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Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 748091, 13 pages
http://dx.doi.org/10.1155/2012/748091
Research Article

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. 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 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.

1. Introduction

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 [1], Sakthivel [2], Ren and Sakthivel [3], Ntouyas and Regan [4], Kang et al. [5], Sakthivel and Mahmudov [6], and Shubov et al. [7]. With the help of fixed point theorem, Luo [8, 9] and Burton [1013] 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 [14]. 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 [15]. 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 [16], Liu and Willms [17], Hernández et al. [18], Prato and Zabczyk [19], and Fattorini [20]. 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: 𝑑𝑥𝑥(𝑡)𝐷𝑡=𝐴𝑥(𝑡)+𝐵𝑢(𝑡)+𝑓𝑡,𝑥𝑡𝑑𝑡+𝑔𝑡,𝑥𝑡[]𝑑𝑤(𝑡),𝑡0,𝑇,𝑡𝑡𝑘𝑡Δ𝑥𝑘=𝐼𝑘𝑥𝑡𝑘,Δ𝑥𝑡𝑘=𝐼𝑘𝑥𝑡𝑘,𝑘=1,,𝑛,𝑥(0)=𝜙,𝑥(0)=𝑦0.(1.1) Here, 𝑥() is a stochastic process taking values in a real separable Hilbert space H with inner product (,) and norm . 𝐴𝐷(𝐴)𝐻𝐻 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 (Ω,𝐹,{𝐹𝑡}𝑡0,𝑃) and K is another separable Hilbert space with inner product (,)𝐾 and norm 𝐾. The fixed time 𝑡𝑘,𝑘=1,,𝑛, satisfies 0<𝑡1<<𝑡𝑛<𝑇, 𝑥(𝑡+𝑘) and 𝑥(𝑡𝑘) denote the right and left limits of 𝑥(𝑡) at 𝑡=𝑡𝑘, and Δ𝑥(𝑡𝑘)=𝑥(𝑡+𝑘)𝑥(𝑡𝑘) represents the jump in the state x at time 𝑡𝑘, where 𝐼𝑘𝐶(𝐻,𝐻)(𝑘=1,1,2,,𝑚) are bounded which determine the size of the jump. Similarly 𝑥(𝑡+𝑘) and 𝑥(𝑡𝑘) denote, respectively, the right and left limits of 𝑥 at 𝑡𝑘. f, B, g are appropriate mappings specified later; 𝑥0 and 𝑦0 are 𝐹0-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.

2. Preliminaries

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 [19] and Fattorini [20] 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 (Ω,𝐹,𝑃) be a complete probability space furnished with a normal filtration{𝐹𝑡}𝑡0. Suppose {𝛽𝑘}𝑘1 is a sequence of real independent one-dimensional standard Brownian motions over (Ω,𝐹,𝑃). Set 𝑊(𝑡)=𝑘=1𝜆𝑘𝛽𝑘(𝑡)𝑒𝑘,𝑡0,(2.1) where {𝑒𝑘}𝑘1 is the complete orthonormal system in K and 𝜆𝑘,𝑘1, a bounded sequence of nonnegative real numbers. Let Q L(K, K) be an operator defined by 𝑄𝑒𝑘=𝜆𝑘𝑒𝑘,𝑘=1,2,, with tr𝑄=𝑘=1𝜆𝑘<. The K-valued stochastic process 𝑊=(𝑊𝑡)𝑡0 is called a Q-Wiener process. Let 𝐿02=𝐿2(𝑄1/2𝐾,𝐻) be the space of all Hilbert-Schmidt operators from 𝑄1/2𝐾 to H with the inner product 𝜑,𝜙𝐿02=tr[𝜑𝑄𝜙].

