#### Abstract

The approximate controllability of semilinear neutral stochastic integrodifferential inclusions with infinite delay in an abstract space is studied. Sufficient conditions are established for the approximate controllability. The results are obtained by using the theory of analytic resolvent operator, the fractional power theory, and the theorem of nonlinear alternative for Kakutani maps. Finally, an example is provided to illustrate the theory.

#### 1. Introduction

Controllability is an important concept which plays a vital role in many areas of applied mathematics. For the last decades, many authors established sufficient conditions for the controllability of nonlinear systems in Banach spaces; see [1–3]. Most of the controllability results for nonlinear systems concern the so-called semilinear control systems that consist of a linear part and a nonlinear part. Chang et al. [4] investigated the controllability of semilinear differential systems with nonlocal initial conditions in Banach spaces.

Integrodifferential equations have been recently proved to be strong tools in the modeling of many phenomena arising from many fields such as electronics. The theory of integrodifferential inclusions in deterministic cases may be found in several papers and monographs; for example, see [5–8]. In fact, many deterministic models often fluctuate because of noise, so we must move from deterministic control problems to stochastic control problems. The applications of stochastic differential equations and stochastic delay differential equations have attracted great interest; see [9–13]. In particular, Balasubramaniam and Ntouyas [14] have studied the controllability of the following neutral stochastic functional differential inclusions with infinite delay in abstract space:where and are, respectively, measurable and multivalued measurable mapping. They achieved controllability for (1) by assuming the semigroup is compact and uniformly bounded and applying the theory of fractional power operators.

The concept of complete controllability is usually too strong which has limited applicability. Compared with complete controllability, approximate controllability is a weaker concept and it is completely adequate in applications; see [15–17]. Recently, Mokkedem and Fu [18] have studied the approximate controllability of the following semilinear neutral integrodifferential systems with finite delay:where and . They studied the problem by applying the theory of fractional power operators and -norm. Meanwhile, they supposed that the operator generates a compact analytic semigroup on and the corresponding linear system of (2) is approximately controllable. They established sufficient conditions of approximate controllability for the neutral integrodifferential control system in Hilbert space under a resolvent condition; in particular, they did not require that the resolvent operator is compact for .

In fact, neutral stochastic integrodifferential inclusions arise in many areas for applied mathematics. The approximate controllability for neutral stochastic integrodifferential inclusions is also an important and interesting topic for mathematics, but it is also a difficult problem. Meanwhile, few researches have been done on the approximate controllability for semilinear neutral stochastic integrodifferential inclusions with infinite delay. Motivated by [14, 18], we will show the approximate controllability of the following semilinear neutral stochastic integrodifferential inclusions with infinite delay in a Hilbert space:where is the infinitesimal generator of an analytic semigroup of bound linear operator , on a separable Hilbert space with inner product and norm . The control function takes values in of admissible control functions for a separable Hilbert space and is a bounded linear operator from into . is a family of closed linear operators to be specified later. Let be another separable Hilbert space with inner product and norm Suppose is a given -valued Wiener process in a finite trace nuclear covariance operator . and are, respectively, measurable and multivalued measurable mapping, where is the family of all nonempty subsets of and denotes the space of all -Hilbert-Schmidt operators from into . The histories , belong to an abstract phase space .

The aim of the present work is to investigate the approximate controllability for (3) by using the resolvent operator theory. Because the nonlinear terms involve frequently special derivatives in many practical models and the history variables of the functions and are only defined on , we cannot discuss the problem on the whole space . We also suppose that there exists a semigroup which generated by is compact analytic on so that the resolvent is also analytic. So, we restrict this integrodifferential inclusion in the and demonstrate the existence of mild solutions by applying and the fractional power theory and then prove the approximate controllability for (3) in space . In particular, we do not require that the resolvent operator be compact for which differs greatly from [14].

The whole paper is arranged as follows: in Section 2, we introduce some concepts, hypotheses, and basic results about resolvent operator and approximate controllability. Section 3 is devoted to studying the existence of mild solution of (3) and proving the approximate controllability. The application of our theoretical results is given in Section 4.

#### 2. Preliminaries

The complete probability space, denoted by , is furnished with a complete family of right continuous increasing sub--algebras satisfying . Let the complete orthonormal basis in be a bounded sequence of nonnegative real numbers such that and , where are independent one-dimensional standard Wiener processes. We assume that is the sub--algebra generated by and . Let be the space of all bounded linear operators from into with the usual operator norm . For defineThen is called a -Hilbert-Schmidt operator. Let be the resolvent set of the operator and ; then the fractional power operator , , is a closed linear operator on in and the expressiondefines a norm on . Denoting the space by , then is a Banach space for each . For , the imbedding is compact. Meanwhile, , the resolvent operator of , is compact. Let be the space of all bounded linear operators from into with the norm and . Let be the Banach space of continuous functions from to with the normand the Banach space of continuous functions from to is defined by with the norm

Keep it simple; let , for any . Let be a -measurable function from into endowed with a seminorm ; we assume that satisfies the following conditions:(a)If: , , is continuous on and in , then for every the following conditions hold:(1) is in ;(2);(3), where is a positive constant and and are continuous and independent of .(b)For the function in (a), is a -valued function on .(c)The space is complete.

We define the set of all elements in which takes values in space by . Since is still a Banach space, we will assume that the subspace also satisfies the following conditions: ()If: , , is continuous on and in , then for every the following conditions hold: (1) is in ; (2) ; (3) , where is a positive constant and and are continuous and independent of . ()For the function in , is a -valued function on . ()The space is complete.

