. Then there is a constant, denoted again by , such that hold for ,; and ,,;.

From now on, we establish the following results on the solvability of (34).

Theorem 6. Let Assumptions B and F be satisfied. Assume that , ,; for . Then, there exists a solution of (34) such that Moreover, there is a constant independent of and the forcing term such that

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 is a closed convex subset of a Banach space . Assume that and are mappings from to such that the following conditions are satisfied:(i),(ii) is a completely continuous mapping,(iii) is a contraction mapping.Then the operator has a fixed point in .

Proof of Theorem 6. Let where is constant in Lemma 4. Let , and choose such that where is constant in Lemma 5. Let Define a mapping ,;,; as It will be shown that the operator has a fixed point in the space ,;. By Assumptions B and F, it is easily seen that is continuous from ,; in itself. Let which is a bounded closed subset of ,;. From (27) it follows that By (21), (25), and assumption B we have By virtue of (29) in Lemma 5, for , it holds that Since (21) and Assumption F the following inequality holds: Let Then there holds Therefore, from (43), (47)–(52) it follows that and hence maps into .
Define mapping on ,; by the formula We can now employ Lemma 7 with . Assume that a sequence of ,; converges weakly to an element ,;; that is, . Then we will show that which is equivalent to the completely continuity of since ,; is reflexive. For a fixed ,, let for every ,;. Then ,; and we have since . Hence, By (21), (25), and assumption B we have Therefore, by Lebesgue’s dominated convergence theorem it holds that that is, . Since ,; is a Hilbert space, the relation (55) holds. Next, we prove that is a contraction mapping on . Indeed, for every and , we have Similar to (49) and (52), we have So by virtue of condition (44) the contraction mapping principle gives that the solution of (34) exists uniquely in ,.
So by virtue of condition (44), is contractive. Thus, Lemma 7 gives that the equation of (34) has a solution in .
From now on we establish a variation of constant formula (41) of solution of (34). Let be a solution of (34) and . Then we have that from (47)-(52) it follows that Taking into account (44) there exists a constant such that which obtain the inequality (41). Since the conditions (43) and (44) are independent of initial value and by (25) by repeating the above process, the solution can be extended to the interval ,.

Corollary 8. If , then the uniqueness of the solution of (34) in is obtained.

Proof. Let . Then instead of condition (44), we can choose such that For every and , we have Similar to (49) and (52), we have So by virtue of condition (64) the contraction mapping principle gives that the solution of (34) exists uniquely in .

Remark 9. Let Assumptions B and F be satisfied and ,,;. Then by the argument of the proof of Theorem 6 term by term, we also obtain that there exists a solution of (34) such that Moreover, there exists a constant such that where is a constant depending on .

The following inequality is refered to as the Young inequality.

Lemma 10 (Young inequality). Let , , and , where , and . Then for every one has

From the following result, we obtain that the solution mapping is continuous, which is useful for physical applications of the given equation.

Theorem 11. Let Assumptions B and F be satisfied and ,,,;. Then the solution of (34) belongs to ,;,; and the mapping is continuous.

Proof. From Theorem 6, it follows that if ,,;, then belongs to . Let ,,,; and let be the solution of (34) with ,, in place of ,, for ,. Let ,. Then as seen in Theorem 6, it holds that So the solution of the above equation is represented by And, hence, it holds that
From (43), we have Hence, repeating this process as seen in Theorem 6, we conclude that the solution mapping is continuous.

For ,;, let be the solution of (34) with instead of .

Theorem 12. Let one assume that the embedding is compact. For ,; let be the solution of (34). Then the mapping is compact from ,; to ,;. Moreover, if one defines the operator by then is also a compact mapping from ,; to ,;.

Proof. If ,,;, then in view of Theorem 6 Since ,;, we have ,,;. Consequently, by (25), we know that ,;. With aid of of Lemma 3, noting that , we have Hence if is bounded in ,;, then so is in ,;,;. Since is compactly embedded in by assumption, the embedding is compact in view of Theorem 2 of Aubin [24]. Hence is compact from ,;. Moreover, we have that is a compact mapping of which is of ,; to ,;.

5. 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 ,; to ,;, which is called a controller. For ,; we set where ,, is strongly continuous. Then it is immediately seen that ,; and hence for because . Since is strong continuous, by the uniform boundedness principle, there exists a constant such that, for any ,

Consider the following neutral control equation Let ;,, be a state value of the system (83) at time corresponding to the operator , the nonlinear term , and the control . We note that is the analytic semigroup generated by . Then the solution ;,, can be written as And in view of Theorem 6,

We define the reachable sets for the system (34) as follows:

Definition 13. The system (83) is said to be approximately controllable on , if for every and there exists a control function ,; such that the solution ;,, of (83) satisfies ;,; that is, , where is the closure of in .

We define the linear operator from ,; to by for ,;.

We need the following hypothesis.

Assumption S
(i) For any and ,;, there exists a ,; such that where is a constant independent of .
(ii) is a nonlinear mapping of , into satisfying the following.
There exists a function such that hold for and .

By virtue of condition (i) of Assumption S we note that for each . Therefore, the system (83) is approximately controllable on , if for any and there exists a control ,; such that where , for . Throughout this section, invoking (85), we can choose a constant such that and set

Lemma 14. Let and be in ,;. Then under the Assumption S, one has that, for , where .

Proof. Let ;,, and ;,,. Then for , we have So we immediately obtain and so it holds that Moreover, we have Thus, from (95) it follows that Therefore, by using Gronwall's inequality this lemma follows.

Theorem 15. Under Assumption S, the system (83) is approximately controllable on ,.

Proof. We will show that ; that is, for given and , there exists ,; such that where As there exists ,; such that for instance, take . Let ,; be arbitrary fixed. Since by Assumption S there exists ,; such that it follows that We can also choose ,; by Assumption S such that and by Assumption S for . Therefore, in view of Lemma 14 and Assumption S Put . We determine such that for . Hence, we have By proceeding with this process and from it follows that Therefore, there exists ,; such that From (104), (105) it follows that By choosing ,; by Assumption B, such that putting , we have Therefore, for there exists integer such that Thus the system (83) is approximately controllable on , as tends to infinity.

Example 16. Let The eigenvalue and the eigenfunction of are and , respectively. Moreover, (a) is an orthogonal basis of ,(b), ,(c)let ; then the fractional power of is given by

In particular, , and .

Consider the following neutral differential control system: where is a real valued function belonging to , which satisfies the following conditions:(i), for ,(ii) and for and . If we present is a mapping from the whole to by Sobolev's imbedding theorem (see [21], Theorem ). As an example of in the above, we can choose ( and are constants). In addition, we need to impose the following conditions (see [7, 25]).(iii)The function is measurable and (iv)The function is measurable, ,,,,, and (v),;,; is a bounded linear operator.We define ,;,; by From (ii) it follows that is bounded linear and where Thus, by (iv) the operator is bounded linear with and we have that and . Therefore from Theorem 6, there exists a solution of (120) such that Moreover, from Theorem 15 the neutral system (120) is approximately controllable on .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012-0007560).

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Copyright © 2014 Jin-Mun Jeong and Seong Ho Cho. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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