The collection of all strongly measurable, square-integrable H-valued random variables, denoted by 𝐿2(Ω,𝐻), is a Banach space equipped with norm 𝑥𝐿2=(𝐸𝑥2)1/2. An important subspace of 𝐿2(Ω,𝐻) is given by 𝐿20𝐿(Ω,𝐻)=2(Ω,𝐻)𝑥is𝐹0.measurable(2.2)

Let []=[]=𝐷(0,𝑇,𝐻)𝑥0,𝑇𝐻,𝑥|(𝑡𝑘,𝑡𝑘+1]𝑡𝐶𝑘,𝑡𝑘+1𝑡,𝐻,andthereexists𝑥+𝑘,for𝑘=1,2,,𝑛=[]=𝐷(0,𝑇,𝐻)𝑥,𝑥|(𝑡𝑘,𝑡𝑘+1]𝐶1𝑡𝑘,𝑡𝑘+1,𝐻,andthereexists𝑥𝑡+𝑘.for𝑘=1,2,,𝑛(2.3) It is obvious that 𝐷([0,𝑇],𝐻) and 𝐷([0,𝑇],𝐻) are Banach spaces endowed with the norm 𝑥=sup[]𝑡0,𝑇𝐸𝑥(𝑡)21/2(2.4) and 𝑥=𝑥+𝑥, respectively.

To simplify the notations, we put 𝑡0=0, 𝑡𝑚+1=𝑇, and for 𝑢=𝐻2, we denote by ̃𝑢𝑘𝐶([𝑡𝑘,𝑡𝑘+1],𝐿2(Ω,𝐻)), 𝑘=0,1,,𝑚, the function given by ̃𝑢𝑘𝑢𝑡(𝑡)=(𝑡),𝑡𝑘,𝑡𝑘+1,𝑢𝑡+𝑘,𝑡=𝑡𝑘.(2.5) Moreover, for 𝐵𝐻2 we denote 𝐵𝑘={̃𝑢𝑘𝑢𝐵},𝑘=1,,𝑚. To prove our results, we need the following lemma introduced in Hernández et al. [18].

Lemma 2.1. A set 𝐵 is relatively compact in if and only if the set 𝐵𝑘 is relatively compact in 𝐶([𝑡𝑘,𝑡𝑘+1],𝐻), for every 𝑘=0,1,,𝑚.

Now, we recall some facts about cosine families of operators, see Fattorini [20] and Travis and Webb [21].

Definition 2.2. (1) The one-parameter family {𝐶(𝑡)𝑡}𝐿(𝐻,𝐻) is said to be a strongly continuous cosine family if the following hold:(1)𝐶(0)=𝐼;(2)C(t)x is continuous in t on for any 𝑥𝐻;(3)𝐶(𝑡+𝑠)+𝐶(𝑡𝑠)=2𝐶(𝑡)𝐶(𝑠) for all 𝑡,𝑠.
(2) The corresponding strongly continuous sine family {𝑆(𝑡):𝑡}𝐿(𝐻,𝐻) is defined by 𝑆(𝑡)𝑥=𝑡0𝐶(𝑠)𝑥𝑑𝑠,𝑡,𝑥𝐻.(2.6)
(3) The (infinitesimal) generator 𝐴𝐻𝐻 of {𝐶(𝑡):𝑡𝑅} is given by 𝑑𝐴𝑥=2𝑑𝑡2𝐶||||(𝑡)𝑥𝑡=0,(2.7) for all 𝑥𝐷(𝐴)={𝑥𝐻𝐶()𝑥𝐶2(,𝐻)}.

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 [21].

Proposition 2.3. Suppose that A is the infinitesimal generator of a cosine family of operators {𝐶(𝑡)𝑡}. Then, the following hold(i)There exist a pair of constants 𝑀𝐴1 and 𝛼0 such that 𝐶(𝑡)𝑀𝐴𝑒𝛼|𝑡| and hence, 𝑆(𝑡)𝑀𝐴𝑒𝛼|𝑡|.(ii)𝐴𝑟𝑠𝑆(𝑢)𝑥𝑑𝑢=[𝐶(𝑟)𝐶(𝑠)]𝑥, for all 0𝑠𝑟<.(iii)There exist 𝑁1 such that 𝑆(𝑠)𝑆(𝑟)𝑁|𝑟𝑠𝑒𝛼|𝑠|𝑑𝑠|, for all 0𝑠𝑟<.The uniform boundedness principle: as a direct consequence we see that both {𝐶(𝑡)𝑡[0,𝑇]} and {𝑆(𝑡)𝑡[0,𝑇]} are uniformly bounded by 𝑀=𝑀𝐴𝑒𝛼|𝑇|.

