On Approximate Controllability of Second-Order Neutral Partial Stochastic Functional Integrodifferential Inclusions with Infinite Delay and Impulsive Effects

Abstract

We discuss the approximate controllability of second-order impulsive neutral partial stochastic functional integrodifferential inclusions with infinite delay under the assumptions that the corresponding linear system is approximately controllable. Using the fixed point strategy, stochastic analysis, and properties of the cosine family of bounded linear operators combined with approximation techniques, a new set of sufficient conditions for approximate controllability of the second-order impulsive partial stochastic integrodifferential systems are formulated and proved. The results in this paper are generalization and continuation of the recent results on this issue. An example is provided to show the application of our result.

Academic Editor:Mark A. McKibben

Department of Mathematics, Hexi University, Zhangye, Gansu 734000, China

Received 20 January 2015; Revised 16 April 2015; Accepted 19 April 2015; 26 August 2015

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.

1. Introduction

Impulsive effects exist widely in many evolution processes in which states are changed abruptly at certain moments of time, involving fields such as physics, chemical technology, population dynamics, biotechnology, and economics; see [1-4] and the references therein. However, in addition to impulsive effects, stochastic effects likewise exist in real systems. A lot of dynamical systems have variable structures subject to stochastic abrupt changes, which may result from abrupt phenomena such as stochastic failures and repairs of the components, changes in the interconnections of subsystems, sudden environment changes, and other areas of science. Therefore, it is necessary and important to consider the impulsive stochastic dynamical systems. Particularly, the authors in [5-7] studied the existence of mild solutions for a class of abstract impulsive neutral stochastic functional differential and integrodifferential equations with infinite delay in Hilbert spaces.

The concept of controllability leads to some very important conclusions regarding the behavior of linear and nonlinear dynamical systems. In the case of infinite-dimensional systems, two basic concepts of controllability can be distinguished. There are exact and approximate controllability. However, the concept of exact controllability is usually too strong [8]. Therefore, approximate controllability problems for deterministic and stochastic dynamical systems in infinite dimensional spaces are well developed using different kind of approaches (see [9, 10]). Stochastic control theory is a stochastic generalization of classic control theory. So significant progress has been made in the approximate controllability of linear and nonlinear stochastic systems in Banach spaces (see, e.g., [9-12]). Several papers [13-16] have appeared on the approximate controllability of nonlinear impulsive stochastic differential systems in Hilbert spaces.

In many cases, it is advantageous to treat the second-order stochastic differential equations directly rather than to convert them to first-order systems. The second-order stochastic differential equations are the right model in continuous time to account for integrated processes that can be made stationary. Recently, based on the fixed point theory, the existence and approximate controllability of mild solutions for various second-order stochastic partial differential equations and impulsive stochastic partial differential equations have been extensively studied. For example, Ren and Sun [17], Cui and Yan [18], and Mahmudov and McKibben [19] proved the approximate controllability of second-order neutral stochastic evolution differential equations. Muthukumar and Balasubramaniam [20] established sufficient conditions for the approximate controllability of a class of second-order nonlinear stochastic functional differential equations of McKean-Vlasov type. Balasubramaniam and Muthukumar in [21] discussed the approximate controllability of second-order neutral stochastic distributed implicit functional differential equations with infinite delay. Sakthivel et al. in [22] studied the approximate controllability of second-order impulsive stochastic differential equations. On the other hand, many systems arising from realistic models can be described as partial stochastic differential or integrodifferential inclusions (see [23-27] and references therein), so it is natural to extend the concept controllability of mild solution for second-order impulsive stochastic evolution equations to second-order impulsive systems represented by stochastic partial differential or integrodifferential inclusions. In this paper, we consider the approximate controllability of the following second-order impulsive neutral partial stochastic functional integrodifferential inclusions with infinite delay in Hilbert spaces of the form [figure omitted; refer to PDF] where the state [figure omitted; refer to PDF] takes values in a separable real Hilbert space [figure omitted; refer to PDF] with inner product [figure omitted; refer to PDF] and norm [figure omitted; refer to PDF] The operator [figure omitted; refer to PDF] is the infinitesimal generator of a strongly continuous cosine family on [figure omitted; refer to PDF] . The control function [figure omitted; refer to PDF] , a Hilbert space of admissible control functions, [figure omitted; refer to PDF] is an integer, and [figure omitted; refer to PDF] is a bounded linear operator from a Banach space [figure omitted; refer to PDF] to [figure omitted; refer to PDF] Let [figure omitted; refer to PDF] be another separable Hilbert space with inner product [figure omitted; refer to PDF] and norm [figure omitted; refer to PDF] Suppose that [figure omitted; refer to PDF] is a given [figure omitted; refer to PDF] -valued Wiener process with a covariance operator [figure omitted; refer to PDF] defined on a complete probability space [figure omitted; refer to PDF] equipped with a normal filtration [figure omitted; refer to PDF] , which is generated by the Wiener process [figure omitted; refer to PDF] . The time history [figure omitted; refer to PDF] given by [figure omitted; refer to PDF] belongs to some abstract phase space [figure omitted; refer to PDF] defined axiomatically; [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] are given functions to be specified later. Moreover, let [figure omitted; refer to PDF] be prefixed points and the symbol [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] represent the right and left limits of [figure omitted; refer to PDF] at [figure omitted; refer to PDF] , respectively. The initial data [figure omitted; refer to PDF] is an [figure omitted; refer to PDF] -adapted, [figure omitted; refer to PDF] -valued random variable independent of the Wiener process [figure omitted; refer to PDF] with finite second moment.

