Mathematical Problems in Engineering

Volume 2015, Article ID 254310, 9 pages

http://dx.doi.org/10.1155/2015/254310

## Controllability of Nonlinear Impulsive Stochastic Evolution Systems Driven by Fractional Brownian Motion

^{1}Chengdu Institute of Computer Application, Chinese Academy of Sciences, Chengdu 610041, China^{2}Guangxi Key Lab of Hybrid Computation and IC Design Analysis, Guangxi University for Nationalities, Nanning 530006, China^{3}College of ASEAN Studies, Guangxi University for Nationalities, Nanning 530006, China

Received 12 June 2015; Accepted 9 July 2015

Academic Editor: Xiaoyu Song

Copyright © 2015 Juxia Xiong et al. 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

We consider the infinite-dimensional dynamical control systems described by nonlinear impulsive stochastic evolution differential equations. Sufficient conditions for the complete controllability of nonlinear impulsive stochastic systems are formulated and proved under the reasonable assumption that the corresponding linear system is completely controllable.

#### 1. Introduction

The impulsive differential systems are valuable tools in the modelling of many processes in which states are changed abruptly at certain moment of time, involving such fields as engineering, physics, and economics, and so forth; see [1–3]. It is well-known that the evolution differential system theory is a generalization of classical theory. So some partial differential systems can be changed into the abstract evolution systems by using semigroup technique. Then the researchers can easily discuss the properties of the partial differential systems by classical differential theory; for more details one can see [4].

The purpose of this paper is to discuss the controllability of the impulsive stochastic evolution systems driven by fractional Brownian motion as the following form:where generates an evolution system on a Hilbert space , . The control function takes value in , and is a Hilbert space, is a linear operator from into , , and is a fBm with Hurst index defined in a completely probability space . Further, and and represent the right and the left limits of at . Also, represents the jump in the state at time with determining, , is continuous for , and exist with , and is a random variable satisfying .

It is well-known that the noise or perturbations of a stochastic system are typically modeled by a Brownian motion, such as Gauss-Markov. This process has independent increments. However, many researchers have found that the standard Brownian motion is not an effective process in modeling through many physical phenomena. A family of process that seems to have wide physical applicability is fractional Brownian motion (fBm). This process was first introduced by Kolmogorov in 1940. Mandelbrot and Van Ness studied the applications of the fBm process soon after. Since then various forms of equations have been studied based on different settings. For example, the case of finite-dimensional equations has been studied by Besalú and Rovira [5], Unterberger [6], and Nguyen [7], and the case of infinite-dimensional equations in a Hilbert space has been considered by Boufoussi and Hajji [8], Caraballo et al. [9], and Ahmed [10].

One of the basic qualitative behaviors of a dynamical system is controllability, which was first researched by Kalman [11] in 1963. It 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. Many researchers have paid close attention to the study of the controllability for dynamical systems since then. There are many different methods for dealing with the controllability problems for various types of nonlinear stochastic systems. Subalakshmi and Balachandran [12] studied the approximate controllability of nonlinear stochastic impulsive systems in Hilbert spaces by using Nussbaum’s fixed point theorem. In [13], by using stochastic Lyapunov-like approach, sufficient conditions for stochastic -controllability are formulated. Balachandran et al. [14] researched the controllability of semilinear stochastic integrodifferential systems by using the Picard type iteration. By using the contraction mapping principle, Mahmudov and Zorlu studied the controllability [15] for nonlinear stochastic systems. Moreover, there are some researchers discussing the controllability for the stochastic system driven by fractional Brownian motion; for example, see [10, 16].

However, the above authors only consider that is an infinitesimal generator of a strongly continuous semigroup. But in our work (1), generates an evolution system on a Hilbert space . If and , the works [7, 8] are the special cases. We only assume that the linear system is completely controllable. By using the Cauchy-Schwarz inequality, Banach fixed point theorem, and so forth, we prove that the nonlinear system is completely controllable.

