Controllability of Second-Order Semilinear Impulsive Stochastic Neutral Functional Evolution Equations

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 [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 [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: 𝑑𝑥^(𝑡)−𝐷𝑥_𝑡=𝐴𝑥(𝑡)+𝐵𝑢(𝑡)+𝑓𝑡,𝑥_𝑡𝑑𝑡+𝑔𝑡,𝑥_𝑡𝑑𝑤(𝑡),𝑡∈[0,𝑇],𝑡≠𝑡_𝑘Δ𝑥𝑡_𝑘=𝐼_𝑘𝑥𝑡_𝑘,Δ𝑥^𝑡_𝑘=𝐼_𝑘𝑥𝑡_𝑘,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 ‖⋅‖. 𝐴∶𝐷(𝐴)⊂𝐻→𝐻 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,{𝐹_𝑡}_𝑡≥0,𝑃) and K is another separable Hilbert space with inner product (⋅,⋅)_𝐾 and norm ‖⋅‖_𝐾. The fixed time 𝑡_𝑘,𝑘=1,…,𝑛, satisfies 0<𝑡₁<⋯<𝑡_𝑛<𝑇, 𝑥(𝑡^+𝑘) and 𝑥(𝑡^−𝑘) denote the right and left limits of 𝑥(𝑡) at 𝑡=𝑡_𝑘, and Δ𝑥(𝑡_𝑘)=𝑥(𝑡^+𝑘)−𝑥(𝑡^−𝑘) represents the jump in the state x at time 𝑡_𝑘, where 𝐼_𝑘∈𝐶(𝐻,𝐻)(𝑘=1,1,2,…,𝑚) are bounded which determine the size of the jump. Similarly 𝑥^(𝑡^+𝑘) and 𝑥^(𝑡^−𝑘) denote, respectively, the right and left limits of 𝑥^ at 𝑡_𝑘. 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{𝐹_𝑡}_𝑡≥0. Suppose {𝛽_𝑘}_𝑘≥1 is a sequence of real independent one-dimensional standard Brownian motions over (Ω,𝐹,𝑃). Set 𝑊(𝑡)=^∞_𝑘=1√𝜆_𝑘𝛽_𝑘(𝑡)𝑒_𝑘,𝑡≥0,(2.1) where {𝑒_𝑘}_𝑘≥1 is the complete orthonormal system in K and 𝜆_𝑘,𝑘≥1, a bounded sequence of nonnegative real numbers. Let Q ∈ L(K, K) be an operator defined by 𝑄𝑒_𝑘=𝜆_𝑘𝑒_𝑘,𝑘=1,2,…, with tr𝑄=∑^∞𝑘=1𝜆_𝑘<∞. The K-valued stochastic process 𝑊=(𝑊_𝑡)_𝑡≥0 is called a Q-Wiener process. Let 𝐿⁰²=𝐿₂(𝑄^1/2𝐾,𝐻) be the space of all Hilbert-Schmidt operators from 𝑄^1/2𝐾 to H with the inner product ⟨𝜑,𝜙⟩_𝐿⁰²=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 ‖𝑥‖_𝐿²=(𝐸‖𝑥‖²)^1/2. An important subspace of 𝐿²(Ω,𝐻) is given by 𝐿²⁰(Ω,𝐻)=𝐿²(Ω,𝐻)∋𝑥is𝐹₀−measurable.(2.2)

Let ℘∶==𝐷𝑥(∶[0[,0𝑇,𝑇],]𝐻→)𝐻,𝑥|_{(𝑡𝑘,𝑡𝑘+1]}∈𝐶𝑡_𝑘,𝑡_𝑘+1,𝐻,andthereexists𝑥𝑡^+𝑘for𝑘=1,2,…,𝑛,℘∶==𝐷𝑥(∈[0℘,𝑇,]𝑥,|𝐻)_{(𝑡𝑘,𝑡𝑘+1]}∈𝐶¹𝑡_𝑘,𝑡_𝑘+1,𝐻,andthereexists𝑥^𝑡^+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_{[]𝑡∈0,𝑇}𝐸‖𝑥(𝑡)‖²^1/2(2.4) and ‖𝑥‖_℘=‖𝑥‖_℘+‖𝑥′‖_℘, respectively.

