Mathematical Problems in Engineering

Volume 2013, Article ID 419156, 13 pages

http://dx.doi.org/10.1155/2013/419156

## Controllability of Nonlinear Neutral Stochastic Differential Inclusions with Infinite Delay

^{1}College of Mathematics Science, Chongqing Normal University, Chongqing 400047, China^{2}School of Computer and Information Science, Southwest University, Chongqing 400715, China

Received 17 November 2012; Revised 20 February 2013; Accepted 15 March 2013

Academic Editor: Yong-Kui Chang

Copyright © 2013 Yong Li and Qiang Zou. 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.

#### Abstract

The paper is concerned with the controllability of nonlinear neutral stochastic differential inclusions with infinite delay in a Hilbert space. Sufficient conditions for the controllability are obtained by using a fixed-point theorem for condensing maps due to O'Regan.

#### 1. Introduction

Let be a separable Hilbert space, and let be a complete probability space furnished with a complete family of right continuous increasing algebras satisfying for . Suppose is a given -valued, adapted Brownian motion with a finite trace nuclear covariance operator . We are interested in the controllability of nonlinear stochastic differential inclusions with infinite delay where is measurable and is the infinitesimal generator of an analytic semigroup of bounded linear operator , , the state takes values in Hilbert space with the norm , is a bounded closed, convex-valued multivalued map, is continuous, and the histories , , and belong to the space . For , define , where is the adjoint of the operator , . furnished with the scalar product is a pre-Hilbert space. The completion of with respect to the topology induced by the norm , where , is a Hilbert space.

At first, we present the abstract phase space . Assume that is continuous function with . Define If is endowed with the norm then is a Banach space.

Controllability is one of the fundamental concepts in mathematical control theory and plays an important role in control systems. The problem of controllability is to show the existence of a control function, which steers the solution of the system from its initial state to a final state, where the initial and final states may vary over the entire space. Controllability generally means that it is possible to steer a dynamical control system from an arbitrary initial state to an arbitrary final state using the set of admissible controls. A standard approach is to transform the controllability problem into a fixed-point problem for an appropriate operator in a functional space. The problem of controllability for functional differential systems has been extensively studied in many papers [1–10]. For example, Sakthivel et al. [5] considered a class of fractional neutral control systems governed by abstract nonlinear fractional neutral differential equations and established a new set of sufficient conditions for the controllability of nonlinear fractional systems by using a fixed-point analysis approach. Using fixed-point technique, fractional calculations, stochastic analysis technique, and methods adopted directly from deterministic control problems, Sakthivel et al. [7] gave a new set of sufficient conditions for approximate controllability of fractional stochastic differential equations. In [8], Sakthivel and Ren investigated the complete controllability property of a nonlinear stochastic control system with jumps in a separable Hilbert space. In [9], Sakthivel et al. studied the approximate controllability for nonlinear dispersion system under the assumption that the corresponding linear control system is approximately controllable.

Benchohra et al. [11] discussed the controllability for one-order, second-order functional differential and integrodifferential inclusions in Banach space with finite delay. Park et al. [12] discussed the controllability for second-order neutral functional differential inclusions in Banach space with the help of some fixed-point theorems. In [13, 14], Liu investigated the controllability of neutral functional differential and integrodifferential inclusions with infinite delay.

History about the research of stochastic differential equations is over fifty years; many important results on the stochastic differential equations have been reported in the literature; see [15, 16]. Recently, stochastic partial differential equations (SPDE) arise in the mathematical modeling of various fields in physics and engineering science cited by Sobczyk [16]. About neutral stochastic differential equations with delay, Mahmudov [17] discussed the existence and uniqueness of neutral stochastic differential equations by using Picard technique. For the controllability of stochastic functional differential inclusions, Balasubramaniam and Ntouyas [18] discussed the controllability of semilinear stochastic delay evolution equations in Hilbert spaces. Balasubramaniam and Dauer [19] obtained the controllability result of stochastic differential inclusions with infinite delay in abstract space. Since many systems arising from realistic models heavily depend on histories (i.e., there is the effect of infinite delay on state equations) [20], there is a real need to discuss partial functional differential systems with infinite delay. So in the present paper, we will concentrate on the case with infinite delay and establish sufficient conditions for the controllability of systems (1) by relying on a fixed-point theorem for condensing maps due to O’Regan [21].

The structure of this paper is as follows. In Section 2 we briefly present some basic notations and preliminaries. The controllability result of system (1) is investigated by means of fixed-point theorem and operator theory in Section 3. Conclusion is given in Section 4.

#### 2. Preliminaries

Let be a Banach space. A multivalued map is convex (closed)-valued, if is convex (closed) for all . is bounded on bounded set if is bounded in for any bounded set of ; that is,

is called upper semicontinuous (Usc) on if for each , the set is nonempty, closed subset of and if for each open set of containing , there exists an open neighborhood of such that .

is said to be completely continuous if is relatively compact, for every bounded subset .