To the best of the author's knowledge, there are no results about the existence and approximate controllability of mild solutions for second-order impulsive second-order neutral partial stochastic functional integrodifferential inclusions with infinite delay, which is expressed in the form of (1). In order to fill this gap, this paper studies this interesting problem. We derive the sufficient conditions for the approximate controllability of system (1) by using the fixed point theorem for multivalued mapping due to Dhage [28] with stochastic analysis and properties of the cosine family of bounded linear operators combined with approximation techniques. The obtained result can be seen as a contribution to this emerging field. Moreover, the operators [figure omitted; refer to PDF] , [figure omitted; refer to PDF] are continuous but without imposing completely continuous and Lipschitz condition. The results shown are also new for deterministic second-order systems with impulsive effects.

The rest of this paper is organized as follows. In Section 2, we introduce some notations and necessary preliminaries. Section 3 verifies the existence of solutions for impulsive stochastic control system (1). In Section 4 we establish the approximate controllability of impulsive stochastic control system (1). Finally in Section 5, an example is given to illustrate our results.

2. Preliminaries

Let [figure omitted; refer to PDF] be a complete probability space equipped with a normal filtration [figure omitted; refer to PDF] Let [figure omitted; refer to PDF] and [figure omitted; refer to PDF] be the separable Hilbert spaces and let [figure omitted; refer to PDF] be a [figure omitted; refer to PDF] -Weiner process on [figure omitted; refer to PDF] with the covariance operator [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] We assume that there exists a complete orthonormal system [figure omitted; refer to PDF] in [figure omitted; refer to PDF] , a bounded sequence of nonnegative real numbers [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] , and a sequence [figure omitted; refer to PDF] of independent Brownian motions such that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is the [figure omitted; refer to PDF] -algebra generated by [figure omitted; refer to PDF] Let [figure omitted; refer to PDF] be the space of all Hilbert-Schmidt operators from [figure omitted; refer to PDF] to [figure omitted; refer to PDF] with the inner product [figure omitted; refer to PDF] Let [figure omitted; refer to PDF] be the Banach space of all [figure omitted; refer to PDF] -measurable [figure omitted; refer to PDF] th power integrable random variables with values in the Hilbert space [figure omitted; refer to PDF] . Let [figure omitted; refer to PDF] be the Banach space of continuous maps from [figure omitted; refer to PDF] into [figure omitted; refer to PDF] satisfying the condition [figure omitted; refer to PDF]

We use the notations that [figure omitted; refer to PDF] is the family of all subsets of [figure omitted; refer to PDF] Let us introduce the following notations: [figure omitted; refer to PDF] Consider [figure omitted; refer to PDF] given by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] Then [figure omitted; refer to PDF] is a metric space and [figure omitted; refer to PDF] is a generalized metric space.

In what follows, we briefly introduce some facts on multivalued analysis. For more details, one can see [29, 30].

A multivalued map [figure omitted; refer to PDF] is convex (closed) valued if [figure omitted; refer to PDF] is convex (closed) for all [figure omitted; refer to PDF] . [figure omitted; refer to PDF] is bounded on bounded sets if [figure omitted; refer to PDF] is bounded in [figure omitted; refer to PDF] for any bounded set [figure omitted; refer to PDF] of [figure omitted; refer to PDF] ; that is, [figure omitted; refer to PDF] .

[figure omitted; refer to PDF] is called upper semicontinuous (u.s.c., in short) on [figure omitted; refer to PDF] , if, for any [figure omitted; refer to PDF] , the set [figure omitted; refer to PDF] is a nonempty, closed subset of [figure omitted; refer to PDF] and if, for each open set [figure omitted; refer to PDF] of [figure omitted; refer to PDF] containing [figure omitted; refer to PDF] , there exists an open neighborhood [figure omitted; refer to PDF] of [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] .

[figure omitted; refer to PDF] is said to be completely continuous if [figure omitted; refer to PDF] is relatively compact for every bounded subset [figure omitted; refer to PDF] of [figure omitted; refer to PDF] If the multivalued map [figure omitted; refer to PDF] is completely continuous with nonempty compact values, then [figure omitted; refer to PDF] is u.s.c. if and only if [figure omitted; refer to PDF] has a closed graph; that is, [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] imply [figure omitted; refer to PDF] .

[figure omitted; refer to PDF] is said to be completely continuous if [figure omitted; refer to PDF] is relatively compact, for every bounded subset [figure omitted; refer to PDF] .