To simplify the notations, we put 𝑡₀=0, 𝑡_𝑚+1=𝑇, and for 𝑢=𝐻₂, we denote by 𝑢_𝑘̃∈𝐶([𝑡_𝑘,𝑡_𝑘+1],𝐿²(Ω,𝐻)), 𝑘=0,1,…,𝑚, the function given by 𝑢_𝑘̃(𝑡)=𝑢(𝑡),𝑡∈𝑡^𝑘𝑢𝑡_+𝑘,𝑡_𝑘+1,,𝑡=𝑡_𝑘.(2.5) Moreover, for 𝐵⊂𝐻₂ we denote 𝐵_𝑘={𝑢_𝑘̃∶𝑢∈𝐵},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 𝐵_𝑘 is relatively compact in 𝐶([𝑡_𝑘,𝑡_𝑘+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∈ℝ}⊂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 ℝ for any 𝑥∈𝐻;(3)𝐶(𝑡+𝑠)+𝐶(𝑡−𝑠)=2𝐶(𝑡)𝐶(𝑠) for all 𝑡,𝑠∈ℝ.
(2) The corresponding strongly continuous sine family {𝑆(𝑡):𝑡∈ℝ}⊂𝐿(𝐻,𝐻) is defined by 𝑆(𝑡)𝑥=^𝑡0𝐶(𝑠)𝑥𝑑𝑠,𝑡∈ℝ,𝑥∈𝐻.(2.6)
(3) The (infinitesimal) generator 𝐴∶𝐻→𝐻 of {𝐶(𝑡):𝑡∈𝑅} is given by 𝐴𝑥=𝑑²𝑑𝑡²𝐶(𝑡)𝑥||||_𝑡=0,(2.7) for all 𝑥∈𝐷(𝐴)={𝑥∈𝐻∶𝐶(⋅)𝑥∈𝐶²(ℝ,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∈ℝ}. Then, the following hold(i)There exist a pair of constants 𝑀_𝐴≥1 and 𝛼≥0 such that ‖𝐶(𝑡)‖≤𝑀_𝐴𝑒^𝛼|𝑡| and hence, ‖𝑆(𝑡)‖≤𝑀_𝐴𝑒^𝛼|𝑡|.(ii)𝐴∫^𝑟𝑠𝑆(𝑢)𝑥𝑑𝑢=[𝐶(𝑟)−𝐶(𝑠)]𝑥, for all 0≤𝑠≤𝑟<∞.(iii)There exist 𝑁≥1 such that ‖𝑆(𝑠)−𝑆(𝑟)‖≤𝑁|∫^𝑟𝑠𝑒^𝛼|𝑠|𝑑𝑠|, for all 0≤𝑠≤𝑟<∞.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^∗=𝑀_𝐴𝑒^𝛼|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 𝐹_𝑡−adapted stochastic process 𝑥(𝑡)∶[0,𝑇]→𝐻 is said to be a mild solution of the system (1.1) if(1)𝑥₀,𝑦₀∈𝐿²⁰(Ω,𝐻);(2)Δ𝑥(𝑡_𝑘)=𝑥(𝑡^+𝑘)−𝑥(𝑡^−𝑘)=𝐼_𝑘(𝑥(𝑡_𝑘)), Δ𝑥′(𝑡_𝑘)=𝑥′(𝑡^+𝑘)−𝑥′(𝑡^−𝑘)=𝐼_𝑘(𝑥(𝑡_𝑘)), 𝑘=1,…,𝑛;(3)𝑥(𝑡) satisfies the following integral equation: 𝑥(𝑡)=𝐶+(𝑡)𝜙(0)+𝑆(𝑡)𝑦₀−𝐷(0,𝜙)+^𝑡0𝐶(𝑡−𝑠)𝐷𝑠,𝑥_𝑠𝑑𝑠_𝑡0+𝑆(𝑡−𝑠)𝐵𝑢(𝑠)𝑑𝑠+^𝑡0𝑆(𝑡−𝑠)𝑓𝑠,𝑥_𝑠𝑑𝑠_𝑡0+𝑆(𝑡−𝑠)𝑔𝑠,𝑥_𝑠𝑑𝑊(𝑠)+_{0<𝑡𝑘<𝑡}𝐶𝑡−𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘_{0<𝑡𝑘<𝑡}𝑆𝑡−𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘.(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 𝑀_𝐷,𝑀_𝑓,𝑀_𝑔, such that 𝐷,𝑓,𝑔 satisfy the following Lipschitz condition: ‖‖𝐷𝑓((𝑡𝑡,,𝜑𝜑))−−𝐷𝑓((𝑡𝑡,,𝜙𝜙))‖‖≤≤𝑀𝑀_𝐷𝑓‖𝜑−𝜙‖,‖𝑔(𝑡,𝜑)−𝑔(𝑡,𝜙)‖≤𝑀_𝑔‖𝜑−𝜙‖,‖𝜑−𝜙‖,(3.3) for all 𝜑,𝜙∈𝐻,𝑘=1,…,𝑛 and 𝑡∈[0,𝑇], and there exist positive constants 𝑀_𝐷,𝑀_𝑓,𝑀_𝑔 that satisfy the following linear growth condition: ‖‖𝐷𝑓((𝑡𝑡,,𝜑𝜑))‖‖²₂≤𝑀_𝐷‖𝜑‖²+1,‖𝑔(𝑡,𝜑)‖₂≤𝑀_𝑓‖𝜑‖²+1,≤𝑀_𝑔‖𝜑‖²+1(3.4) for all 𝜑,𝜙∈𝐻,𝑘=1,…,𝑛 and 𝑡∈[0,𝑇].(A3)𝐼_𝑘,𝐼_𝑘∶𝐻→𝐻 are continuous and there exist positive constants 𝑀_𝑘, 𝑁_𝑘 such that ‖‖𝐼_𝑘(𝑥)−𝐼_𝑘(𝑦)‖‖≤𝑀_𝑘‖𝑥−𝑦‖²,‖‖𝐼_𝑘(𝑥)−𝐼_𝑘(𝑦)‖‖≤𝑁_𝑘‖𝑥−𝑦‖²(3.5) for each 𝑥,𝑦∈𝐻,𝑘=1,…,𝑛.(A4)𝐵 is a continuous operator from Ω to 𝐻 and the linear operator 𝑊∶𝐿²⁰(Ω,𝐻)→𝑋 defined by 𝑊𝑢=^𝑇0𝑆(𝑇−𝑠)𝐵𝑢(𝑠)𝑑𝑠(3.6) has a bounded invertible operator 𝑊⁻¹ which takes values in 𝐿²⁰(Ω,𝐻)/ker𝑊 such that ||𝐵||≤𝑀₁,||𝑊⁻¹||≤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,𝑦₀∈𝐿²⁰(Ω,𝐻), then the system (1.1) is controllable on [0,𝑇] provided that 8𝑀²𝑇𝑀^2𝐷+𝑇𝑀^2𝑓+tr(𝑄)𝑀^2𝑔+2𝑀^2𝑛_𝑘=1𝑀_𝑘+2𝑀^2𝑛_𝑘=1𝑁_𝑘+8𝑀₂𝑇𝑀^2𝐷+𝑇𝑀^2𝑓+tr(𝑄)𝑀^2𝑔+2𝑀^2𝑛_𝑘=1𝑀_𝑘+2𝑀^2𝑛_𝑘=1𝑁_𝑘<1.(3.7)