The rest of this paper is organized as follows. In the next section, we will introduce some useful preliminaries. In Section 3, some sufficient conditions are established to guarantee the existence and uniqueness of mild solutions of system (1). In Section 4, we will study the completely controllability for nonlinear impulsive stochastic evolution systems. Finally, we present an example to illustrate our main results.

#### 2. Preliminaries

Now we introduce some basic definitions, preliminaries, and notations which are used throughout this paper.

Let be a complete probability space with probability measure on . denotes the Hilbert space of all -measurable square integrable random variables with values in . is the Hilbert space of all square integrable and -measurable processes with values in . denotes the Banach space of continuous maps from into satisfying .

In order to define the solution of problem (1), we introduce the space formed by all -adapted, -valued processes such that is continuous at and and exist with .

In this paper, we assume that is endowed with the norm Then, is a Banach space (see [17]).

We also introduce some basic definitions on fractional Brownian motion (fBm).

Let be a complete probability space with a filtration satisfying the standard conditions.

*Definition 1. *The fractional Brownian motion (fBm) with Hurst index is a Gaussian process , having the properties , , and

Let , for a linear space ; there exists -valued step function on , such that where , and For , the Wiener integral with respect to can be defined as

Let be a Hilbert space, which is defined as the closure of with respect to the scalar product Then the mapping is an isometry between and the linear space span , which can be extended to an isometry between and the first Wiener chaos of the fBm (see [18]). The image of an element by this isometry is called the Wiener integral of with respect to

Next we give an explicit expression of this integral.

Let us consider the Kernel where ( denote the Beta function) and . It is easily shown that

Let be the linear operator, which is defined as Then , and is an isometry between and which can be extended to .

We denote , since ; then we get Moreover, the following lemma holds.

Lemma 2 (see [19]). *For ,*

*Let and be separable Hilbert spaces. denotes the space of all bounded linear operator from to and is a nonnegative self-adjoint operator. Denote by the space of all such that is a Hilbert-Schmidt operator; the norm is given by Then is a -Hilbert-Schmidt operator from to .*

*
Let be a sequence of two-side one-dimensional fBm, which is mutually independent on the complete probability space be a complete orthonormal basis in . One defines the -valued stochastic process as If is a nonnegative self-adjoint trace class operator, then converges in the space ; that is, it holds that . Then, we can say that is a -valued -cylindrical fBm with covariance operator .*

*Definition 3. *Let such thatThen for , its stochastic integral with respect to the fBm is defined as where is a Wiener process.

*Notice that ifthen in particular (15) holds, which follows immediately from (13).*

*The following lemma is obtained as a simple application of Lemma 2.*

*Lemma 4 (see [19]). For any such that is uniformly convergent for , and for any with ,Thenwhere .*

*In the following, let us give some basic properties of the operator .*

*Let be a family of linear operators and satisfy the following:*

* The domain of is dense in and independent of , and is a closed linear operator.*

* For each , the resolvent exists for all with and there is a constant such that .*

* For , there exist constants and such that *

*To establish the framework for our main controllability results, we will introduce the following definitions.*

*Definition 5 (see [4]). *A two-parameter family of bounded linear operators on is called an evolution system if the following two conditions are satisfied:(i) for .(ii) is strongly continuous for .

*Definition 6. *-valued process is called a mild solution of (1), if , ; for each and , the following integral equation holds:

*The following lemmas are of great importance in the proof of our main result.*

*Lemma 7 (see [4], Theorem 6.1 in Chapter 5). Under the assumptions , there is a unique evolution system on , satisfying the following:(i) for .(ii)For and is strongly differentiable in . The derivative and it is strongly continuous on . Moreover, (iii)For every is differentiable with respect to on , and *

*3. Existence Result*

*3. Existence Result*

*In this section, we will give the existence results for system (1). We will assume the following conditions:*

* There exist constants such that for a.e. , for all .*

* There exists a constant such thatfor all and a.e. .*

* Function → satisfies .*

* There exist constants such that, for each , *

*Now, let us consider the existence result for system (1).*

*Theorem 8. Assume that hypotheses hold. Then for any the impulsive stochastic system (1) has a unique mild solution in provided that *

*Proof. *Define an operator byBy using the Banach contraction mapping principle, we will show that the operator has a unique fixed point. To prove that, we divide the subsequent proof into two steps:*Step 1*. For any , let us show that is continuous on in the -sense.

