#### 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.

#### 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 formwhere the state takes values in a separable real Hilbert space with inner product and norm The operator is the infinitesimal generator of a strongly continuous cosine family on . The control function , a Hilbert space of admissible control functions, is an integer, and is a bounded linear operator from a Banach space to Let be another separable Hilbert space with inner product and norm Suppose that is a given -valued Wiener process with a covariance operator defined on a complete probability space equipped with a normal filtration , which is generated by the Wiener process . The time history given by belongs to some abstract phase space defined axiomatically; , , , , are given functions to be specified later. Moreover, let be prefixed points and the symbol , where and represent the right and left limits of at , respectively. The initial data is an -adapted, -valued random variable independent of the Wiener process 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 , 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 be a complete probability space equipped with a normal filtration Let and be the separable Hilbert spaces and let be a -Weiner process on with the covariance operator such that We assume that there exists a complete orthonormal system in , a bounded sequence of nonnegative real numbers such that , and a sequence of independent Brownian motions such that and , where is the -algebra generated by Let be the space of all Hilbert-Schmidt operators from to with the inner product Let be the Banach space of all -measurable th power integrable random variables with values in the Hilbert space . Let be the Banach space of continuous maps from into satisfying the condition

We use the notations that is the family of all subsets of Let us introduce the following notations: Consider given by where and Then is a metric space and 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 is convex (closed) valued if is convex (closed) for all . is bounded on bounded sets if is bounded in for any bounded set of ; that is, .

is called upper semicontinuous (u.s.c., in short) on , if, for any , the set is a 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 of If the multivalued map is completely continuous with nonempty compact values, then is u.s.c. if and only if has a closed graph; that is, , , and imply .

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

A multivalued map is said to be measurable if, for each , the function defined by is measurable.

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

Lemma 1 (see [32]). *Suppose that is the infinitesimal generator of a cosine family of operators . Then, the following hold: *(a)*There exist and such that and hence *(b)*Consider for all *(c)*There exists such that for all *

The existence of solutions of the second-order linear abstract Cauchy problemwhere 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 given by is called a mild solution of (5) and if , the function is continuously differentiable and

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

In this paper, we assume that the phase space is a seminormed linear space of -measurable functions mapping into and satisfying the following fundamental axioms due to Hale and Kato (see, e.g., [34]).(A)If , such that and , then for every the following conditions hold:(i) is in ;(ii);(iii), where is a constant; is continuous and is locally bounded, and , , and are independent of (B)For the function in (A), the function is continuous from into (C)The space is complete.

*Example 2. *The phase space . Let , and let 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 is locally integrable and there is a nonnegative, locally bounded function on such that for all and , where is a set whose Lebesgue measure is zero. We denote by the set consisting of all classes of functions such that is Lebesgue measurable on , and is Lebesgue integrable on The seminorm is given by The space satisfies axioms (A)–(C). Moreover, when and , we can take , , and , for (see [35, Theorem ] for details).

*Remark 3 (see [4]). *In retarded functional differential equations without impulses, the axioms of the abstract phase space include the continuity of the function 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

*Remark 4. *In the rest of this paper and are the constants defined by and

For , we denote by , , the unique continuous function such that Moreover, for we denote by , , the set The notation stands for the closed ball with center at and radius in

Lemma 5. *A set is relatively compact in if and only if the set is relatively compact in , for every *

Furthermore, we need the following result.

Lemma 6 (see [36]). *Let be an integrable function such that Then the function belongs to , the function is integrable on , and , .*

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

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

*Definition 7. *An -adapted stochastic process is called a mild solution of system (1) if , , and the impulsive conditions and , are satisfied and (i) is adapted to .(ii) has càdlàg paths on a.s. and, for each , satisfies the integral equationwhere

*Definition 8. *System (1) is said to be approximately controllable on the interval if

It is convenient at this point to define operators where denotes the adjoint of and is the adjoint of It is straightforward that the operator is a linear bounded operator:(S1), as in the strong operator topology.

Lemma 9. *Assumption (S1) holds if and only if , as in the strong operator topology.*

The proof of this lemma is a straightforward adaptation of the proof of [9, Theorem ].

Lemma 10 (see [10]). *For any there exists such that .*

Now for any and we define the control functionwhere

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

Lemma 11. *Let be an -adapted measurable process such that -adapted process and ; then *

Lemma 12 (see [37]). *For any and for arbitrary -valued predictable process , *

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

Lemma 13. *Let be a Hilbert space, and let and be two multivalued operators satisfying that*(a)* is a contraction*(b)* is completely continuous.**Then either *(i)*the operator inclusion has a solution or*(ii)*the set is unbounded for *

#### 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) is the infinitesimal generator of a strongly continuous cosine family on and the corresponding sine family satisfies the conditions , , and , for some constants , , , and (H2), is compact.(H3)The function is continuous and there exist such that for all , and with (H4)The function satisfies the following conditions:(i)For each the function is continuous and, for each , the function is strongly measurable.(ii)There exists a continuous function , such that for a.e. , where is a continuous nondecreasing function.(H5)The multivalued map satisfies the following conditions:(i)For each , the function is u.s.c. and, for each , the function is measurable and the set is nonempty.(ii)There exist a continuous function and a continuous nondecreasing function such that a.e. , with where(H6)The functions are continuous and there are constants , , , such that for every

Lemma 14 (see [38]). *Let be a compact interval and let be a Hilbert space. Let be a multivalued map satisfying (H5) (i) and let be a linear continuous operator from to Then the operator is a closed graph in *

*Remark 15. *In what follows, we set , , and

Theorem 16. *If assumptions (H1)–(H6) are satisfied, further, suppose that, for all , system (1) has at least one mild solution on , provided that*

*Proof. *Let endowed with the norm of Thus is Banach space. Now we can define the multivalued map by the set of such that whereand , and is such that and on In what follows, we aim to show that the operator has a fixed point, which is a solution of problem (1).

Let be a decreasing sequence in such that To prove the above theorem, we consider the following problem:We will show that the problem has at least one mild solution

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

Let be the extension of such that on Now, we consider the following multivalued operators and defined by It is clear that The problem of finding mild solutions of (27) is reduced to find the solutions of the operator inclusion In what follows, we show that operators and satisfy the conditions of Lemma 13.*Step 1. * is a contraction on

Let and From (H3) and Lemma 11, we haveBy Lemma 6, we have Similarly, for any , we have Taking supremum over , it follows that where Hence, is a contraction on .*Step 2. * has compact, convex values and it is completely continuous.

is convex for each .

In fact, if , belong to , then there exist such that Let For each we have Since is convex (because has convex values) we have .

maps bounded sets into bounded sets in

Indeed, it is enough to show that there exists a positive constant such that, for each , one has . If , then there exists such that, for each ,