Proof. Define the control process with final value 𝜉=𝑥(𝑇)𝑢^𝑇𝑥(𝑡)=𝑊⁻¹⎧⎪⎨⎪⎩𝜉−𝑆(𝑇)𝑦₀−−𝐷(0,𝜙)−𝐶(𝑇)𝜙(0)−^𝑇0𝐶(𝑇−𝑠)𝐷𝑠,𝑥_𝑠𝑑𝑠_𝑇0−𝑆(𝑇−𝑠)𝑓𝑠,𝑥_𝑠𝑑𝑠−^𝑇0𝑆(𝑇−𝑠)𝑔𝑠,𝑥_𝑠𝑑𝑊(𝑠)_{0<𝑡𝑘<𝑡}𝐶𝑇−𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘−_{0<𝑡𝑘<𝑡}𝑆𝑇−𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘⎫⎪⎬⎪⎭(𝑡).(3.8) Let 𝐵_𝑁={𝑥∈𝐻₂∶‖𝑥‖^2℘≤𝑁}, for every positive integer N. It is clear that 𝐵_𝑁 is a bounded closed convex set in 𝐻₂ for each N. Define an operator 𝜋∶𝐻₂→𝐻₂ by (𝜋𝑥)(𝑡)=𝐶+(𝑡)𝜙(0)+𝑆(𝑡)𝑦₀−𝐷(0,𝜙)+^𝑡0𝐶(𝑡−𝑠)𝐷𝑠,𝑥_𝑠𝑑𝑠+^𝑡0𝑆(𝑡−𝑠)𝐵𝑢(𝑠)𝑑𝑠_𝑡0+𝑆(𝑡−𝑠)𝑓𝑠,𝑥_𝑠𝑑𝑠+^𝑡0𝑆(𝑡−𝑠)𝑔𝑠,𝑥_𝑠𝑑𝑊(𝑠)_{0<𝑡𝑘<𝑡}𝐶𝑡−𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝑆𝑡−𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘.(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(𝐵_𝑁)⊂𝐵_𝑁.

Proof. This proof can be done by contradiction. In fact, if it is not true, then for each positive number N and 𝑡^𝑁∈[0,𝑇], there exists a function 𝑥^𝑁∈𝐵_𝑁, but 𝜋(𝑥^𝑁)(𝑡^𝑁)∉𝐵_𝑁. That is, 𝐸‖𝜋(𝑥^𝑁)(𝑡^𝑁)‖²>𝑁. By applying assumptions (A1)–(A4) one can obtain the following estimates: 