At the end of this section we recall the fixed point theorem of Sadovskii [22] which is used to estimate the controllability of the mild solution to the system (1.1).

Lemma 2.4. Let Φ be a condensing operator on a Banach space H. If Φ(𝑁)𝑁 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 𝐹𝑡adapted stochastic process 𝑥(𝑡)[0,𝑇]𝐻 is said to be a mild solution of the system (1.1) if(1)𝑥0,𝑦0𝐿20(Ω,𝐻);(2)Δ𝑥(𝑡𝑘)=𝑥(𝑡+𝑘)𝑥(𝑡𝑘)=𝐼𝑘(𝑥(𝑡𝑘)), Δ𝑥(𝑡𝑘)=𝑥(𝑡+𝑘)𝑥(𝑡𝑘𝐼)=𝑘(𝑥(𝑡𝑘)), 𝑘=1,,𝑛;(3)𝑥(𝑡) satisfies the following integral equation: 𝑦𝑥(𝑡)=𝐶(𝑡)𝜙(0)+𝑆(𝑡)0+𝐷(0,𝜙)𝑡0𝐶(𝑡𝑠)𝐷𝑠,𝑥𝑠+𝑑𝑠𝑡0𝑆(𝑡𝑠)𝐵𝑢(𝑠)𝑑𝑠+𝑡0𝑆(𝑡𝑠)𝑓𝑠,𝑥𝑠+𝑑𝑠𝑡0𝑆(𝑡𝑠)𝑔𝑠,𝑥𝑠𝑑𝑊(𝑠)+0<𝑡𝑘<𝑡𝐶𝑡𝑡𝑘𝐼𝑘𝑥𝑡𝑘+0<𝑡𝑘<𝑡𝑆𝑡𝑡𝑘𝐼𝑘𝑥𝑡𝑘.(3.1)
In this paper, we will work under the following assumptions.(A1) The cosine family of operators {𝐶(𝑡)𝑡[0,𝑇]} on H and the corresponding sine family {𝑆(𝑡):𝑡[0,𝑇]} are compact for t > 0, and there exists a positive constant M such that 𝐶(𝑡)𝑀,𝑆(𝑡)𝑀.(3.2)(A2)𝐷,𝑓,𝑔 are continuous functions, and there exist some positive constants 𝑀𝐷,𝑀𝑓,𝑀𝑔, such that 𝐷,𝑓,𝑔 satisfy the following Lipschitz condition: 𝐷(𝑡,𝜑)𝐷(𝑡,𝜙)𝑀𝐷𝜑𝜙,𝑓(𝑡,𝜑)𝑓(𝑡,𝜙)𝑀𝑓𝜑𝜙,𝑔(𝑡,𝜑)𝑔(𝑡,𝜙)𝑀𝑔𝜑𝜙,(3.3) for all 𝜑,𝜙𝐻,𝑘=1,,𝑛 and 𝑡[0,𝑇], and there exist positive constants 𝑀𝐷,𝑀𝑓,𝑀𝑔 that satisfy the following linear growth condition: (𝐷𝑡,𝜑)2𝑀𝐷𝜑2,(+1𝑓𝑡,𝜑)2𝑀𝑓𝜑2,(+1𝑔𝑡,𝜑)2𝑀𝑔𝜑2+1(3.4) for all 𝜑,𝜙𝐻,𝑘=1,,𝑛 and 𝑡[0,𝑇].(A3)𝐼𝑘,𝐼𝑘𝐻𝐻 are continuous and there exist positive constants 𝑀𝑘, 𝑁𝑘 such that 𝐼𝑘(𝑥)𝐼𝑘(𝑦)𝑀𝑘𝑥𝑦2,𝐼𝑘𝐼(𝑥)𝑘(𝑦)𝑁𝑘𝑥𝑦2(3.5) for each 𝑥,𝑦𝐻,𝑘=1,,𝑛.(A4)𝐵 is a continuous operator from Ω to 𝐻 and the linear operator 𝑊𝐿20(Ω,𝐻)𝑋 defined by 𝑊𝑢=𝑇0𝑆(𝑇𝑠)𝐵𝑢(𝑠)𝑑𝑠(3.6) has a bounded invertible operator 𝑊1 which takes values in 𝐿20(Ω,𝐻)/ker𝑊 such that ||𝐵||𝑀1,||𝑊1||𝑀2, for some positive constants 𝑀1,𝑀2.

