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Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 748091, 13 pages
Controllability of Second-Order Semilinear Impulsive Stochastic Neutral Functional Evolution Equations
1College of Information Sciences and Technology, Donghua University, Shanghai 201620, China
2Engineering Research Center of Digitized Textile & Fashion Technology, Ministry of Education, Donghua University, Shanghai 201620, China
3Department of Mathematics, Donghua University, Shanghai 201620, China
Received 17 June 2012; Accepted 7 August 2012
Academic Editor: Bo Shen
Copyright © 2012 Lei Zhang et al.
We consider a class of impulsive neutral second-order stochastic functional evolution equations. The Sadovskii fixed point theorem and the theory of strongly continuous cosine families of operators are used to investigate the sufficient conditions for the controllability of the system considered. An example is provided to illustrate our results.
Controllability, as a fundamental concept of control theory, plays an important role both in stochastic and deterministic control problems. The study of controllability of linear and nonlinear systems represented by infinite-dimensional systems in Banach spaces has been raised by many authors recently, see Chang , Sakthivel , Ren and Sakthivel , Ntouyas and Regan , Kang et al. , Sakthivel and Mahmudov , and Shubov et al. . With the help of fixed point theorem, Luo [8, 9] and Burton [10–13] have investigated the problem of controllability of the systems in Banach spaces.
Recently, stochastic partial differential equations (SPDEs) arise in the mathematical modeling of various fields in physics and engineering science cited by Sobczyk . Among them, several properties of SPDEs such as existence, controllability, and stability are studied for the first-order equations. But in many situations, it is useful to investigate the second-order abstract differential equations directly rather than to convert them to first-order systems introduced by Fitzgibbon . The second-order stochastic differential equations are the right model in continuous time to account for integrated processes that can be made stationary. For instance, it is useful for engineers to model mechanical vibrations or charge on a capacitor or condenser subjected to white noise excitation by second-order stochastic differential equations. A useful tool for the study of abstract second-order equations is the fixed point theory and the theory of strongly continuous cosine families.
In the past decades, the theory of impulsive differential equations or inclusions is emerging as an active area of investigation due to the application in area such as mechanics, electrical engineering, medicine biology, and ecology, see Benchohra and Henderson , Liu and Willms , Hernández et al. , Prato and Zabczyk , and Fattorini . As an adequate model, impulsive differential equations are used to study the evolution of processes that are subject to sudden changes in their states.
The focus of this paper is the controllability of mild solutions for a class of impulsive neutral second-order stochastic evolution equations of the form: Here, is a stochastic process taking values in a real separable Hilbert space H with inner product and norm . is the infinitesimal generator of a strongly continuous cosine family on H. W is a given K-valued Wiener process with a finite trace nuclear covariance operator Q ≥ 0 defined on a filtered complete probability space and K is another separable Hilbert space with inner product and norm . The fixed time , satisfies , and denote the right and left limits of at , and represents the jump in the state x at time , where are bounded which determine the size of the jump. Similarly and denote, respectively, the right and left limits of at . f, B, g are appropriate mappings specified later; and are -measurable random variables with finite second moment. The main contributions are as follows. The Sadovskii fixed point theorem and the theory of strongly continuous cosine families of operators are used to investigate the sufficient conditions for the controllability of the system considered. The differences of using the fixed point theorem between our proposed method and others are that Sadovskii fixed point theorem is much easier in application, and the condition is easier to be satisfied than other fixed point theorem. To our best knowledge, there are few works about the controllability for mild solutions to second-order semilinear impulsive stochastic neutral functional evolution equations, motivated by the previous problems, our current consideration is on second-order semilinear impulsive stochastic neutral functional evolution equations. We will apply the Sadovskii fixed point theorem to investigate the controllability of mild solution of this class of equations.
The rest of this paper is arranged as follows. In Section 2, we briefly present some basic notations and preliminaries. Section 3 is devoted to the controllability of mild solutions for the system (1.1) and an example is given to illustrate our results in Section 4. Conclusion is given in Section 5.
In this section, we briefly recall some basic definitions and results for stochastic equations in infinite dimensions and cosine families of operators. We refer to Prato and Zabczyk  and Fattorini  for more details. Throughout this paper, let L(K,H) be the set of all linear bounded operators from K into H, equipped with the usual operator norm . Let be a complete probability space furnished with a normal filtration. Suppose is a sequence of real independent one-dimensional standard Brownian motions over . Set where is the complete orthonormal system in K and , a bounded sequence of nonnegative real numbers. Let Q ∈ L(K, K) be an operator defined by , with . The K-valued stochastic process is called a Q-Wiener process. Let be the space of all Hilbert-Schmidt operators from to H with the inner product .
The collection of all strongly measurable, square-integrable H-valued random variables, denoted by , is a Banach space equipped with norm . An important subspace of is given by
Let It is obvious that and are Banach spaces endowed with the norm and , respectively.