Let , here , and be sufficiently small. Then we obtainThen, by the strong continuous of and the Lebesgue’s dominated convergence theorem, we know that the right hand of (27) tends to 0 as . Hence, is continuous on in the -sense.*Step 2*. We prove that is a contraction mapping. Let be two mild solutions of (1); thenInequality (28) equates toSince , we know that is a contraction mapping. Hence a unique fixed point in exists, which is the mild solution of problem (1).

*4. Controllability Result*

*4. Controllability Result*

*In this section, we discuss the controllability results for system (1). Before starting, we consider the following assumption:*

* The linear operator is defined byand has an inverse operator which takes values in , where , and there are positive constants such that .*

*To the readers’ convenience, we give the definitions of controllability as follows.*

*Definition 9. *System (1) is said to be completely controllable on the interval ifthat is, all the points in can be exactly reached from arbitrary initial condition and at time .

*Theorem 10. Assume that hypotheses hold. Then the impulsive stochastic system (1) is completely controllable on , if*

*Proof. *Fix and let be the Banach space of all functions from into , endowed with the supremum norm Let us consider the set We easily know that is a closed subset of equipped with norm .

By condition , we choose the feedback control function as The operator defined on byhas a fixed point on .

To prove that, we divide the subsequent proof into two steps.*Step 1*. For any , let us show that is continuous on in the -sense.

Let , here , and be sufficiently small. Then we obtain Then, by the strong continuous of and the Lebesgue’s dominated convergence theorem, we know that the right hand of (27) tends to 0 as . Hence, is continuous on in the -sense.*Step 2*. We prove that is a contraction mapping.

Let be two mild solutions of (1); then We can easily show that Since we haveHence From above inequalities, we obtainSince therefore is a contraction mapping. A unique fixed point exists in , which is the mild solution of problem (1). Problem (1) is completely controllable on .

*5. An Example*

*5. An Example*

*Consider the following semilinear stochastic differential equations of the following form:where is a bounded open set in are nonlinear functions, measurable with respect to and almost everywhere continuous with respect to , and continuous in the last two variables.*

*We define the function as follows:(i).(ii) is Lipschitz continuous with respect to and continuous differentiable in , and .*

*Let and ; then is dense in . By [20], the operator , defined byThen, system (45) can be rewritten as For the operator , we can obtain that there exist two constants , such that It is shown that are satisfied. Therefore, if we impose suitable conditions on and to ensure are satisfied, system (45) will be completely controllable by Theorem 10.*

*Conflict of Interests*

*Conflict of Interests*

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

*Acknowledgments*

*Acknowledgments*

*The research is supported by Guangxi Science and Technology Research Project for Universities no. LX2014105, no. LX2014106, and no. ZD2014044, Guangxi Universities for Nationalities Youth Project no. 2013MDQN043, the Chinese NSF no. 61402121, no. 11371003, and no. 11461006, Guangxi NSF no. 2013GXNSFAA019342, no. 2011GXNSFA018154, and no. 2012GXNSFGA060003, the Bagui scholarship project of Guangxi, the Science Research Project 2014 of the China-ASEAN Study Center (Guangxi Science Experiment Center) of Guangxi University for Nationalities, the Innovation Project of Guangxi Graduate Education (gxun-YCSZ2014121), the Science and Technology Foundation of Guangxi under Grant no. 10169-1, and the Scientific Research Project no. 201012MS274 from Guangxi Education Department.*