A multivalued map [figure omitted; refer to PDF] is said to be measurable if, for each [figure omitted; refer to PDF] , the function [figure omitted; refer to PDF] defined by [figure omitted; refer to PDF] is measurable.

In this paper, [figure omitted; refer to PDF] is the infinitesimal generator of a strongly continuous cosine function of bounded linear operators [figure omitted; refer to PDF] on [figure omitted; refer to PDF] . The corresponding strongly continuous sine family [figure omitted; refer to PDF] is defined by [figure omitted; refer to PDF] , [figure omitted; refer to PDF] The generator [figure omitted; refer to PDF] of [figure omitted; refer to PDF] is given by [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] It is well known that the infinitesimal generator [figure omitted; refer to PDF] is a closed densely defined operator on [figure omitted; refer to PDF] As usual we denote by [figure omitted; refer to PDF] the domain of operator [figure omitted; refer to PDF] endowed with the graph norm [figure omitted; refer to PDF] Moreover, the notation [figure omitted; refer to PDF] stands for the space formed by the vectors [figure omitted; refer to PDF] for which [figure omitted; refer to PDF] is of class [figure omitted; refer to PDF] on [figure omitted; refer to PDF] It was proved by Kisynski [31] that [figure omitted; refer to PDF] endowed with the norm [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , is a Banach space. Such cosine and corresponding sine families and their generators satisfy that the following properties.

Lemma 1 (see [32]).

Suppose that [figure omitted; refer to PDF] is the infinitesimal generator of a cosine family of operators [figure omitted; refer to PDF] . Then, the following hold:

(a) There exist [figure omitted; refer to PDF] and [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] and hence [figure omitted; refer to PDF]

(b) Consider [figure omitted; refer to PDF] for all [figure omitted; refer to PDF]

(c) There exists [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] for all [figure omitted; refer to PDF]

The existence of solutions of the second-order linear abstract Cauchy problem [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is an integrable function, has been discussed in [33]. Similarly, the existence of solutions for semilinear second-order abstract Cauchy problem has been treated in [32]. We only mention here that the function [figure omitted; refer to PDF] given by [figure omitted; refer to PDF] is called a mild solution of (5) and if [figure omitted; refer to PDF] , the function [figure omitted; refer to PDF] is continuously differentiable and [figure omitted; refer to PDF]

A function [figure omitted; refer to PDF] is said to be normalized piecewise continuous function on [figure omitted; refer to PDF] if [figure omitted; refer to PDF] is piecewise continuous and left continuous on [figure omitted; refer to PDF] . We denote by [figure omitted; refer to PDF] the space formed by the normalized piecewise continuous, [figure omitted; refer to PDF] -adapted measurable processes from [figure omitted; refer to PDF] into [figure omitted; refer to PDF] . In particular, we introduce the space [figure omitted; refer to PDF] formed by all [figure omitted; refer to PDF] -adapted measurable, [figure omitted; refer to PDF] -valued stochastic processes [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] is continuous at [figure omitted; refer to PDF] and [figure omitted; refer to PDF] exists for [figure omitted; refer to PDF] Similarly, [figure omitted; refer to PDF] formed by all [figure omitted; refer to PDF] -adapted measurable, [figure omitted; refer to PDF] -valued stochastic processes [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] is continuous at [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] exists for [figure omitted; refer to PDF] In this paper, we always assume that [figure omitted; refer to PDF] is endowed with the norm [figure omitted; refer to PDF] Then [figure omitted; refer to PDF] is a Banach space. Next, for [figure omitted; refer to PDF] , we represent by [figure omitted; refer to PDF] the right derivative at [figure omitted; refer to PDF] and by [figure omitted; refer to PDF] the right derivative at zero. It is easy to see that [figure omitted; refer to PDF] is provided with the norm [figure omitted; refer to PDF] being a Banach space.

In this paper, we assume that the phase space [figure omitted; refer to PDF] is a seminormed linear space of [figure omitted; refer to PDF] -measurable functions mapping [figure omitted; refer to PDF] into [figure omitted; refer to PDF] and satisfying the following fundamental axioms due to Hale and Kato (see, e.g., [34]).

(A) If [figure omitted; refer to PDF] , such that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , then for every [figure omitted; refer to PDF] the following conditions hold:

(i) [figure omitted; refer to PDF] is in [figure omitted; refer to PDF] ;

(ii) [figure omitted; refer to PDF] ;

(iii): [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is a constant; [figure omitted; refer to PDF] is continuous and [figure omitted; refer to PDF] is locally bounded, and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are independent of [figure omitted; refer to PDF]

(B) For the function [figure omitted; refer to PDF] in (A), the function [figure omitted; refer to PDF] is continuous from [figure omitted; refer to PDF] into [figure omitted; refer to PDF]

(C) The space [figure omitted; refer to PDF] is complete.

Example 2.

