Abstract
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 KrasnoselskiiSchaefertype 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.
Keywords:
approximate controllability; fixed point principle; fractional impulsive neutral stochastic differential equations; mild solution; nonlocal conditions1 Introduction
The purpose of this paper is to prove the existence and approximate controllability of mild solutions for a class of fractional impulsive neutral stochastic differential equations with nonlocal conditions described in the form
where
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.[1], Mill and Ross [2], Podlubny [3] and the references therein. There is also an extensive literature concerned with the fractional differential equations. For example, Benchohra et al. in [4] considered the VIP for a particular class of fractional neutral functional differential equations with infinite delay. Zhou in [5] 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 [6] investigated the controllability of infinite dimensional linear stochastic systems, and in [7] Dauer and Mahmudov extended the results to semilinear stochastic evolution equations with finite delay. Park, Balasubramaniam and Kumaresan [8] 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 [913].
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.
2 Preliminaries
In this section, we introduce some notations and preliminary results, needed to establish
our results. Throughout this paper, let U and H be two real separable Hilbert spaces, and we denote by
where Q is a positive, selfadjoint, trace class operator on U. Let
For the construction of stochastic integral in Hilbert space, see Da Prato and Zabczyk
[14]. Let A be the infinitesimal generator of an analytic semigroup
Now, we present the abstract space
then
Now, we consider the space
where
Lemma 2.1 (see [17])
Assume that
where
Definition 2.1 The fractional integral of order α with the lower limit 0 for a function f is defined as
provided the right side is pointwise defined on
Definition 2.2 The Caputo derivative of order α with the lower limit 0 for a function f can be written as
Definition 2.3 A stochastic process
(i)
(ii)
(iii)
Lemma 2.2[18]
The operatorsandhave the following properties:
(i) For any fixed
(ii)
(iii) For every
In order to study the approximate controllability for the fractional control system (1), we introduce the following linear fractional differential system
The controllability operator associated with (2) is defined by
where
Let
Definition 2.4 The system (1) is said to be approximately controllable on J if
Lemma 2.3[19]
The linear fractional control system (2) is approximately controllable onJif and only if
Lemma 2.4 ([18] Krasnoselskii’s fixed point theorem)
LetNbe a Banach space, let
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.
(H1) The functions
and
for every
(H2) There exists a positive
(H3) The function
(H4) μ is continuous, and there exists some constant
(H5) The linear stochastic system (2) is approximately controllable on
The following lemma is required to define the control function.
Lemma 3.1[6]
For any
Now, for any
Theorem 3.1Assume that the assumptions (H1)(H4) hold. Then for each
and
Proof For any
We shall show that the operator Φ has a fixed point in the space
Then
Set
Thus,
For the sake of convenience, we divide the proof into several steps.
Step 1. We claim that there exists a positive number r such that
By using (H1)(H4), Lemma 2.1 and Hölder’s inequality, we obtain
where
Dividing both sides by r and taking the limit as
which is a contradiction to our assumption. Thus, for each
Next, we show that the operator Φ is condensing, for convenience, we decompose Φ as
Step 2. We prove that
where
Step 3.
Therefore, for each
Step 4. The map
Noting the fact that for every
The right hand of the inequality above tends to 0 as
Step 5. The set
Then from the compactness of
Theorem 3.2Assume that (H1)(H5) are satisfied, and the conditions of Theorem 3.1 hold. Further, if the functionsfandgare uniformly bounded, and
Proof Let
In view of the assumptions that f and g are uniformly bounded on J, hence, there is a subsequence still denoted by
This gives the approximate controllability of (1), the proof is complete. □
4 An example
As an application, we consider an impulsive neutral stochastic partial differential equation with the following form
Let
We define the operator A by
Define the operators
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
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the manuscript.
Acknowledgements
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).
References

Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam (2006)

Miller, KS, Ross, B: An Introduction to the Fractional Calculus and Differential Equations, Wiley, New York (1993)

Podlubny, I: Fractional Differential Equations, Academic Press, San Diego (1999)

Benchohra, M, Henderson, J, Ntouyas, SK, Ouahab, A: Existence results for fractional order functional differential equations with infinite delay. J. Math. Anal. Appl.. 338, 1340–1350 (2008). Publisher Full Text

Zhou, Y, Jiao, F, Li, J: Existence and uniqueness for fractional neutral differential equations with infinite delay. Nonlinear Anal.. 71, 3249–3256 (2009). Publisher Full Text

Mahmudov, NI: Controllability of linear stochastic systems in Hilbert spaces. J. Math. Anal. Appl.. 259, 64–82 (2001). Publisher Full Text

Dauer, JP, Mahmudov, NI: Controllability of stochastic semilinear functional differential equations in Hilbert spaces. J. Math. Anal. Appl.. 290, 373–394 (2004). PubMed Abstract  Publisher Full Text

Park, JY, Balasubramaniam, P, Kumaresan, N: Controllability for neutral stochastic functional integrodifferential infinite delay systems in abstract space. Numer. Funct. Anal. Optim.. 28, 1369–1386 (2007). Publisher Full Text

Li, CX, Sun, JT, Sun, RY: Stability analysis of a class of stochastic differential delay equations with nonlinear impulsive effects. J. Franklin Inst.. 347, 1186–1198 (2010). Publisher Full Text

Sakthivel, R, Mahmudov, NI, Lee, SG: Controllability of nonlinear impulsive stochastic systems. Int. J. Control. 82, 801–807 (2009). Publisher Full Text

Shen, LJ, Shi, JP, Sun, JT: Complete controllability of impulsive stochastic integrodifferential systems. Automatica. 46, 1068–1073 (2010). Publisher Full Text

Shen, LJ, Sun, JT: Approximate controllability of stochastic impulsive functional systems with infinite delay. Automatica. 48, 2705–2709 (2012). Publisher Full Text

Subalakshmi, R, Balachandran, K: Approximate controllability of nonlinear stochastic impulsive intergrodifferential systems in Hilbert spaces. Chaos Solitons Fractals. 42, 2035–2046 (2009). Publisher Full Text

Da Prato, G, Zabczyk, J: Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge (1992)

Ren, Y, Sun, DD: Secondorder neutral stochastic evolution equations with infinite delay under Caratheodory conditions. J. Optim. Theory Appl.. 147, 569–582 (2010). Publisher Full Text

Ren, Y, Zhou, Q, Chen, L: Existence, uniqueness and stability of mild solutions for timedependent stochastic evolution equations with Poisson jumps and infinite delay. J. Optim. Theory Appl.. 149, 315–331 (2011). Publisher Full Text

Chang, YK: Controllability of impulsive functional differential systems with infinite delay in Banach space. Chaos Solitons Fractals. 33, 1601–1609 (2007). Publisher Full Text

Zhou, Y, Jiao, F: Existence of mild solution for fractional neutral evolution equations. Comput. Math. Appl.. 59, 1063–1077 (2010). Publisher Full Text

Mahmudov, NI, Denker, A: On controllability of linear stochastic systems. Int. J. Control. 73, 144–151 (2000). Publisher Full Text