We formulate and prove conditions for the approximate controllability of semilinear control differential systems

Theorem 3.2. Assume that (A1)–(A4) are satisfied and 𝑥0,𝑦0𝐿20(Ω,𝐻), then the system (1.1) is controllable on [0,𝑇] provided that 8𝑀2𝑇𝑀2𝐷+𝑇𝑀2𝑓+tr(𝑄)𝑀2𝑔+2𝑀2𝑛𝑘=1𝑀𝑘+2𝑀2𝑛𝑘=1𝑁𝑘+8𝑀2𝑇𝑀2𝐷+𝑇𝑀2𝑓+tr(𝑄)𝑀2𝑔+2𝑀2𝑛𝑘=1𝑀𝑘+2𝑀2𝑛𝑘=1𝑁𝑘<1.(3.7)

Proof. Define the control process with final value 𝜉=𝑥(𝑇)𝑢𝑇𝑥(𝑡)=𝑊1𝑦𝜉𝑆(𝑇)0𝐷(0,𝜙)𝐶(𝑇)𝜙(0)𝑇0𝐶(𝑇𝑠)𝐷𝑠,𝑥𝑠𝑑𝑠𝑇0𝑆(𝑇𝑠)𝑓𝑠,𝑥𝑠𝑑𝑠𝑇0𝑆(𝑇𝑠)𝑔𝑠,𝑥𝑠𝑑𝑊(𝑠)0<𝑡𝑘<𝑡𝐶𝑇𝑡𝑘𝐼𝑘𝑥𝑡𝑘0<𝑡𝑘<𝑡𝑆𝑇𝑡𝑘𝐼𝑘𝑥𝑡𝑘(𝑡).(3.8) Let 𝐵𝑁={𝑥𝐻2𝑥2𝑁}, for every positive integer N. It is clear that 𝐵𝑁 is a bounded closed convex set in 𝐻2 for each N. Define an operator 𝜋𝐻2𝐻2 by (𝑦𝜋𝑥)(𝑡)=𝐶(𝑡)𝜙(0)+𝑆(𝑡)0+𝐷(0,𝜙)𝑡0𝐶(𝑡𝑠)𝐷𝑠,𝑥𝑠𝑑𝑠+𝑡0+𝑆(𝑡𝑠)𝐵𝑢(𝑠)𝑑𝑠𝑡0𝑆(𝑡𝑠)𝑓𝑠,𝑥𝑠𝑑𝑠+𝑡0𝑆(𝑡𝑠)𝑔𝑠,𝑥𝑠+𝑑𝑊(𝑠)0<𝑡𝑘<𝑡𝐶𝑡𝑡𝑘𝐼𝑘𝑥𝑡𝑘+0<𝑡𝑘<𝑡𝑆𝑡𝑡𝑘𝐼𝑘𝑥𝑡𝑘.(3.9) Now let us show that π has a fixed point in 𝐻2 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 𝜋(𝐵𝑁)𝐵𝑁.

