- Open Access
Approximate controllability of fractional impulsive neutral stochastic differential equations with nonlocal conditions
© Zang and Li; licensee Springer 2013
Received: 16 April 2013
Accepted: 12 August 2013
Published: 28 August 2013
In this paper, the approximate controllability of fractional impulsive neutral stochastic differential equations with nonlocal conditions and infinite delay in Hilbert spaces is studied. By using the Krasnoselskii-Schaefer-type fixed point theorem and stochastic analysis theory, some sufficient conditions are given for the approximate controllability of the system. At the end, an example is given to illustrate the application of our result.
MSC:65C30, 93B05, 34K40, 34K45.
where is the Caputo fractional derivative of order ; the state variable takes values in the real separable Hilbert space H; is the infinitesimal generator of a strongly continuous semigroup of a bounded linear operators , , in the Hilbert space H. The history , , , belongs to an abstract phase space . The control function is given in , U is a Hilbert space; B is a bounded linear operator from U into H. The functions f, h, g, are appropriate functions to be specified later. The process is a given U-valued Wiener process with a finite trace nuclear covariance operator defined on a complete probability space . Here , , and represent the right and left limits of at , respectively. The initial data is an -measurable, -valued random variable independent of with finite second moments.
In the past few decades, the theory of fractional differential equations has received a great deal of attention, and they play an important role in many applied fields, including viscoelasticity, electrochemistry, control, porous media, electromagnetic and so on. We refer the reader to the monographs of Kilbas et al. , Mill and Ross , Podlubny  and the references therein. There is also an extensive literature concerned with the fractional differential equations. For example, Benchohra et al. in  considered the VIP for a particular class of fractional neutral functional differential equations with infinite delay. Zhou in  discussed the existence and uniqueness for fractional neutral differential equations with infinite delay.
In practice, deterministic systems often fluctuate due to environmental noise. So it is important and necessary for us to discuss the stochastic differential systems. On the other hand, the control theory is one of the important topics in mathematics. Roughly speaking, controllability generally means that it is possible to steer a dynamical control system from an arbitrary initial state to an arbitrary final state using the set of admissible controls. As a result of its widespread use, the controllability of stochastic or deterministic systems all have received extensive attention. Mahmudov  investigated the controllability of infinite dimensional linear stochastic systems, and in  Dauer and Mahmudov extended the results to semilinear stochastic evolution equations with finite delay. Park, Balasubramaniam and Kumaresan  gave the controllability of neutral stochastic functional infinite delay systems. Besides the environmental noise, sometimes, we have to consider the impulsive effects, which exist in many evolution processes, because the impulsive effects may bring an abrupt change at certain moments of time. For the literatures on controllability of stochastic system with impulsive effect, we can see [9–13].
However, to the best of our knowledge, it seems that little is known about approximate controllability of fractional impulsive neutral stochastic differential equations with infinite delay and nonlocal conditions. The aim of this paper is to study this interesting problem. The rest of the paper is organized as follows. In Section 2, we introduce some preliminaries such as definitions of fractional calculus and some useful lemmas. In Section 3, we prove our main results. Finally in Section 4, an example is given to demonstrate the application of our results.
For the construction of stochastic integral in Hilbert space, see Da Prato and Zabczyk . Let A be the infinitesimal generator of an analytic semigroup of uniformly bounded linear operators on H, and in this paper, we always assume that is compact.
Lemma 2.1 (see )
provided the right side is pointwise defined on , where is the gamma function.
is measurable and -adapted, for each ;
- (ii)has càdlàg paths on a.s., and satisfies the following integral equation
- (iii)on satisfying , where
Lemma 2.2 
- (i)For any fixed , and are linear and bounded operators, i.e., for any ,
- (ii)and are strongly continuous, which means that for every and , we have
For every , and are also compact operators if is compact for every .
Let be the state value of (1) at terminal time T, corresponding to the control u and the initial value φ. Denote by the reachable set of system (1) at terminal time T, its closure in H is denoted by .
Definition 2.4 The system (1) is said to be approximately controllable on J if .
Lemma 2.3 
The linear fractional control system (2) is approximately controllable on J if and only if as in the strong operator topology.
Lemma 2.4 ( Krasnoselskii’s fixed point theorem)
Let N be a Banach space, let be a bounded closed and convex subset of N, and let , be maps of into N such that for every pair . If is a contraction and is completely continuous, then the equation has a solution on .
3 Main results
In this section, we formulate sufficient conditions for the approximate controllability of system (1). For this purpose, we first prove the existence of solutions for system (1). Second, in Theorem 3.2, we shall prove that system (1) is approximately controllable under certain assumptions. In order to prove our main results, we need the following assumptions.
for every , .
(H5) The linear stochastic system (2) is approximately controllable on .
The following lemma is required to define the control function.
Lemma 3.1 
For any , there exists such that .
For the sake of convenience, we divide the proof into several steps.
which is a contradiction to our assumption. Thus, for each , there exists some positive number r such that .
where , hence is a contraction.
Therefore, for each , we get .
The right hand of the inequality above tends to 0 as and , hence the set is equicontinuous.
Then from the compactness of , we obtain that is relatively compact in H for every ϵ, . Moreover, for , we can easily prove that is convergent to in as and , hence the set is also relatively compact in . Thus, by Arzela-Ascoli theorem is completely continuous. Consequently, from Lemma 2.4, Φ has a fixed point, which is a mild solution of (1). □
Theorem 3.2 Assume that (H1)-(H5) are satisfied, and the conditions of Theorem 3.1 hold. Further, if the functions f and g are uniformly bounded, and is compact, then the system (1) is approximately controllable on .
This gives the approximate controllability of (1), the proof is complete. □
4 An example
Let and , with . To study the approximate controllability of (3), assume that is measurable and continuous on and thus bounded by . is measurable and continuous with finite .
We define the operator A by with domain . It is well known that A generates an analytic semigroup given by , , and , , is the orthogonal set of eigenvectors of A.
With the choice of A, h, f, g, (3) can be rewritten as the abstract form of system (1). Thus, under the appropriate conditions on the functions h, f, g and as those in (H1)-(H5), system (3) is approximately controllable.
We are very grateful to the anonymous referee and the associate editor for their careful reading and helpful comments. This work was substantially supported by the National Natural Sciences Foundation of China (No. 11071259), Research Fund for the Doctoral Program of Higher Education of China (No. 20110162110060).
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