𝐸‖‖‖‖_{0<𝑡𝑘<𝑡𝑁}𝑆𝑡^𝑁−𝑡_𝑘𝐼_𝑘𝑥^𝑁𝑡_𝑘‖‖‖‖²≤𝑁𝑀²_0<_{𝑡𝑘<𝑇}≤2𝑁𝑀₂𝐸‖‖𝐼_𝑘𝑥^𝑁𝑡_𝑘−𝐼_𝑘(0)+𝐼_𝑘(0)‖‖^𝑁_𝑘=1𝑁_𝑘𝐸‖‖𝑥^𝑁𝑡_𝑘‖‖²+^𝑁_𝑘=1‖‖𝐼_𝑘(0)‖‖²𝐸‖‖‖‖,(3.10)_{0<𝑡𝑘<𝑡𝑁}𝐶𝑡^𝑁−𝑡_𝑘𝐼_𝑘𝑥^𝑁𝑡_𝑘‖‖‖‖²≤𝑁𝑀²_0<_{𝑡𝑘<𝑇}≤2𝑁𝑀₂𝐸‖‖𝐼_𝑘𝑥^𝑁𝑡_𝑘−𝐼_𝑘(0)+𝐼_𝑘(0)‖‖^𝑁_𝑘=1𝑀_𝑘𝐸‖‖𝑥^𝑁𝑡_𝑘‖‖²+^𝑁_𝑘=1‖‖𝐼_𝑘(0)‖‖²𝐸‖‖‖‖,(3.11)_𝑡𝑁0𝑆𝑡^𝑁−𝑠𝑔𝑠,𝑥_𝑠𝑑𝑊(𝑠)‖‖‖‖²≤tr(𝑄)𝑀²^𝑡𝑁₀≤tr(𝑄)𝑀₂𝐸‖‖𝑔𝑠,𝑥_𝑠‖‖²𝑑𝑠𝑀^2𝑔^𝑡𝑁₀𝐸‖‖‖‖𝐸‖𝜑‖²+1𝑑𝑠,(3.12)_𝑡𝑁0𝐶𝑡^𝑁−𝑠𝐷𝑥_𝑠𝑑𝑠‖‖‖‖²≤𝑇𝑀²𝑀^2𝐷^𝑡𝑁₀𝐸‖‖‖‖𝐸‖𝜑‖²+1𝑑𝑠,(3.13)_𝑡𝑁0𝑆𝑡^𝑁−𝑠𝑓𝑠,𝑥_𝑠𝑑𝑠‖‖‖‖²≤𝑇𝑀²𝑀^2𝑓^𝑡𝑁₀𝐸‖‖‖‖𝐸‖𝜑‖²+1𝑑𝑠,(3.14)_𝑡𝑁0𝑆𝑡^𝑁−𝑠𝐵𝑢(𝑠)𝑑𝑠‖‖‖‖²≤8𝑀₂𝑀²‖𝜉‖²+‖𝜑(0)‖²+𝑦²⁰+(𝑇+1)𝑀^2𝐷^𝑡𝑁₀+𝑇𝐸‖𝜑‖²+1𝑑𝑠𝑀^2𝑓^𝑡𝑁₀+𝐸‖𝜑‖²+1𝑑𝑠𝑀^2𝑔^𝑡𝑁₀+2𝑁_𝑁𝐸‖𝜑‖²+1𝑑𝑠_𝑘=1𝑁_𝑘𝐸‖‖𝑥^𝑁𝑡_𝑘‖‖²+2𝑁^𝑁_𝑘=1𝑀_𝑘𝐸‖‖𝑥^𝑁𝑡_𝑘‖‖²∶=𝑀²𝑈(3.15) which gives 𝑁≤𝐸‖‖𝜋𝑥^𝑁𝑡^𝑁‖‖²≤8𝐸‖‖𝐶𝑡^𝑁[𝜑(0)]‖‖²+8𝐸‖‖𝑆𝑡^𝑁𝑦₀−𝐷(0,𝜑)‖‖²+8𝐸‖‖‖‖^𝑡𝑁₀𝐶𝑡^𝑁−𝑠𝐷(𝑠,𝜑)𝑑𝑠‖‖‖‖²+8𝐸‖‖‖‖^𝑡𝑁₀𝑆𝑡^𝑁−𝑠𝑓(𝑠,𝜑)𝑑𝑠‖‖‖‖²+8𝐸‖‖‖‖^𝑡𝑁₀𝑆𝑡^𝑁−𝑠𝑔(𝑠,𝜑)𝑑𝑊(𝑠)‖‖‖‖²+8𝐸‖‖‖‖_{0<𝑡𝑘<𝑡𝑁}𝐶𝑡^𝑁−𝑡_𝑘𝐼_𝑘𝑥^𝑁𝑡_𝑘‖‖‖‖²+8𝐸‖‖‖‖_{0<𝑡𝑘<𝑡𝑁}𝑆𝑡^𝑁−𝑡_𝑘𝐼_𝑘𝑥^𝑁𝑡_𝑘‖‖‖‖²+8𝐸‖‖‖‖^𝑡𝑁₀𝑆𝑡^𝑁−𝑠𝐵𝑢(𝑠)𝑑𝑠‖‖‖‖²≤𝐿+8𝑀²𝑇𝑀^2𝐷𝑁+𝑇𝑀^2𝑓𝑁+tr(𝑄)𝑀^2𝑔𝑁+2𝑁𝑀^2𝑛_𝑘=1𝑀_𝑘+2𝑁𝑀^2𝑛_𝑘=1𝑁_𝑘+8𝑀₂𝑇𝑀^2𝐷𝑁+𝑇𝑀^2𝑓𝑁+tr(𝑄)𝑀^2𝑔𝑁+2𝑁𝑀^2𝑛_𝑘=1𝑀_𝑘+2𝑁𝑀^2𝑛_𝑘=1𝑁_𝑘,(3.16) where 𝐿=8𝑀²𝐸‖‖𝑥⁰‖‖²+𝐸‖‖𝑦₀‖‖²+𝑇𝑀^2𝐷+𝑇𝑀^2𝑓+tr(𝑄)𝑀^2𝑔+2𝑀^2𝑛_𝑘=1𝑀_𝑘+2𝑀^2𝑛_𝑘=1𝑁_𝑘+8𝑀₂𝑇𝑀^2𝐷+𝑇𝑀^2𝑓+tr(𝑄)𝑀^2𝑔+2𝑀^2𝑛_𝑘=1𝑀_𝑘+2𝑀^2𝑛_𝑘=1𝑁_𝑘.(3.17) Dividing both sides of (3.16) by N and taking limit as 𝑁→∞, we obtain that 8𝑀²𝑇𝑀^2𝐷+𝑇𝑀^2𝑓+tr(𝑄)𝑀^2𝑔+2𝑀^2𝑛_𝑘=1𝑀_𝑘+2𝑀^2𝑛_𝑘=1𝑁_𝑘+8𝑀₂𝑇𝑀^2𝐷+𝑇𝑀^2𝑓+tr(𝑄)𝑀^2𝑔+2𝑀^2𝑛_𝑘=1𝑀_𝑘+2𝑀^2𝑛_𝑘=1𝑁_𝑘≥1(3.18) which is a contradiction by (3.7). Thus, 𝜋(𝐵_𝑁)⊂𝐵_𝑁, for some positive number N.