Proof. This proof can be done by contradiction. In fact, if it is not true, then for each positive number N and 𝑡𝑁[0,𝑇], there exists a function 𝑥𝑁𝐵𝑁, but 𝜋(𝑥𝑁)(𝑡𝑁)𝐵𝑁. That is, 𝐸𝜋(𝑥𝑁)(𝑡𝑁)2>𝑁. By applying assumptions (A1)–(A4) one can obtain the following estimates: 𝐸0<𝑡𝑘<𝑡𝑁𝑆𝑡𝑁𝑡𝑘𝐼𝑘𝑥𝑁𝑡𝑘2𝑁𝑀20<𝑡𝑘<𝑇𝐸𝐼𝑘𝑥𝑁𝑡𝑘𝐼𝑘𝐼(0)+𝑘(0)2𝑁𝑀2𝑁𝑘=1𝑁𝑘𝐸𝑥𝑁𝑡𝑘2+𝑁𝑘=1𝐼𝑘(0)2,𝐸(3.10)0<𝑡𝑘<𝑡𝑁𝐶𝑡𝑁𝑡𝑘𝐼𝑘𝑥𝑁𝑡𝑘2𝑁𝑀20<𝑡𝑘<𝑇𝐸𝐼𝑘𝑥𝑁𝑡𝑘𝐼𝑘(0)+𝐼𝑘(0)2𝑁𝑀2𝑁𝑘=1𝑀𝑘𝐸𝑥𝑁𝑡𝑘2+𝑁𝑘=1𝐼𝑘(0)2,𝐸(3.11)𝑡𝑁0𝑆𝑡𝑁𝑔𝑠𝑠,𝑥𝑠𝑑𝑊(𝑠)2tr(𝑄)𝑀2𝑡𝑁0𝐸𝑔𝑠,𝑥𝑠2𝑑𝑠tr(𝑄)𝑀2𝑀2𝑔𝑡𝑁0𝐸𝜑2𝐸+1𝑑𝑠,(3.12)𝑡𝑁0𝐶𝑡𝑁𝐷𝑥𝑠𝑠𝑑𝑠2𝑇𝑀2𝑀2𝐷𝑡𝑁0𝐸𝜑2𝐸+1𝑑𝑠,(3.13)𝑡𝑁0𝑆𝑡𝑁𝑓𝑠𝑠,𝑥𝑠𝑑𝑠2𝑇𝑀2𝑀2𝑓𝑡𝑁0𝐸𝜑2𝐸+1𝑑𝑠,(3.14)𝑡𝑁0𝑆𝑡𝑁𝑠𝐵𝑢(𝑠)𝑑𝑠28𝑀2𝑀2𝜉2+𝜑(0)2+𝑦20+(𝑇+1)𝑀2𝐷𝑡𝑁0𝐸𝜑2+1𝑑𝑠+𝑇𝑀2𝑓𝑡𝑁0𝐸𝜑2++1𝑑𝑠𝑀2𝑔𝑡𝑁0𝐸𝜑2+1𝑑𝑠+2𝑁𝑁𝑘=1𝑁𝑘𝐸𝑥𝑁𝑡𝑘2+2𝑁𝑁𝑘=1𝑀𝑘𝐸𝑥𝑁𝑡𝑘2=𝑀2𝑈(3.15) which gives 𝑁𝐸𝜋𝑥𝑁𝑡𝑁2𝐶𝑡8𝐸𝑁[]𝜑(0)2𝑆𝑡+8𝐸𝑁𝑦0𝐷(0,𝜑)2+8𝐸𝑡𝑁0𝐶𝑡𝑁𝐷𝑠(𝑠,𝜑)𝑑𝑠2+8𝐸𝑡𝑁0𝑆𝑡𝑁𝑓𝑠(𝑠,𝜑)𝑑𝑠2+8𝐸𝑡𝑁0𝑆𝑡𝑁𝑠𝑔(𝑠,𝜑)𝑑𝑊(𝑠)2+8𝐸0<𝑡𝑘<𝑡𝑁𝐶𝑡𝑁𝑡𝑘𝐼𝑘𝑥𝑁𝑡𝑘2+8𝐸0<𝑡𝑘<𝑡𝑁𝑆𝑡𝑁𝑡𝑘𝐼𝑘𝑥𝑁𝑡𝑘2+8𝐸𝑡𝑁0𝑆𝑡𝑁𝑠𝐵𝑢(𝑠)𝑑𝑠2𝐿+8𝑀2𝑇𝑀2𝐷𝑁+𝑇𝑀2𝑓𝑁+tr(𝑄)𝑀2𝑔𝑁+2𝑁𝑀2𝑛𝑘=1𝑀𝑘+2𝑁𝑀2𝑛𝑘=1𝑁𝑘+8𝑀2𝑇𝑀2𝐷𝑁+𝑇𝑀2𝑓𝑁+tr(𝑄)𝑀2𝑔𝑁+2𝑁𝑀2𝑛𝑘=1𝑀𝑘+2𝑁𝑀2𝑛𝑘=1𝑁𝑘,(3.16) where 𝐿=8𝑀2𝐸𝑥02𝑦+𝐸02+𝑇𝑀2𝐷+𝑇𝑀2𝑓+tr(𝑄)𝑀2𝑔+2𝑀2𝑛𝑘=1𝑀𝑘+2𝑀2𝑛𝑘=1𝑁𝑘+8𝑀2𝑇𝑀2𝐷+𝑇𝑀2𝑓+tr(𝑄)𝑀2𝑔+2𝑀2𝑛𝑘=1𝑀𝑘+2𝑀2𝑛𝑘=1𝑁𝑘.(3.17) Dividing both sides of (3.16) by N and taking limit as 𝑁, we obtain that 8𝑀2𝑇𝑀2𝐷+𝑇𝑀2𝑓+tr(𝑄)𝑀2𝑔+2𝑀2𝑛𝑘=1𝑀𝑘+2𝑀2𝑛𝑘=1𝑁𝑘+8𝑀2𝑇𝑀2𝐷+𝑇𝑀2𝑓+tr(𝑄)𝑀2𝑔+2𝑀2𝑛𝑘=1𝑀𝑘+2𝑀2𝑛𝑘=1𝑁𝑘1(3.18) which is a contradiction by (3.7). Thus, 𝜋(𝐵𝑁)𝐵𝑁, for some positive number N.
In what follows, we aim to show that the operator π has a fixed point on 𝐵𝑁, which implies that (1.1) is controllable. To this end, we decompose π as follows: 𝜋=𝜋1+𝜋2,(3.19) where 𝜋1, 𝜋2 are defined on 𝐵𝑁, respectively, by 𝜋1𝑥(𝑦𝑡)=𝑆(𝑡)0+𝐷(0,𝜑)𝑡0𝐶(𝑡𝑠)𝐷(0,𝜑)𝑑𝑠+𝑡0𝑆(𝑡𝑠)𝑓𝑠,𝑥𝑠+𝑑𝑠0<𝑡𝑘<𝑡𝐶𝑡𝑡𝑘𝐼𝑘𝑥𝑡𝑘+0<𝑡𝑘<𝑡𝑆𝑡𝑡𝑘𝐼𝑘𝑥𝑡𝑘,𝜋(3.20)2𝑥(𝑡)=𝐶(𝑡)𝜙(0)+𝑡0𝑆(𝑡𝑠)𝑔𝑠,𝑥𝑠𝑑𝑊(𝑠)+𝑡0𝑆(𝑡𝑠)𝐵𝑢(𝑠)𝑑𝑠.(3.21)