The phase space [figure omitted; refer to PDF] . Let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] and let [figure omitted; refer to PDF] be a nonnegative measurable function which satisfies conditions (h-5) and (h-6) in the terminology of Hino et al. [35]. Briefly, this means that [figure omitted; refer to PDF] is locally integrable and there is a nonnegative, locally bounded function [figure omitted; refer to PDF] on [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is a set whose Lebesgue measure is zero. We denote by [figure omitted; refer to PDF] the set consisting of all classes of functions [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] is Lebesgue measurable on [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] is Lebesgue integrable on [figure omitted; refer to PDF] The seminorm is given by [figure omitted; refer to PDF] The space [figure omitted; refer to PDF] satisfies axioms (A)-(C). Moreover, when [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , we can take [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , for [figure omitted; refer to PDF] (see [35, Theorem [figure omitted; refer to PDF] ] for details).

Remark 3 (see [4]).

In retarded functional differential equations without impulses, the axioms of the abstract phase space [figure omitted; refer to PDF] include the continuity of the function [figure omitted; refer to PDF] Due to the impulsive effect, this property is not satisfied in impulsive delay systems and, for this reason, has been eliminated in our abstract description of [figure omitted; refer to PDF]

Remark 4.

In the rest of this paper [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are the constants defined by [figure omitted; refer to PDF] and [figure omitted; refer to PDF]

For [figure omitted; refer to PDF] , we denote by [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , the unique continuous function [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] Moreover, for [figure omitted; refer to PDF] we denote by [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , the set [figure omitted; refer to PDF] The notation [figure omitted; refer to PDF] stands for the closed ball with center at [figure omitted; refer to PDF] and radius [figure omitted; refer to PDF] in [figure omitted; refer to PDF]

Lemma 5.

A set [figure omitted; refer to PDF] is relatively compact in [figure omitted; refer to PDF] if and only if the set [figure omitted; refer to PDF] is relatively compact in [figure omitted; refer to PDF] , for every [figure omitted; refer to PDF]

Furthermore, we need the following result.

Lemma 6 (see [36]).

Let [figure omitted; refer to PDF] be an integrable function such that [figure omitted; refer to PDF] Then the function [figure omitted; refer to PDF] belongs to [figure omitted; refer to PDF] , the function [figure omitted; refer to PDF] is integrable on [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

Let [figure omitted; refer to PDF] be the state value of system (1) at terminal time [figure omitted; refer to PDF] corresponding to the control [figure omitted; refer to PDF] and the initial value [figure omitted; refer to PDF] Introduce the set [figure omitted; refer to PDF] which is called the reachable set of system (1) at terminal time [figure omitted; refer to PDF] , and its closure in [figure omitted; refer to PDF] is denoted by [figure omitted; refer to PDF]

Now we give the definitions of mild solutions and approximate controllability for system (1).

Definition 7.

An [figure omitted; refer to PDF] -adapted stochastic process [figure omitted; refer to PDF] is called a mild solution of system (1) if [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and the impulsive conditions [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , are satisfied and

(i) [figure omitted; refer to PDF] is adapted to [figure omitted; refer to PDF] .

(ii) [figure omitted; refer to PDF] has cadlag paths on [figure omitted; refer to PDF] a.s. and, for each [figure omitted; refer to PDF] , [figure omitted; refer to PDF] satisfies the integral equation [figure omitted; refer to PDF]

where [figure omitted; refer to PDF]

Definition 8.

System (1) is said to be approximately controllable on the interval [figure omitted; refer to PDF] if [figure omitted; refer to PDF]

It is convenient at this point to define operators [figure omitted; refer to PDF] where [figure omitted; refer to PDF] denotes the adjoint of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is the adjoint of [figure omitted; refer to PDF] It is straightforward that the operator [figure omitted; refer to PDF] is a linear bounded operator:

(S1) [figure omitted; refer to PDF] , as [figure omitted; refer to PDF] in the strong operator topology.

Lemma 9.

Assumption (S1) holds if and only if [figure omitted; refer to PDF] , as [figure omitted; refer to PDF] in the strong operator topology.

The proof of this lemma is a straightforward adaptation of the proof of [9, Theorem [figure omitted; refer to PDF] ].

Lemma 10 (see [10]).

For any [figure omitted; refer to PDF] there exists [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] .

Now for any [figure omitted; refer to PDF] and [figure omitted; refer to PDF] we define the control function [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

The next result is a consequence of the phase space axioms.

Lemma 11.

Let [figure omitted; refer to PDF] be an [figure omitted; refer to PDF] -adapted measurable process such that [figure omitted; refer to PDF] -adapted process [figure omitted; refer to PDF] and [figure omitted; refer to PDF] ; then [figure omitted; refer to PDF]

Lemma 12 (see [37]).

For any [figure omitted; refer to PDF] and for arbitrary [figure omitted; refer to PDF] -valued predictable process [figure omitted; refer to PDF] , [figure omitted; refer to PDF]

The consideration of this paper is based on the following fixed point theorem due to Dhage [28].

Lemma 13.

Let [figure omitted; refer to PDF] be a Hilbert space, and let [figure omitted; refer to PDF] and [figure omitted; refer to PDF] be two multivalued operators satisfying that

(a) [figure omitted; refer to PDF] is a contraction

(b) [figure omitted; refer to PDF] is completely continuous.