In what follows, we aim to show that the operator π has a fixed point on 𝐵_𝑁, 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 𝐵_𝑁, respectively, by 𝜋₁𝑥(𝑡)=𝑆+(𝑡)𝑦₀−𝐷(0,𝜑)+^𝑡0𝐶(𝑡−𝑠)𝐷(0,𝜑)𝑑𝑠+^𝑡0𝑆(𝑡−𝑠)𝑓𝑠,𝑥_𝑠𝑑𝑠_{0<𝑡𝑘<𝑡}𝜋𝐶𝑡−𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝑆𝑡−𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘,(3.20)₂𝑥(𝑡)=𝐶(𝑡)𝜙(0)+^𝑡0𝑆(𝑡−𝑠)𝑔𝑠,𝑥_𝑠𝑑𝑊(𝑠)+^𝑡0𝑆(𝑡−𝑠)𝐵𝑢(𝑠)𝑑𝑠.(3.21)

Lemma 3.4. The operator π₁ as above is contractive.

Proof. Let 𝑥,𝑦∈𝐵_𝑁. It follows from assumptions (A1)–(A4) and Hölder’s inequality that 𝐸‖‖𝜋₁𝑥(𝑡)−𝜋₁𝑦(𝑡)‖‖²≤5𝐸‖𝑆(𝑡)[𝐷(0,𝜑)−𝐷(0,𝜙)]‖²+5𝐸‖‖‖^𝑡0𝐶(𝑡−𝑠)[𝐷(0,𝜑)−𝐷(0,𝜙)]𝑑𝑠‖‖‖²+5𝐸‖‖‖^𝑡0𝑆(𝑡−𝑠)[𝑓(𝑠,𝜑)−𝑓(𝑠,𝜙)]𝑑𝑠‖‖‖²+5𝐸‖‖‖‖_{0<𝑡𝑘<𝑡}𝐶𝑡−𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘−𝐼_𝑘𝑦𝑡_𝑘‖‖‖‖²+5𝐸‖‖‖‖_{0<𝑡𝑘<𝑡}𝑆𝑡−𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘−𝐼_𝑘𝑦𝑡_𝑘‖‖‖‖²≤5𝑀²𝑀^2𝐷sup_{[]𝑠∈0,𝑇}𝐸‖𝑥(𝑠)−𝑦(𝑠)‖²+5𝑇𝑀²𝑀^2𝐷sup_{[]𝑠∈0,𝑇}𝐸‖𝑥(𝑠)−𝑦(𝑠)‖²+5𝑇𝑀²𝑀^2𝑓sup_{[]𝑠∈0,𝑇}𝐸‖𝑥(𝑠)−𝑦(𝑠)‖²+5𝑛𝑀²_0<_{𝑡𝑘<𝑡}𝑀_𝑘𝐸‖‖𝑥𝑡_𝑘−𝑦𝑡_𝑘‖‖²+5𝑛𝑀²_0<_{𝑡𝑘<𝑡}𝑁_𝑘𝐸‖‖𝑥𝑡_𝑘−𝑦𝑡_𝑘‖‖²(3.22) which deduces sup_{[]𝑠∈0,𝑇}𝐸‖‖𝜋₁𝑥(𝑠)−𝜋₁𝑦(𝑠)‖‖²≤5𝑀²𝑀^2𝐷+𝑇𝑀^2𝐷+𝑇𝑀^2𝑓+𝑛^𝑛_𝑖=0𝑀_𝑘+𝑛^𝑛_𝑖=0𝑁_𝑘sup_{[]𝑠∈0,𝑇}𝐸‖𝑥(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(𝐵_𝑁)⊂𝐵_𝑁.