Lemma 3.4. The operator π1 as above is contractive.

Proof. Let 𝑥,𝑦𝐵𝑁. It follows from assumptions (A1)–(A4) and Hölder’s inequality that 𝐸𝜋1𝑥𝜋(𝑡)1𝑦(𝑡)2[]5𝐸𝑆(𝑡)𝐷(0,𝜑)𝐷(0,𝜙)2+5𝐸𝑡0[]𝐶(𝑡𝑠)𝐷(0,𝜑)𝐷(0,𝜙)𝑑𝑠2+5𝐸𝑡0[]𝑆(𝑡𝑠)𝑓(𝑠,𝜑)𝑓(𝑠,𝜙)𝑑𝑠2+5𝐸0<𝑡𝑘<𝑡𝐶𝑡𝑡𝑘𝐼𝑘𝑥𝑡𝑘𝐼𝑘𝑦𝑡𝑘2+5𝐸0<𝑡𝑘<𝑡𝑆𝑡𝑡𝑘𝐼𝑘𝑥𝑡𝑘𝐼𝑘𝑦𝑡𝑘25𝑀2𝑀2𝐷sup[]𝑠0,𝑇𝐸𝑥(𝑠)𝑦(𝑠)2+5𝑇𝑀2𝑀2𝐷sup[]𝑠0,𝑇𝐸𝑥(𝑠)𝑦(𝑠)2+5𝑇𝑀2𝑀2𝑓sup[]𝑠0,𝑇𝐸𝑥(𝑠)𝑦(𝑠)2+5𝑛𝑀20<𝑡𝑘<𝑡𝑀𝑘𝐸𝑥𝑡𝑘𝑡𝑦𝑘2+5𝑛𝑀20<𝑡𝑘<𝑡𝑁𝑘𝐸𝑥𝑡𝑘𝑡𝑦𝑘2(3.22) which deduces sup[]𝑠0,𝑇𝐸𝜋1𝑥𝜋(𝑠)1𝑦(𝑠)25𝑀2𝑀2𝐷+𝑇𝑀2𝐷+𝑇𝑀2𝑓+𝑛𝑛𝑖=0𝑀𝑘+𝑛𝑛𝑖=0𝑁𝑘sup[]𝑠0,𝑇𝐸𝑥(𝑠)𝑦(𝑠)2(3.23) and the lemma follows.