Then either

(i) the operator inclusion [figure omitted; refer to PDF] has a solution or

(ii) the set [figure omitted; refer to PDF] is unbounded for [figure omitted; refer to PDF]

3. Existence of Solutions for Impulsive Stochastic Control System

In this section, we prove the existence of solutions for impulsive stochastic control system (1). We make the following hypotheses:

(H1) [figure omitted; refer to PDF] is the infinitesimal generator of a strongly continuous cosine family [figure omitted; refer to PDF] on [figure omitted; refer to PDF] and the corresponding sine family [figure omitted; refer to PDF] satisfies the conditions [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] for some constants [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF]

(H2) [figure omitted; refer to PDF] , is compact.

(H3) The function [figure omitted; refer to PDF] is continuous and there exist [figure omitted; refer to PDF] such that [figure omitted; refer to PDF]

: for all [figure omitted; refer to PDF] , and [figure omitted; refer to PDF]

: with [figure omitted; refer to PDF]

(H4) The function [figure omitted; refer to PDF] satisfies the following conditions:

(i) For each [figure omitted; refer to PDF] the function [figure omitted; refer to PDF] is continuous and, for each [figure omitted; refer to PDF] , the function [figure omitted; refer to PDF] is strongly measurable.

(ii) There exists a continuous function [figure omitted; refer to PDF] , such that [figure omitted; refer to PDF]

: for a.e. [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is a continuous nondecreasing function.

(H5) The multivalued map [figure omitted; refer to PDF] satisfies the following conditions:

(i) For each [figure omitted; refer to PDF] , the function [figure omitted; refer to PDF] is u.s.c. and, for each [figure omitted; refer to PDF] , the function [figure omitted; refer to PDF] is measurable and the set [figure omitted; refer to PDF]

: is nonempty.

(ii) There exist a continuous function [figure omitted; refer to PDF] and a continuous nondecreasing function [figure omitted; refer to PDF] such that [figure omitted; refer to PDF]

: a.e. [figure omitted; refer to PDF] , with [figure omitted; refer to PDF]

: where [figure omitted; refer to PDF]

(H6) The functions [figure omitted; refer to PDF] are continuous and there are constants [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] such that [figure omitted; refer to PDF]

: for every [figure omitted; refer to PDF]

Lemma 14 (see [38]).

Let [figure omitted; refer to PDF] be a compact interval and let [figure omitted; refer to PDF] be a Hilbert space. Let [figure omitted; refer to PDF] be a multivalued map satisfying (H5) (i) and let [figure omitted; refer to PDF] be a linear continuous operator from [figure omitted; refer to PDF] to [figure omitted; refer to PDF] Then the operator [figure omitted; refer to PDF] is a closed graph in [figure omitted; refer to PDF]

Remark 15.

In what follows, we set [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF]

Theorem 16.

If assumptions (H1)-(H6) are satisfied, further, suppose that, for all [figure omitted; refer to PDF] , system (1) has at least one mild solution on [figure omitted; refer to PDF] , provided that [figure omitted; refer to PDF]

Proof.

Let [figure omitted; refer to PDF] endowed with the norm of [figure omitted; refer to PDF] Thus [figure omitted; refer to PDF] is Banach space. Now we can define the multivalued map [figure omitted; refer to PDF] by [figure omitted; refer to PDF] the set of [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] is such that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] on [figure omitted; refer to PDF] In what follows, we aim to show that the operator [figure omitted; refer to PDF] has a fixed point, which is a solution of problem (1).

Let [figure omitted; refer to PDF] be a decreasing sequence in [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] To prove the above theorem, we consider the following problem: [figure omitted; refer to PDF] We will show that the problem has at least one mild solution [figure omitted; refer to PDF]

For fixed [figure omitted; refer to PDF] , set the multivalued map [figure omitted; refer to PDF] by [figure omitted; refer to PDF] the set of [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] It is easy to see that the fixed point of [figure omitted; refer to PDF] is a mild solution of the Cauchy problem (27).

Let [figure omitted; refer to PDF] be the extension of [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] on [figure omitted; refer to PDF] Now, we consider the following multivalued operators [figure omitted; refer to PDF] and [figure omitted; refer to PDF] defined by [figure omitted; refer to PDF] It is clear that [figure omitted; refer to PDF] The problem of finding mild solutions of (27) is reduced to find the solutions of the operator inclusion [figure omitted; refer to PDF] In what follows, we show that operators [figure omitted; refer to PDF] and [figure omitted; refer to PDF] satisfy the conditions of Lemma 13.

Step 1. [figure omitted; refer to PDF] is a contraction on [figure omitted; refer to PDF]

Let [figure omitted; refer to PDF] and [figure omitted; refer to PDF] From (H3) and Lemma 11, we have [figure omitted; refer to PDF] By Lemma 6, we have [figure omitted; refer to PDF] Similarly, for any [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] Taking supremum over [figure omitted; refer to PDF] , it follows that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Hence, [figure omitted; refer to PDF] is a contraction on [figure omitted; refer to PDF] .

