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Mathematical Problems in Engineering
Volume 2013 (2013), Article ID 316906, 10 pages
Controllability of Fractional Neutral Stochastic Integro-Differential Systems with Infinite Delay
1College of Information Science and Technology, Donghua University, 2999 North Renmin Road, Songjiang, Shanghai 201620, China
2Department of Mathematics, Donghua University, 2999 North Renmin Road, Songjiang, Shanghai 201620, China
3Department of Mathematics, Anhui Normal University, 1 East Beijing Road, Wuhu 241000, China
Received 28 October 2012; Revised 20 January 2013; Accepted 27 January 2013
Academic Editor: Engang Tian
Copyright © 2013 Xichao Sun 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.
This paper is concerned with the controllability of a class of fractional neutral stochastic integro-differential systems with infinite delay in an abstract space. By employing fractional calculus and Sadovskii's fixed point principle without assuming severe compactness condition on the semigroup, a set of sufficient conditions are derived for achieving the controllability result.
It is well known that the fractional calculus is a classical mathematical notion and is a generalization of ordinary differentiation and integration to arbitrary (noninteger) order. Nowadays, studying fractional-order calculus has become an active research field [1–7]. Much effort has been devoted to apply the fractional calculus to networks control. For example, Chen et al. , Delshad et al. , and Wang and Zhang  studied the synchronization for fractional-order complex dynamical networks; Zhang et al.  investigated a fractional order three-dimensional Hopfield neural network and pointed out that chaotic behaviors can emerge in a fractional network; Kaslik and Sivasundaram  discussed the local stability for fractional-order neural networks of Hopfield type by applying the linear stability theory of fractional-order system.
One of the emerging branches of this study is the theory of fractional evolution equations, say, evolution equations, where the integer derivative with respect to time is replaced by a derivative of fractional order. The increasing interest in this class of equations is motivated both by their application to problems from fluid dynamic traffic model, viscoelasticity, heat conduction in materials with memory, electrodynamics with memory, and also because they can be employed to approach nonlinear conservation laws (see  and references therein). In addition, neutral stochastic differential equations with infinite delay have become important in recent years as mathematical models of phenomena in both science and engineering, for instance, in the theory development in Gurtin and Pipkin  and Nunziato  for the description of heat conduction in materials with fading memory. It should be pointed out that the deterministic models often fluctuate due to noise, which is random or at least appears to be so. Therefore, we must move from deterministic problems to stochastic ones. We mention here the recent papers [16, 17] concerning the existence of mild solutions of fractional stochastic systems.
As one of the fundamental concepts in mathematical control theory, controllability plays an important role both in deterministic and stochastic control problems such as stabilization of unstable systems by feedback control. 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. Controllability problems for different nonlinear stochastic systems in infinite dimensional spaces have been extensively studied in many papers; see [18–22] and references therein. We would also like to mention that the controllability for stochastic systems with infinite delay has been investigated by Balasubramaniam et al. [23, 24] and Ren et al.  using some abstract spaces. Nevertheless, to the best of our knowledge, it seems that little is known about the controllability of fractional neutral stochastic differential equations with infinite delay, and the aim of this paper is to close this gap.
In this paper, we are interested in the controllability of a class of fractional neutral stochastic integro-differential systems with infinite delay of the followin form: Here, , . takes value in a real separable Hilbert space with inner product and norm . The fractional derivative , is understood in the Caputo sense. is the infinitesimal generator of a analytic semigroup of a bounded linear operator , , on . Let be another separable Hilbert space with inner product and norm . is a given -valued Wiener process with a finite trace nuclear covariance operator defined on a filtered complete probability space . The control function takes value in of admissible control functions for a separable Hilbert space , and is a bounded linear operator from into . The histories defined by belong to the phase space , which will be defined in Section 2. The initial data is an -measurable, -valued random variable independent of with finite second moments, and , are appropriate mappings specified later (here, denotes the space of all -Hilbert-Schmidt operators from into , which is going to be defined later).