Lemma 3.5. The operator π2 is compact.

Proof. Let 𝑁>0 be such that 𝜋2(𝐵𝑁)𝐵𝑁.
We first need to prove that the set of functions 𝜋2(𝐵𝑁) is equicontinuous on [0, T]. Let 0<𝜀<𝑡<𝑇 and 𝛿>0 such that 𝑆(𝑠)𝑥𝑆(𝑠)𝑥2<𝜀 and 𝐶(𝑠)𝑥𝐶(𝑠)𝑥2<𝜀, for every 𝑠,𝑠[0,𝑇] with |𝑠𝑠|𝛿. For 𝑥𝐵𝑁 and 0<||<𝛿 with 𝑡+[0,𝑇] we have 𝐸𝜋2𝑥𝜋(𝑡+)2𝑥(𝑡)2[]3𝐸𝐶(𝑡+)𝐶(𝑡)𝜙(0)2+3𝐸𝑡0[]𝑔𝑆(𝑡+𝑠)𝑆(𝑡𝑠)𝑠,𝑥𝑠𝑑𝑊(𝑠)𝑡𝑡+𝑆(𝑡+𝑠)𝑔𝑠,𝑥𝑠𝑑𝑊(𝑠)2+3𝐸𝑡0[]𝑆(𝑡+𝑠)𝑆(𝑡𝑠)𝐵𝑢(𝑠)𝑑𝑠𝑡𝑡+𝑆(𝑡+𝑠)𝐵𝑢(𝑠)𝑑𝑠23𝜀𝐸𝜙(0)2+6tr(𝑄)𝑀2𝑡𝑡+𝐸𝑔𝑠,𝑥(𝑠),𝑥𝑠2𝑑𝑠+6𝑀2𝑡𝑡+𝐸𝐵𝑢(𝑠)2𝑑𝑠+6𝑀2𝑡0𝐸𝐵𝑢(𝑠)2𝑑𝑠+6tr(𝑄)𝑡0𝐸[]𝑔𝑆(𝑡+𝑠)𝑆(𝑡𝑠)𝑠,𝑥𝑠2𝑥𝑑𝑠4𝜀𝐸02+4𝜀𝐸𝑔(𝑥)2+4𝜀tr(𝑄)𝑡0𝐸𝑔𝑠,𝑥𝑠2𝑑𝑠+4tr(𝑄)𝑀2𝑡𝑡+𝐸𝑔(𝑠,𝑠𝑥(𝑠))2𝑑𝑠.(3.24) Noting that 𝐸𝑔(𝑠,𝑠𝑥(𝑠))2𝑁(𝑠)𝐿1([0,𝑇]), we see that 𝜋2(𝐵𝑁) is equicontinuous on [0, T].
We next need to prove that 𝜋2 maps 𝐵𝑁 into a precompact set in 𝐵𝑁. That is, for every fixed 𝑡[0,𝑇], the set 𝑉(𝑡)={(𝜋2𝑥)(𝑡)𝑥𝐵𝑁} is precompact in 𝐵𝑁. It is obvious that 𝑉(0)={(𝜋2𝑥)(0)} is precompact. Let 0<𝑡𝑇 be fixed and 0<𝜀<𝑡. For 𝑥𝐵𝑁, define 𝜋𝜀2𝑥(𝑡)=𝐶(𝑡)𝜙(0)+0𝑡𝜀𝑆(𝑡𝑠)𝑔𝑠,𝑥𝑠𝑑𝑊(𝑠)+0𝑡𝜀𝑆(𝑡𝑠)𝐵𝑢(𝑠)𝑑𝑠=𝐶(𝑡)𝜙(0)+𝑆(𝜀)0𝑡𝜀𝑆(𝑡𝜀𝑠)𝑔𝑠,𝑥𝑠𝑑𝑊(𝑠)+𝑆(𝜀)0𝑡𝜀𝑆(𝑡𝜀𝑠)𝐵𝑢(𝑠)𝑑𝑠.(3.25) Since 𝐶(𝑡),𝑆(𝑡),𝑡>0, are compact, it follows that 𝑉𝜀(𝑡)={(𝜋𝜀2𝑥)(𝑡)𝑥𝐵𝑁} is precompact in H for every 0<𝜀<𝑡. Moreover, for each 𝑥𝐵𝑁, we have 𝐸𝜋2𝑥(𝜋𝑡)𝜀2𝑥(𝑡)22tr(𝑄)𝑀2𝑡𝑡𝜀𝐸𝑔𝑠,𝑥𝑠2𝑑𝑠+2𝑀2𝑡𝑡𝜀𝐸𝐵𝑢(𝑠)2𝑑𝑠𝜀2𝑀2tr(𝑄)𝐸𝜑2+1+𝑈0𝑎𝑠𝜀0+(3.26) which means that there are precompact sets arbitrary close to the set 𝑉(𝑡). Thus, 𝑉(𝑡) is precompact in 𝐵𝑁.
Finally, from the assumptions on g, it is obvious that 𝜋2 is continuous. Thus, Arzelá-Ascoli theorem yields that 𝜋2 is compact. Therefore, π is a condensing map on 𝐵𝑁.