Step 2. [figure omitted; refer to PDF] has compact, convex values and it is completely continuous.

[figure omitted; refer to PDF] is convex for each [figure omitted; refer to PDF] .

In fact, if [figure omitted; refer to PDF] , [figure omitted; refer to PDF] belong to [figure omitted; refer to PDF] , then there exist [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] Let [figure omitted; refer to PDF] For each [figure omitted; refer to PDF] we have [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] is convex (because [figure omitted; refer to PDF] has convex values) we have [figure omitted; refer to PDF] .

[figure omitted; refer to PDF] maps bounded sets into bounded sets in [figure omitted; refer to PDF]

Indeed, it is enough to show that there exists a positive constant [figure omitted; refer to PDF] such that, for each [figure omitted; refer to PDF] , one has [figure omitted; refer to PDF] . If [figure omitted; refer to PDF] , then there exists [figure omitted; refer to PDF] such that, for each [figure omitted; refer to PDF] , [figure omitted; refer to PDF] However, on the other hand, from the condition (H6), we conclude that there exist positive constants [figure omitted; refer to PDF] such that, for all [figure omitted; refer to PDF] , [figure omitted; refer to PDF] Let [figure omitted; refer to PDF] Therefore [figure omitted; refer to PDF] If [figure omitted; refer to PDF] , from Lemma 11, it follows that [figure omitted; refer to PDF] By (H1)-(H5) and (40)-(41), from (37) we have for [figure omitted; refer to PDF] [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] By Lemma 6, we have [figure omitted; refer to PDF] Similarly, for any [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] Take [figure omitted; refer to PDF] Then for each [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] .

[figure omitted; refer to PDF] is a compact multivalued map.

To this end, we decompose [figure omitted; refer to PDF] by [figure omitted; refer to PDF] , where the map [figure omitted; refer to PDF] is defined by [figure omitted; refer to PDF] and the set [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] and the map [figure omitted; refer to PDF] is defined by [figure omitted; refer to PDF] , and the set [figure omitted; refer to PDF] such that [figure omitted; refer to PDF]

First, [figure omitted; refer to PDF] is a compact multivalued map. We begin by showing that [figure omitted; refer to PDF] is equicontinuous. Let [figure omitted; refer to PDF] Then, we have, for each [figure omitted; refer to PDF] , [figure omitted; refer to PDF] where [figure omitted; refer to PDF] The fact of the compactness of [figure omitted; refer to PDF] for [figure omitted; refer to PDF] implies the continuity in the uniform operator topology. So as [figure omitted; refer to PDF] , with [figure omitted; refer to PDF] being sufficiently small, the right-hand side of the above inequality is independent of [figure omitted; refer to PDF] and tends to zero. The equicontinuities for the cases [figure omitted; refer to PDF] or [figure omitted; refer to PDF] are very simple. Thus the set [figure omitted; refer to PDF] is equicontinuous.

We now prove that [figure omitted; refer to PDF] is relatively compact for every [figure omitted; refer to PDF] Let [figure omitted; refer to PDF] be fixed and let [figure omitted; refer to PDF] be a real number satisfying [figure omitted; refer to PDF] For [figure omitted; refer to PDF] , we define [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Using the compactness of [figure omitted; refer to PDF] for [figure omitted; refer to PDF] , we deduce that the set [figure omitted; refer to PDF] is relatively compact in [figure omitted; refer to PDF] for every [figure omitted; refer to PDF] , [figure omitted; refer to PDF] Moreover, for every [figure omitted; refer to PDF] we have [figure omitted; refer to PDF] There are relatively compact sets arbitrarily close to the set [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] is a relatively compact in [figure omitted; refer to PDF] Hence, the Arzela-Ascoli theorem shows that [figure omitted; refer to PDF] is a compact multivalued map.

Secondly, [figure omitted; refer to PDF] is a compact multivalued map. We begin by showing that [figure omitted; refer to PDF] is equicontinuous. For each [figure omitted; refer to PDF] , [figure omitted; refer to PDF] is fixed, [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , such that [figure omitted; refer to PDF] Next, for [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , we have, using the property of compact operator, [figure omitted; refer to PDF] As [figure omitted; refer to PDF] , the right-hand side of the above inequality tends to zero independently of [figure omitted; refer to PDF] due to the sets [figure omitted; refer to PDF] which are relatively compact in [figure omitted; refer to PDF] and the strong continuity of [figure omitted; refer to PDF] So [figure omitted; refer to PDF] , are equicontinuous.

Now we prove that [figure omitted; refer to PDF] , is relatively compact for every [figure omitted; refer to PDF]

From the following relations [figure omitted; refer to PDF] we conclude that [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , is relatively compact for every [figure omitted; refer to PDF] By Lemma 5, we infer that [figure omitted; refer to PDF] is relatively compact. Moreover, using the continuity of the operators [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , for all [figure omitted; refer to PDF] , we conclude that operator [figure omitted; refer to PDF] is also a compact multivalued map.

