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 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 Uvalued 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.[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 the set of all linear bounded operators from U into H. For convenience, we will use the same notation to denote the norms in U, H and , and use to denote the inner product of U and H without any confusion. Let be a complete probability space with a filtration satisfying the usual conditions (i.e., it is increasing and right continuous, while contains all Pnull sets). Let be a QWiener process defined on with the covariance operator Q, that is
where Q is a positive, selfadjoint, trace class operator on U. Let be the space of all QHilbertSchmidt operators from U to H with the norm
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 of uniformly bounded linear operators on H, and in this paper, we always assume that is compact.
Now, we present the abstract space . Assume that with is a continuous function. The abstract phase space is defined by = {, for any , is a bounded and measurable function on and }. If is endowed with the norm
then is a Banach space [15,16].
Now, we consider the space
where is the restriction of x to , . We endow a seminorm on , it is defined by
Lemma 2.1 (see [17])
Assume that, then for, . Moreover,
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 , where is the gamma function.
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 is called a mild solution of the system (1) if
(i) is measurable and adapted, for each ;
(ii) has càdlàg paths on a.s., and satisfies the following integral equation
is a probability density function defined on , that is,
Lemma 2.2[18]
The operatorsandhave the following properties:
(i) For any fixed, andare linear and bounded operators, i.e., for any,
(ii) andare strongly continuous, which means that for everyand, we have
(iii) For every, andare also compact operators ifis compact 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 and denote the adjoint of B and , respectively.
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[19]
The linear fractional control system (2) is approximately controllable onJif and only ifasin the strong operator topology.
Lemma 2.4 ([18] Krasnoselskii’s fixed point theorem)
LetNbe a Banach space, letbe a bounded closed and convex subset ofN, and let, be maps ofintoNsuch thatfor every pair. Ifis a contraction andis completely continuous, then the equationhas 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.
(H1) The functions are continuous, and there exist two positive constants and such that the function satisfies that
and
(H2) There exists a positive such that
(H3) The function is continuous, and there exists continuous nondecreasing function such that, for each ,
(H4) μ is continuous, and there exists some constant such that
(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, there existssuch that.
Now, for any and , we define the control function
Theorem 3.1Assume that the assumptions (H1)(H4) hold. Then for each, the system (1) has a mild solution on, provided that
and
Proof For any , define the operator by
We shall show that the operator Φ has a fixed point in the space , which is the mild solution of (1). Let , , where is defined by
Then , and it is clear that x satisfies (1) if and only if z satisfies and
Thus, is a Banach space. Let for some , then , for each r, is a bounded, closed subset of H. Moreover, for , by lemma 2.1, we have
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 . If this is not true, then, for each positive integer r, there exists such that for , t may depending upon r. However, on the other hand, we have
By using (H1)(H4), Lemma 2.1 and Hölder’s inequality, we obtain
Dividing both sides by r and taking the limit as , we obtain
which is a contradiction to our assumption. Thus, for each , there exists some positive number r such that .
Next, we show that the operator Φ is condensing, for convenience, we decompose Φ as , where
Step 2. We prove that is a contraction on . Let and , we have
where , hence is a contraction.
Step 3. maps bounded sets into bounded sets in ,
Therefore, for each , we get .
Step 4. The map is equicontinuous. Let and . Then, we have
Noting the fact that for every , there exists a such that, whenever for every , and . Therefore, when , we have
The right hand of the inequality above tends to 0 as and , hence the set is equicontinuous.
Step 5. The set is relatively compact in . Let be fixed and . For , , we define
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 ArzelaAscoli theorem is completely continuous. Consequently, from Lemma 2.4, Φ has a fixed point, which is a mild solution of (1). □
Theorem 3.2Assume that (H1)(H5) are satisfied, and the conditions of Theorem 3.1 hold. Further, if the functionsfandgare uniformly bounded, andis compact, then the system (1) is approximately controllable on.
Proof Let be a solution of (1), then we can easily get that
In view of the assumptions that f and g are uniformly bounded on J, hence, there is a subsequence still denoted by and , which converges weakly to say in H, and in . On the other hand, by assumption (H5), the operator strongly as for all , and, moreover, . Thus, the Lebesgue dominated convergence theorem and the compactness of yield
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 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.
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