Discrete Dynamics in Nature and Society
Volume 2015 (2015), Article ID 420826, 16 pages
http://dx.doi.org/10.1155/2015/420826

Research Article

Approximate Controllability of Semilinear Neutral Stochastic Integrodifferential Inclusions with Infinite Delay

Meili Li and Man Liu

School of Science, Donghua University, Shanghai 201620, China

Received 29 July 2015; Revised 23 October 2015; Accepted 28 October 2015

Academic Editor: Chris Goodrich

Copyright © 2015 Meili Li and Man Liu. 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 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 define

Then 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 expression

defines 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 norm

and 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 by

where

, 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 for

if

(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 , and

where 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 operator

is 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 that

where

Proof. Let be the space of all function , such that . The restriction is continuous and is a seminorm in which is defined by

Let . We define the multivalued map by and the set of such that

We will prove that the operator has a fixed point, which then is a solution of (3). For , let be the function defined by

Set . It is clear that satisfies (26) if and only if satisfies and

Let . For any , we have

So, if , then is a Banach space. For any , set

then is uniformly bounded, and we have

For , we can get

Let the operator : be defined by , the set of , such that

We divide the proof into five steps.
Step  1. is convex for each .
If and belong to , then there exists , such that

Let , because the operators , , are linear, ; we have

Since 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 that

by hypotheses , for each , we have

Then, we define the function , . We have

If , 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 , satisfies

where

Or we obtain that

Then, we have

Now, for small and ,

The operator is compactness and the function is uniformly bounded. So

is relatively compact in and . Then, there exists such that

So, we can get , , and by the same way, such that

respectively, as in space . Clearly, is bounded, and there exists a positive constant such that . According to , we have

So holds in , and we obtain the approximate controllability of (3).
The proof is finished.

4. Example

In this section, we study the following neutral inclusion which arises in the study of heat flow in materials of the so-called retarded type [25, 26] as an application to Theorem 10:

where represents the temperature of the point at time , and , , and will be described below.

In the space , stands for cylindrical Wiener process in defined on a probability space . Define the operator as , where are absolutely continuous, and , .

generates a strongly continuous semigroup which is analytic, compact, and self-adjoint. Moreover, the resolvent operator is compact when it exists. The complete orthonormal set of eigenvectors of with also exists. Then, the operator is given by on the space and the following properties hold:

(i)If , then .(ii)If , then .(iii)The operator is given by .

Then, rewrite the above system as follows:

Here we take and and define the phase space

and let . Then is a Banach space which satisfies (a)–(c). Meanwhile, the subspace is defined by

endowed with the norm . Thus for , where , . Clearly, it satisfies the axioms . Let and define the functions and for the infinite delay as follows:

We assume that the following conditions hold:

(i) with primitive . is nonpositive, nondecreasing and .(ii)The function and there exists a function such that

(iii)The function is continuous in ,

(iv)The function is continuous, , for , where is continuous and nondecreasing.

Moreover, it is clear that is a bounded and linear operator which satisfies , and satisfies condition . By condition (ii), we have

so we get and

Then, we have

which implies that and satisfies the assumption . Clearly, according to the definition of , it satisfies the assumption and

which satisfies the assumption .

To sum up, the conditions stated in Theorem 10 have been satisfied in the above case; therefore the inclusion is approximately controllable.

5. Conclusion

In this paper, the approximate controllability of semilinear neutral stochastic integrodifferential inclusions with infinite delay has been studied. A new set of sufficient conditions are formulated for the approximately controllable (3). The family are closed linear operators and the standard semigroup approaches cannot be used here. The sufficient conditions are established in Hilbert space by applying the theory of analytic resolvent operator, the fractional power theory, and the theorem of nonlinear alternative for Kakutani maps under the assumption that the corresponding linear inclusion is approximately controllable. Moreover, an application is provided to illustrate the obtained theoretical results.

Finally, we have to point out here that, like most related work in the literature of distributed parameter control systems, the definition of controllability for inputs in an infinitely dimensional space is not so practical and there is no indication on how to find the controller to achieve the desired state for the system. We expect some important improvement on these issues to appear in the future work. As the impulsive perturbations and state-dependent delay are very common in many real life phenomena, in future we can study the approximate controllability of integrodifferential systems with impulsive effects and state-dependent delay.

Appendices

A. The Proof of Step 2

In fact, we only need to prove that, for each and , there exists a positive constant such that .

If , then there exists , such that, for each ,

From (26), we have

According to hypotheses ()–(), for , we get

here is chosen from Definition 4. Furthermore,

Then, we have , for each .

B. The Proof of Step 3

For each and , let ; there exists such that

It is easy to see that the right-hand side of inequality above tends to as is independent of . Since and are continuous in the uniform operator topology on by Lemmas 2 and 3, thus, the set is equicontinuous in .

Then, we will demonstrate that is relatively compact in . In fact, if , then is relatively compact. Let be fixed and ; we have

where and

Since is fixed, then is a single point in and we only need to certify that is relatively compact. Observe that, for ,

where . It implies that is bounded in . Since the operator and the imbedding is compact, we infer that is relatively compact in . Hence, is also relatively compact in .

C. The Proof of Step 4

Let , , , ; we will prove that . Let ; we have

where is obtained from (26) by replacing . We will certify that there exists such that and respective holds. Let

Since ,  , and are continuous, then for . We have

Consider the linear continuous operator:

Obviously, is linear and continuous; is a closed graph operator from Lemma 5. Moreover,

Since , it follows from Lemma 5 that

for some . Therefore is a completely continuous multivalued map, u.s.c. with convex closed values.

Conflict of Interests

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

Acknowledgment

This paper is supported by NNSF of China (no. 11371087).

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