[figure omitted; refer to PDF] has a closed graph.

Let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] From axiom (A), it is easy to see that [figure omitted; refer to PDF] uniformly for [figure omitted; refer to PDF] as [figure omitted; refer to PDF] We prove that [figure omitted; refer to PDF] Now [figure omitted; refer to PDF] means that there exists [figure omitted; refer to PDF] such that, for each [figure omitted; refer to PDF] , [figure omitted; refer to PDF] where [figure omitted; refer to PDF] We must prove that there exists [figure omitted; refer to PDF] such that, for each [figure omitted; refer to PDF] , [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , are continuous, we obtain that [figure omitted; refer to PDF] Consider the linear continuous operator [figure omitted; refer to PDF] , [figure omitted; refer to PDF] From Lemma 14, it follows that [figure omitted; refer to PDF] is a closed graph operator. Also, from the definition of [figure omitted; refer to PDF] , we have that, for every [figure omitted; refer to PDF] , [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] , for some [figure omitted; refer to PDF] it follows that, for every [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] Therefore, [figure omitted; refer to PDF] is a completely continuous multivalued map, u.s.c. with convex closed, compact values.

Step 3. We will show that the set [figure omitted; refer to PDF] is bounded on [figure omitted; refer to PDF] .

Let [figure omitted; refer to PDF] , and then there exists [figure omitted; refer to PDF] such that we have [figure omitted; refer to PDF] It also follows from Lemma 6 that [figure omitted; refer to PDF] This implies by (H1)-(H5) and (41) that for each [figure omitted; refer to PDF] we have [figure omitted; refer to PDF] Similarly, for any [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] By Lemma 11, it follows that [figure omitted; refer to PDF] Consider the function defined by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] For each [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] , it follows that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] , we obtain [figure omitted; refer to PDF] Denoting by [figure omitted; refer to PDF] the right-hand side of the above inequality, we have [figure omitted; refer to PDF] [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] Let [figure omitted; refer to PDF] ; then [figure omitted; refer to PDF] , and for each [figure omitted; refer to PDF] we have [figure omitted; refer to PDF] where [figure omitted; refer to PDF] This implies that [figure omitted; refer to PDF] This inequality shows that there is a constant [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and hence [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] depends only on [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] and on the functions [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] and [figure omitted; refer to PDF] This indicates that [figure omitted; refer to PDF] is bounded on [figure omitted; refer to PDF] Consequently, by Lemma 13, we deduce that [figure omitted; refer to PDF] has a fixed point [figure omitted; refer to PDF] , which is a mild solution of problem (27). Then, we have [figure omitted; refer to PDF] for [figure omitted; refer to PDF] , and some [figure omitted; refer to PDF]

Next we will show that the set [figure omitted; refer to PDF] is relatively compact in [figure omitted; refer to PDF] . We consider the decomposition [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] for some [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] Step 4. [figure omitted; refer to PDF] is relatively compact in [figure omitted; refer to PDF] .

[figure omitted; refer to PDF] is equicontinuous on [figure omitted; refer to PDF]

For [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , there exists a constant [figure omitted; refer to PDF] such that, for all [figure omitted; refer to PDF] and [figure omitted; refer to PDF] with [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] Using the compact operator property, we can choose [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] By (83) one has [figure omitted; refer to PDF] Therefore, [figure omitted; refer to PDF] is equicontinuous for [figure omitted; refer to PDF] Clearly [figure omitted; refer to PDF] is equicontinuous.

[figure omitted; refer to PDF] is relatively compact in [figure omitted; refer to PDF]

Let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ; there exists [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] for some [figure omitted; refer to PDF] By the compactness of [figure omitted; refer to PDF] , [figure omitted; refer to PDF] for [figure omitted; refer to PDF] , we see that the set [figure omitted; refer to PDF] is relatively compact in [figure omitted; refer to PDF] Combining the above inequality, one has [figure omitted; refer to PDF] which is relatively compact in [figure omitted; refer to PDF] .

Step 5. [figure omitted; refer to PDF] is relatively compact in [figure omitted; refer to PDF] .

[figure omitted; refer to PDF] is equicontinuous on [figure omitted; refer to PDF]

For any [figure omitted; refer to PDF] , [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] , [figure omitted; refer to PDF] are compact operators, we find that the sets [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are relatively compact in [figure omitted; refer to PDF] From the strong continuity of [figure omitted; refer to PDF] , for [figure omitted; refer to PDF] , we can choose [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] when [figure omitted; refer to PDF] . For each [figure omitted; refer to PDF] , [figure omitted; refer to PDF] is fixed and [figure omitted; refer to PDF] , such that [figure omitted; refer to PDF] As [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is sufficiently small, the right-hand side of the above inequality tends to zero independently of [figure omitted; refer to PDF] , so [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , are equicontinuous.

[figure omitted; refer to PDF] is relatively compact in [figure omitted; refer to PDF] .