The structure of this paper is as follows. In Section 2, we briefly present some basic notations and preliminaries. The controllability result of system (1) is investigated by means of Sadovskii’s fixed point theorem and operator theory in Section 3. Conclusion is given in Section 4.
For more details in this section, we refer the reader to Pazy , Da Prato and Zabczyk , and Samko et al. . Throughout this paper, and denote two real separable Hilbert spaces. We denote by the set of all linear bounded operators from into , equipped with the usual operator norm . In this paper, we use the symbol to denote norms of operators regardless of the spaces potentially involved when no confusion possibly arises.
Let be a filtered complete probability space satisfying the usual condition, which means that the filtration is a right continuous increasing family and contains all -null sets. is a -Wiener process defined on with covariance operator such that . We assume that there exist a complete orthonormal system in , a bounded sequence of nonnegative real numbers such that , , and a sequence of independent Brownian motions such that Let be the space of all Hilbert-Schmidt operators from to with the inner product .
Suppose that , where is the resolvent set of , then the semigroup is uniformly bounded. That is to say, , , for some constant . Then, for , it is possible to define the fractional power operator as a closed linear operator on its domain . Furthermore, the subspace is dense in , and the expression defines a norm on . The following properties are well known.
Lemma 1 (Pazy ). Suppose that the preceding conditions are satisfied.(a)If , then and the embedding is compact whenever the resolvent operator of is compact.(b)For every , there exists a positive constant such that
Let now us recall some basic definitions and results of fractional calculus.
Definition 2. The fractional integral of order with the lower limit for a function is defined as provided the right-hand side is pointwise defined on , where is the gamma function.
Definition 3. The Caputo derivative of order with the lower limit for a function can be written as
If is an abstract function with values in , then the integrals that appear in the previous definitions are taken in Bochner’s sense.
Assume that with is a continuous function. Recall that the abstract phase space is defined by If is endowed with the norm then is a Banach space (see Liu et al. ).
At the end of this section, we recall the fixed point theorem of Sadovskii .
Lemma 4. Let be a condensing operator on a Banach space ; that is, is continuous, and take bounded sets into bounded sets, and for every bounded set of with . If for a convex, closed, and bounded set of , then has a fixed point in (where denotes Kuratowski's measure of noncompactness.)
3. Main Results
In this section, we obtain controllability of system (1). We first present the definition of mild solutions.
Definition 5. An -valued stochastic process is said to be a mild solution of system (1) if (i) is -adapted and measurable for each ; (ii) is continuous on almost surely and for each , the function is integrable such that the following stochastic integral equation is verified: (iii) on with , where with a probability density function defined on .
Definition 6. System (1) is said to be controllable on the interval , if for every initial stochastic process defined on , there exists a stochastic control , which is adapted to the filtration such that the mild solution of (1) satisfies , where and are preassigned terminal state and time, respectively.
The following properties of and that appeared in Zhou and Jiao  are useful.
Lemma 7. Under previous assumptions on , , and , then (i)for any fixed , and are linear and bounded operators such that for any , , ; (ii) and are strongly continuous; (iii)for any , , and , one has
In this paper, we will work under the following assumptions. is the infinitesimal generator of an analytic semigroup of bounded linear operators in ; , , is continuous in the uniform operator topology and for some . The function is continuous, and there exist some constants , with , such that is -valued and satisfies the following: for each , is continuous and for each , is strongly measurable; there is a positive integrable function and a continuous nondecreasing function such that for every , we have for any , , there exists a positive constant such that The linear operator from into defined by has an invertible operator defined on (see ), and there exist a pair of constants such that There exists a compact set such that for all , and with finite second moment. Assume that the following relationship holds: where
Denote by the space of all continuous -valued stochastic processes . Let Set to be a seminorm defined by We have the following useful lemma that appeared in Liu et al. .
Lemma 8. Assume that ; then, for all , . Moreover, where .
The main object of this paper is to explain and prove the following theorem.