Let be the collection of all strongly measurable, square-integrable -valued stochastic variables with the norm . The setting is the collection of all -valued stochastic process and let be the collection of all strongly measurable, square-integrable -valued stochastic variables with the norm .

The space consists of measurable and -adapted processes such that and the restriction is continuous. The closed subspace of all continuous process that belongs to is defined by and the seminorm in is defined bywhere, and . It is not hard to prove that is a Banach space.

Now, we introduce some facts on multivalues analysis:(i)If, for each , the set is a nonempty, closed subset of and if, for each open set of containing , there exists an open neighborhood of such that , then is upper semicontinuous (u.s.c.) on .(ii)If, for every bounded subset , is relatively compact, then is completely continuous.(iii)If the multivalued map is completely continuous with nonempty compact values, then is upper semicontinuous (u.s.c.) if and only if has a closed graph (i.e., , imply ).

For each , define the set of selections of by

*Definition 1 (see [19]). *A family of bounded linear operators for is called resolvent operator forif(i) and for some , ;(ii)for all , is strongly continuous in on ;(iii), for , where is the Banach space from endowed with the graph norm. Moreover for and for , the following formula holds:See from [18–20] that the resolvent operator for the above linear system exists which is given by , andwhere is contour of the type used to obtain an analytic semigroup. is also analytic and there exists , , such that

Lemma 2 (see [21, Lemma 2.3]). * is continuous for in the uniform operator topology of .*

Lemma 3 (see [22, Lemma 2.3]). * is continuous for in the uniform operator topology of .*

*Definition 4. *The multivalued map is said to be -*Carathéodory* if(i) is measurable for each ;(ii) is u.s.c. for almost all ;(iii)for each , there exists such that

Lemma 5 (see [23]). *Let be a compact interval and a Hilbert space. Let be an -Carathéodory multivalued map with and let be a linear continuous mapping from to . Then the operatoris a closed graph operator in . Here denotes the family of nonempty, bound, close, compact convex subset of .*

*Definition 6. *A -adapted stochastic process is the mild solution of (3) if is a continuous stochastic process on , the function is integrable for each , and is a selection of such that

Theorem 7 (nonlinear alternative for Kakutani maps). *Let be a Hilbert space, a closed convex subset of , an open subset of , and . Suppose that is an upper semicontinuous compact map; here denotes the family of nonempty, compact convex subsets of . Then either* (i)* has a fixed point in , or* (ii)*there are and with .*

*Definition 8. *Equation (3) is said to be approximately controllable on the interval if is dense in ; that is,where .

#### 3. Approximate Controllability

We introduce the following operators.(1)The controllability operator Grammian is as follows:(2)The resolvent operator is as follows:where and are the adjoints of the operators and , respectively. We also assume that the operator satisfies the following hypothesis:

In this section, we discuss the approximate controllability of (3). Firstly, we compare approximate controllability of the semilinear equation (3) with approximate controllability of the associated linear system. Hypothesis () is equivalent to the fact that the linear control system corresponding to (3)is approximately controllable on . For further details for the linear part of semilinear system, we can look up [24]. Secondly, we demonstrate that, for any given and , we choose a proper control to prove that there exists a mild solution of (3). At last, we prove that in which implies that (3) is approximately controllable on . Assume is fixed: () is a bounded linear operator from to and . ()For each , there exists a constant with , such that . And there exists a number , such that ()The function is completely continuous. Moreover, there exists a positive, nondecreasing function such that () is an -*Carathéodory* function. ()There exists a continuous nondecreasing function such that

For any , let and , and we take the control function , simply denoted by , as follows:

We define the operator on by using this control as follows:

Theorem 9. *Let . If the hypotheses are satisfied, then, for each , the operator has a fixed point in provided that there exists a constant so thatwhere*

*Proof. *Let be the space of all function , such that . The restriction is continuous and is a seminorm in which is defined byLet . We define the multivalued map by and the set of such thatWe will prove that the operator has a fixed point, which then is a solution of (3). For , let be the function defined bySet . It is clear that satisfies (26) if and only if satisfies andLet . For any , we haveSo, if , then is a Banach space. For any , setthen is uniformly bounded, and we haveFor , we can getLet the operator : be defined by , the set of , such thatWe divide the proof into five steps.*Step 1*. is convex for each .

If and belong to , then there exists , such thatLet , because the operators , , are linear, ; we haveSince has convex values, is convex. Then .*Step 2. * is bounded.

The proof can be found in Appendix A.*Step 3.* We prove that is equicontinuous.

The proof can be found in Appendix B.

As a consequence of Steps and , we can find that : is compact multivalued map.*Step 4. * has a closed graph.

The proof can be found in Appendix C.

We will prove that has a fixed point.*Step 5.* We will prove that there exists an open set and , such that , for .

Let for ; then there exists such thatby hypotheses , for each , we haveThen, we define the function , . We haveIf , then and the previous inequality holds. Consequently,Then, according to the hypotheses, there exists such that and set . It is clear that there is no for such that . So we deduce that has a fixed point by Theorem 7 and (3) has a mild solution.

In the end, we prove that (3) is approximately controllable on .

Theorem 10. *Assume that the hypotheses ()–() are satisfied, and the functions and are uniformly bounded in the space ; then (3) is approximately controllable on .*

*Proof. *Let , and is a mild solution of (3) which is got from Theorem 9. From the definition of , satisfieswhereOr we obtain thatThen, we have