For [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , we have that there exists [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is a closed ball of radius [figure omitted; refer to PDF] . One has [figure omitted; refer to PDF] , which is relatively compact for every [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] is relatively compact in [figure omitted; refer to PDF] .

These facts imply the relatively compact of [figure omitted; refer to PDF] in [figure omitted; refer to PDF] . Therefore, without loss of generality, we may suppose that [figure omitted; refer to PDF] Obviously, [figure omitted; refer to PDF] ; taking limits in (79) one has [figure omitted; refer to PDF] for [figure omitted; refer to PDF] , and some [figure omitted; refer to PDF] , which implies that [figure omitted; refer to PDF] is a mild solution of the problem (1) and the proof of Theorem 16 is complete.

4. Approximate Controllability of Impulsive Stochastic Control System

In this section, we present our main result on approximate controllability of system (1). To do this, we also need the following assumptions:

(B1) The function [figure omitted; refer to PDF] is continuous and there exists a constant [figure omitted; refer to PDF] such that [figure omitted; refer to PDF]

: for [figure omitted; refer to PDF] .

(B2) There exists a constant [figure omitted; refer to PDF] such that [figure omitted; refer to PDF]

: for [figure omitted; refer to PDF] , where [figure omitted; refer to PDF]

Theorem 17.

Assume that assumptions of Theorem 16 hold and, in addition, hypotheses (S1), (B1), and (B2) are satisfied. Then system (1) is approximately controllable on [figure omitted; refer to PDF] .

Proof.

Let [figure omitted; refer to PDF] be a fixed point of [figure omitted; refer to PDF] in [figure omitted; refer to PDF] . By Theorem 16, any fixed point of [figure omitted; refer to PDF] is a mild solution of system (1). This means that there is [figure omitted; refer to PDF] ; that is, there is [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and by using the stochastic Fubini theorem, it is easy to see that [figure omitted; refer to PDF] By conditions (B1) and (B2), we get that the sequences [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are uniformly bounded on [figure omitted; refer to PDF] Thus there are subsequences, still denoted by [figure omitted; refer to PDF] and [figure omitted; refer to PDF] that converge weakly to, say, [figure omitted; refer to PDF] in [figure omitted; refer to PDF] and [figure omitted; refer to PDF] in [figure omitted; refer to PDF] , respectively. The compactness of [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , implies that [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . On the other hand, by Lemma 9, for all [figure omitted; refer to PDF] , [figure omitted; refer to PDF] strongly as [figure omitted; refer to PDF] and [figure omitted; refer to PDF] Therefore, by the Lebesque dominated convergence theorem it follows that [figure omitted; refer to PDF] So [figure omitted; refer to PDF] holds, which shows that system (1) is approximately controllable and the proof is complete.

5. Example

Consider the following impulsive partial stochastic neutral differential inclusions of the form [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is a strictly increasing sequence of positive numbers and [figure omitted; refer to PDF] is a real function of bounded variation on [figure omitted; refer to PDF] . [figure omitted; refer to PDF] denotes a standard cylindrical Wiener process in [figure omitted; refer to PDF] defined on a stochastic space [figure omitted; refer to PDF] .

Let [figure omitted; refer to PDF] with the norm [figure omitted; refer to PDF] and define the operator [figure omitted; refer to PDF] by [figure omitted; refer to PDF] with the domain [figure omitted; refer to PDF] It is well known that [figure omitted; refer to PDF] is the infinitesimal generator of a strongly continuous cosine family [figure omitted; refer to PDF] in [figure omitted; refer to PDF] and is given by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the orthogonal set of eigenvalues of [figure omitted; refer to PDF] The associated sine family [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , is compact and is given by [figure omitted; refer to PDF]

Additionally, we will assume the following:

(i) The functions [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , are continuous, and [figure omitted; refer to PDF] .

(ii) The functions [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , are continuous and there exist continuous functions [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , such that [figure omitted; refer to PDF]

: with [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

(iii): The functions [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , are continuous.

(iv) The functions [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , are continuous, [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] for every [figure omitted; refer to PDF] .

Take [figure omitted; refer to PDF] which is the space introduced in Example 2. Set [figure omitted; refer to PDF] , defining the maps [figure omitted; refer to PDF] , [figure omitted; refer to PDF] by [figure omitted; refer to PDF] Using these definitions, we can represent system (99) in the abstract form (1). Moreover, it is easy to see that [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are continuous, and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are bounded linear operators with [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . Further, we can impose some suitable conditions on the above-defined functions to verify the assumptions on Theorem 16. Therefore, assumptions (H1)-(H6), (B1), and (B2) all hold, and the associated linear system of (99) is not exactly controllable but it is approximately controllable. Hence by Theorems 16 and 17, system (99) is approximately controllable on [figure omitted; refer to PDF] .

Acknowledgments

The author thanks the referees for their valuable comments and suggestions which improved their paper. This work is supported by the National Natural Science Foundation of China (Grant no. 11461019) and is supported by the President Found of Scientific Research Innovation and Application of Hexi University (Grant no. xz2013-10).

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

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