Theorem 9. Assume that assumptions – hold. Then, system (1) is controllable on provided that
Proof. Using assumption , for an arbitrary process , define the control process
We transform (1) into a fixed point problem. Consider the operator defined by
It follows from Hölder inequality, Lemma 7, and assumption that
which deduces that is integrable on by Bochner's theorem (see ), where is defined in (19).
In what follows, we will show that using the control , the operator has a fixed point, which is then a mild solution for system (1).
For , define Then, . Set , . It is easy to check that satisfies (1) if and only if and where is obtained by replacing by in (24). Let For each , we have Thus, is a Banach space. For , set then, for each , is clearly a bounded closed convex set in . For , from Lemma 8, we see that
Consider the map defined by A similar argument as (26), we can show that is well defined on for each . Note that the operator with a fixed point is equivalent to show that the operator has fixed point. To this end, we decompose as , where the operators and are defined on , respectively, by Thus, Theorem 9 follows from the next theorem.
Proof. The proof is followed by the several steps.
Step 1. There exists a positive number such that . If it is not true, then for each positive number , there exists a function , but ; that is, for some . An elementary calculation can show that We estimate each term on the right hand side of (35). By virtue of Lemma 8, assumptions and (32), we obtain in view of Lemma 7 and Hölder inequality, we have where and are defined in (18) and (19), respectively.
Applying Burkhölder-Davis-Gundy's inequality and assumptions , we get
On the other hand, in view of (24) and , we have thus, by the same procedure as (36)–(38), it follows that where and are defined in (18) and (19), respectively.
Combining these estimates (35) to (40) yields where Dividing both sides of (42) by and taking , we obtain that which is a contradiction by assumption . Thus, for some positive number , .
Step 2. is a contractive mapping. Let . From the assumptions on and , it is easy to verify that the following inequality holds: Thus, by the assumptions, we have where we have used the fact that . Hence, so, is a contraction by (23).
Step 3. We show that the operator is compact. Let be such that . The proof will be divided into the following claims.
Claim 1. maps bounded sets into bounded sets in .
Indeed, it is enough to show that there exists a positive constant such that for each , , we have . If , from (32), it follows that By the similar argument as before, we get Therefore, for each , it holds that .
Claim 2. The set is an equicontinuous family of functions on . Let and such that , for every with . For , , , we have Applying assumption , Hölder inequality, together with Lemma 7, we get with . Therefore, for sufficiently small, the right hand side of (51) tends to zero as . Thus, the set is equicontinuous by assumption .
Claim 3. maps into a precompact set in . Let be fixed and a real number satisfying . For , define an operator on by Since belongs to the compact set , the set is precompact in for every , . Moreover, for each , we have An application of Hölder inequality together with Burkhölder-Davis-Gundy's inequality, it follows that From (54), we deduce that for each , Therefore, there are relatively compact sets arbitrary close to the set ; hence, the set is also precompact in .
Thus, from Arzelá-Ascoli theorem together with assumptions on and , we conclude that is a compact operator. Therefore, is a condensing map on . This completes the proof.
Remark 11. In order to describe various real-world problems in physical and engineering sciences subject to abrupt changes at certain instants during the evolution process, impulsive differential equations have been used to model the system. The technique used here can be extended to establish the controllability of neutral fractional stochastic integro-differential systems with impulsive effect and infinite delay. The controllability result can be obtained by suitably introducing the impulsive effects defined in .
In this paper, we have studied the controllability of fractional neutral stochastic integro-differential systems with infinite delay in an abstract space. Through fractional calculus and Sadovskii's fixed point principle, we have investigated the sufficient conditions for the controllability of the system considered.
This work is supported by the National Natural Science Foundation of China (no. 11171062), Innovation Program of Shanghai Municipal Education Commission (no. 12ZZ063), Natural Science Foundation of Anhui Province (no. 1308085QA14), and Key Natural Science Foundation of Anhui Education Commission (no. KJ2013A133).
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