If the multivalued map is completely continuous with nonempty compact values, then is Usc if and only if has a closed graph (i.e., , , imply ).

Let denote the sets of all the sets of all nonempty, bounded, closed, and convex subsets of . For more details on multivalued maps see the books of Deimling [22] and Hu and Papageorgiou [23].

An upper semicontinuous map is said to be condensing if for any subset with , we have , where denotes the Kuratowski measure of noncompactness. It is easy to see that completely continuous multivalued map is a condensing map.

If is a uniformly bounded and analytic semigroup with infinitesimal generator such that then it is possible to define the fractional power , for , as a closed linear operator on its domain . Furthermore, the subspace is dense in and the expression defines a norm on . Hereafter we represent by the space endowed with the norm . Then the following properties are well known [24].

Lemma 1 (see [24]). *Suppose that the preceding conditions are satisfied.*(a)*Let . Then is a Banach space.*(b)*If , then and the imbedding is compact whenever the resolvent operator of is compact.*(c)*For every , there exists a positive constant such that
The key tool in our approach is following fixed-point theorem [21].*

Theorem 2 (see [21]). *Let be a Hilbert space, and let be an open, convex subset of and . If *(a)* has closed graph,*(b)* is condensing operator and is bounded set of ,**then either*(i)* has fixed-point in ; or*(ii)*there exist and such that .*

#### 3. Main Result

In the following, we will apply Theorem 2 to study the existence of solutions for system (1).

*Definition 3. *A function is called a mild solution of system (1) if the following holds: let be measurable -valued stochastic processes on and the integral equation
is satisfied, where

*Definition 4. *System (1) is said to be controllable on the interval if for every continuous initial random process , , there exists a control such that the mild solution of (1) satisfies .

To investigate the controllability of system (1), we use the following hypotheses. is the infinitesimal generator of an analytic semigroup of bounded linear operators in , , for ; there exist constants such that . The linear operator defined by
has an induced inverse operator which takes values in and there exist positive constants such that and . There exist constants , and such that and
where . ; is measurable with respect to for each , Usc with respect to for each , and for each fixed , the set
is nonempty. , , , , and is a continuous nondecreasing function. There exists a positive constant such that
where
The operator with values , , is completely continuous in and for any bounded set , the set is equicontinuous in . There exist constants and such that is continuous and for , , such that , .

Lemma 5 (Lasota and Opial [25]). *Let be a compact real interval and let be a Banach space. Let be a multivalued map satisfying and let be a linear continuous mapping from to . Then the operator
**
is a closed graph operator in .*

Lemma 6 (see [26]). *Let be continuous functions. If is nondecreasing and there are constants , such that
**
then
**
for every and every such that , and is the Gamma function.*

Now we consider the space ; let be the space of all -valued stochastic processes , , such that , let be a seminorm in defined by

Lemma 7 (see [27]). *Suppose ; then for , . Moreover
**
where .*

Now, consider the multivalued map defined by the set of such that where .

We will show that the operator has fixed-points, which are then a solution of system (1). For , we define by then . Set It is clear to see that satisfies (7) if and only if satisfies and Let . For any , Thus is a Banach space. Set for some ; then is uniformly bounded, for any ; from Lemma 7, we have Define the multivalued map defined by the set of such that

Theorem 8. *Assume that hypotheses hold; then system (1) is controllable on . *

*Proof. *We divide the proof into several steps.*Step 1.* We show that there exists an open set and there does not exist , for some , such that .

Let and let . Then there exists such that

Since
where , and are defined in .

Consequently

So
Then by there exists such that ; that is, . Define the set: . From the choice of , we know that there does not exist , for some , such that .*Step 2.* is convex for each .

In fact, if , belong to , then there exist , such that for each we have
Let ; we have
Since is convex, we have .*Step 3. * maps bounded set into bounded set in . Indeed, it is enough to show that there exists a positive constant such that for each , one has . If , then there exist , such that for each ,
So, from Lemma 1(c) for
then for each , we have
*Step* * 4. * has a closed graph.

Let , , and . We will prove that . Indeed, means that there exists , such that where We must prove that there exists such that where Set Since , are continuous, then for , Consider the linear continuous operator From Lemma 5, it follows that is a closed graph. Moreover, we have Since , it follows from Lemma 5 that for some .

*Step 5.*Multivalued function is a condensing operator.

From the decomposition of operator , we define two operators on space . For operator , , for each , defined by the set of such that

For operator , , for each , defined by the set of such that So, operator . Next, we will prove that operator is contraction operator and is completely continuous operator.

First, we prove operator is contractive. For each , we have So operator is contraction operator.

Next, we will prove that operator is completely continuous.

For , , if , , then there exist such that for arbitrary ,