*References*

*References*

- D. Bainov and P. Simeonov,
*Impulsiv Differential Equations: Peridic Solutions and Applictiona*, Longman, Essex, UK, 1993. - V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov,
*Theory of Impulsive Differential Equations*, World Scientific, Singapore, 1989. View at Publisher · View at Google Scholar · View at MathSciNet - Y. V. Rogovchenko, “Nonlinear impulse evolution systems and applications to population models,”
*Journal of Mathematical Analysis and Applications*, vol. 207, no. 2, pp. 300–315, 1997. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - A. Pazy,
*Semigroups of Linear Operators and Applications to Partial Differential Equations*, Springer, New York, NY, USA, 1983. View at Publisher · View at Google Scholar · View at MathSciNet - M. Besalú and C. Rovira, “Stochastic delay equations with non-negativity constraints driven by fractional Brownian motion,”
*Bernoulli*, vol. 18, no. 1, pp. 24–45, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - J. Unterberger, “Stochastic calculus for fractional Brownian motion with Hurst exponent $H>1/4$: a rough path method by analytic extension,”
*The Annals of Probability*, vol. 37, no. 2, pp. 565–614, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - D. T. Nguyen, “Mackey-Glass equation driven by fractional Brownian motion,”
*Physica A: Statistical Mechanics and Its Applications*, vol. 391, no. 22, pp. 5465–5472, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - B. Boufoussi and S. Hajji, “Neutral stochastic functional differential equations driven by a fractional Brownian motion in a Hilbert space,”
*Statistics & Probability Letters*, vol. 82, no. 8, pp. 1549–1558, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - T. Caraballo, M. J. Garrido-Atienza, and T. Taniguchi, “The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 74, no. 11, pp. 3671–3684, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - H. M. Ahmed, “Controllability of impulsive neutral stochastic differential equations with fractional Brownian motion,”
*IMA Journal of Mathematical Control and Information*, 2014. View at Publisher · View at Google Scholar - R. E. Kalman, “Controllability of linear dynamical systems,”
*Contributions to the Theory of Differential Equations*, vol. 1, pp. 190–213, 1963. View at Google Scholar - R. Subalakshmi and K. Balachandran, “Approximate controllability of nonlinear stochastic impulsive integrodifferential systems in Hilbert spaces,”
*Chaos, Solitons & Fractals*, vol. 42, no. 4, pp. 2035–2046, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - J. Klamka and L. Socha, “Some remarks about stochastic controllability,”
*IEEE Transactions on Automatic Control*, vol. 22, no. 5, pp. 880–881, 1977. View at Google Scholar · View at MathSciNet - K. Balachandran, S. Karthikeyan, and J.-H. Kim, “Controllability of semilinear stochastic integrodifferential systems,”
*Kybernetika*, vol. 43, no. 1, pp. 31–44, 2007. View at Google Scholar · View at MathSciNet · View at Scopus - N. I. Mahmudov and S. Zorlu, “Controllability of non-linear stochastic systems,”
*International Journal of Control*, vol. 76, no. 2, pp. 95–104, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - H. M. Ahmed, “Approximate controllability of impulsive neutral stochastic differential equations with fractional Brownian motion,”
*Advances in Difference Equations*, vol. 113, pp. 1–11, 2014. View at Google Scholar - Z. M. Yan and X. X. Yan, “Existence of solutions for a impulsive nonlocal stochastic functional integrodifferential inclusion in Hilbert spaces,”
*Zeitschrift für angewandte Mathematik und Physik*, vol. 64, no. 3, pp. 573–590, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - S. Tindel, C. A. Tudor, and F. Viens, “Stochastic evolution equations with fractional Brownian motion,”
*Probability Theory and Related Fields*, vol. 127, no. 2, pp. 186–204, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - Y. Mishura,
*Stochastic Calculus for Fractional Brownian Motion and Related Processes*, Springer, Berlin, Germany, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - W. M. Bian, “Controllability of nonlinear evolution systems with preassigned responses,”
*Journal of Optimization Theory and Applications*, vol. 100, no. 2, pp. 265–285, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus

*
*