4. Applications

In this section, we now give an example to illustrate the theory obtained. Considering the following impulsive neutral second-order stochastic differential equation: 𝑑𝜕𝑥(𝑡,𝑧)=𝜕𝜕𝑡+𝑎(𝑡)𝑥(𝑡,𝑧)2𝜕𝑧2[][],𝑥(𝑡,𝑧)𝑑𝑡+𝜎(𝑡,𝑥(𝑡,𝑧))𝑑𝑊(𝑡),𝑡0,1𝑥(𝑡,0)=𝑥(𝑡,𝜋)=0,𝑡0,1𝜕𝑥(0,𝑧)𝜕𝑡=𝑥1[]𝑡(𝑧),𝑧0,𝜋Δ𝑥𝑘(𝑧)=𝐼𝑘𝑥𝑡𝑘(𝑧),Δ𝑥𝑡𝑘𝐼(𝑧)=𝑘𝑥𝑡𝑘(𝑧),𝑡=𝑡𝑘,(4.1) to rewrite (4.1) into the abstract form of (1.1), let 𝐻=𝐿2[0,𝜋], 𝐴𝐻𝐻 be an operator by 𝐴𝑥=𝑥 with domain 𝐷(𝐴)=𝑥𝐻𝑥,𝑥areabsolutelycontinuous,𝑥𝐻,𝑥(0)=𝑥(𝜋)=0.(4.2) It is well known that 𝐴 is the infinitesimal generator of a strongly continuous cosine family {𝐶(𝑡)𝑡𝑅} in 𝐻 and is given by 𝐶(𝑡)𝑥=𝑛=1cos(𝑛𝑡)𝑥,𝑒𝑛𝑒𝑛,𝑥𝐻,(4.3) where 𝑒𝑛(𝜉)=2/𝜋sin(𝑛𝜉)and𝑖=1,2, is the orthogonal set of eigenvalues of 𝐴. The associated sine family {𝑆(𝑡)𝑡>0} is compact and is given by 𝑆(𝑡)𝑥=𝑛=11𝑛sin(𝑛𝑡)𝑥,𝑒𝑛𝑒𝑛,𝑥𝐻.(4.4) Thus, we can impose some suitable conditions on the above functions to verify the condition in Theorem 3.2.

5. Conclusions

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.

Acknowledgments

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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