To simplify the notations, we put , , and for , we denote by , , the function given by Moreover, for we denote . To prove our results, we need the following lemma introduced in Hernández et al. .
Lemma 2.1. A set is relatively compact in if and only if the set is relatively compact in , for every .
Definition 2.2. The one-parameter family is said to be a strongly continuous cosine family if the following hold:(1);(2)C(t)x is continuous in t on for any ;(3) for all .
The corresponding strongly continuous sine family : is defined by
The (infinitesimal) generator of : is given by for all .
It is known that the infinitesimal generator A is a closed, densely defined operator on H, and the following properties hold, see Travis and Webb .
Proposition 2.3. Suppose that A is the infinitesimal generator of a cosine family of operators . Then, the following hold(i)There exist a pair of constants and such that and hence, .(ii), for all .(iii)There exist such that , for all .The uniform boundedness principle: as a direct consequence we see that both and are uniformly bounded by .
Lemma 2.4. Let Φ be a condensing operator on a Banach space H. If for a convex, closed, and bounded set N of H, then Φ has a fixed point in H.
3. Main Results
In this section we consider the system (1.1). We first present the definition of mild solutions for the system.
Definition 3.1. An adapted stochastic process is said to be a mild solution of the system (1.1) if(1);(2), , ;(3) satisfies the following integral equation:
In this paper, we will work under the following assumptions.(A1) The cosine family of operators on H and the corresponding sine family : are compact for t > 0, and there exists a positive constant M such that (A2) are continuous functions, and there exist some positive constants , such that satisfy the following Lipschitz condition: for all and , and there exist positive constants that satisfy the following linear growth condition: for all and .(A3) are continuous and there exist positive constants , such that for each .(A4) is a continuous operator from to and the linear operator defined by has a bounded invertible operator which takes values in such that , for some positive constants .
We formulate and prove conditions for the approximate controllability of semilinear control differential systems
Theorem 3.2. Assume that (A1)–(A4) are satisfied and , then the system (1.1) is controllable on provided that
Proof. Define the control process with final value Let , for every positive integer N. It is clear that is a bounded closed convex set in for each N. Define an operator by Now let us show that π has a fixed point in which is a solution of (1.1) by Lemma 2.4. This will be done in the next lemmas.
Lemma 3.3. There exists a positive integer N such that .
Proof. This proof can be done by contradiction. In fact, if it is not true, then for each positive number N and , there exists a function , but . That is, . By applying assumptions (A1)–(A4) one can obtain the following estimates:
Dividing both sides of (3.16) by N and taking limit as , we obtain that
which is a contradiction by (3.7). Thus, , for some positive number N.
In what follows, we aim to show that the operator π has a fixed point on , which implies that (1.1) is controllable. To this end, we decompose π as follows: where , are defined on , respectively, by
Lemma 3.4. The operator π1 as above is contractive.
Proof. Let . It follows from assumptions (A1)–(A4) and Hölder’s inequality that which deduces and the lemma follows.
Lemma 3.5. The operator π2 is compact.
Proof. Let be such that .
We first need to prove that the set of functions is equicontinuous on [0, T]. Let and such that and , for every with . For and with we have Noting that , we see that is equicontinuous on [0, T].
We next need to prove that maps into a precompact set in . That is, for every fixed , the set is precompact in . It is obvious that is precompact. Let be fixed and . For , define Since , are compact, it follows that is precompact in H for every . Moreover, for each , we have which means that there are precompact sets arbitrary close to the set . Thus, is precompact in .
Finally, from the assumptions on g, it is obvious that is continuous. Thus, Arzelá-Ascoli theorem yields that is compact. Therefore, π is a condensing map on .
In this section, we now give an example to illustrate the theory obtained. Considering the following impulsive neutral second-order stochastic differential equation: to rewrite (4.1) into the abstract form of (1.1), let , be an operator by with domain It is well known that is the infinitesimal generator of a strongly continuous cosine family in and is given by where is the orthogonal set of eigenvalues of . The associated sine family is compact and is given by Thus, we can impose some suitable conditions on the above functions to verify the condition in Theorem 3.2.
In this paper, we have studied the controllability of second-order impulsive evolution equations. Through the Sadovskii fixed point theorem and the theory of strongly continuous cosine families of operators, we have investigated the sufficient conditions for the controllability of the system considered. At last, an example is provided to show the usefulness and effectiveness of proposed controllability results.
This work was supported in part by the Key Project of the National Nature Science Foundation of China (no. 61134009), the National Nature Science Foundation of China (no. 60975059), Specialized Research Fund for the Doctoral Program of Higher Education from Ministry of Education of China (no. 20090075110002), Specialized Research Fund for Shanghai Leading Talents, and Project of the Shanghai Committee of Science and Technology (nos. 11XD1400100, 11JC1400200, 10JC1400200).
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