We first need to prove that the set of functions 𝜋₂(𝐵_𝑁) is equicontinuous on [0, T]. Let 0<𝜀<𝑡<𝑇 and 𝛿>0 such that ‖𝑆(𝑠)𝑥−𝑆(𝑠^)𝑥‖²<𝜀 and ‖𝐶(𝑠)𝑥−𝐶(𝑠^)𝑥‖²<𝜀, for every 𝑠,𝑠^∈[0,𝑇] with |𝑠−𝑠^|≤𝛿. For 𝑥∈𝐵_𝑁 and 0<|ℎ|<𝛿 with 𝑡+ℎ∈[0,𝑇] we have 𝐸‖‖𝜋₂𝑥(𝑡+ℎ)−𝜋₂𝑥(𝑡)‖‖²≤3𝐸‖[𝐶(𝑡+ℎ)−𝐶(𝑡)]𝜙(0)‖²+3𝐸‖‖‖^𝑡0[𝑆(𝑡+ℎ−𝑠)−𝑆(𝑡−𝑠)]𝑔𝑠,𝑥_𝑠𝑑𝑊(𝑠)−^𝑡𝑡+ℎ𝑆(𝑡+ℎ−𝑠)𝑔𝑠,𝑥_𝑠𝑑𝑊(𝑠)‖‖‖²+3𝐸‖‖‖^𝑡0[𝑆(𝑡+ℎ−𝑠)−𝑆(𝑡−𝑠)]𝐵𝑢(𝑠)𝑑𝑠−^𝑡𝑡+ℎ𝑆(𝑡+ℎ−𝑠)𝐵𝑢(𝑠)𝑑𝑠‖‖‖²≤3𝜀𝐸‖𝜙(0)‖²+6tr(𝑄)𝑀²^𝑡𝑡+ℎ𝐸‖‖𝑔𝑠,𝑥^(𝑠),𝑥_𝑠‖‖²𝑑𝑠+6𝑀²^𝑡𝑡+ℎ+6𝑀₂𝐸‖𝐵𝑢(𝑠)‖²𝑑𝑠^𝑡0≤4𝜀𝐸‖‖𝑥₀𝐸‖𝐵𝑢(𝑠)‖²𝑑𝑠+6tr(𝑄)^𝑡0𝐸‖‖[𝑆(𝑡+ℎ−𝑠)−𝑆(𝑡−𝑠)]𝑔𝑠,𝑥_𝑠‖‖²𝑑𝑠+4tr(𝑄‖‖)²𝑀+₂4𝜀𝐸‖𝑔(𝑥)‖²+4𝜀tr(𝑄)^𝑡0𝐸‖‖𝑔𝑠,𝑥_𝑠‖‖²𝑑𝑠^𝑡𝑡+ℎ𝐸‖𝑔(𝑠,𝑠^𝑥(𝑠))‖²𝑑𝑠.(3.24) Noting that 𝐸‖𝑔(𝑠,𝑠^𝑥(𝑠))‖²≤ℎ_𝑁(𝑠)∈𝐿¹([0,𝑇]), we see that 𝜋₂(𝐵_𝑁) is equicontinuous on [0, T].
We next need to prove that 𝜋₂ maps 𝐵_𝑁 into a precompact set in 𝐵_𝑁. That is, for every fixed 𝑡∈[0,𝑇], the set 𝑉(𝑡)={(𝜋₂𝑥)(𝑡)∶𝑥∈𝐵_𝑁} is precompact in 𝐵_𝑁. It is obvious that 𝑉(0)={(𝜋₂𝑥)(0)} is precompact. Let 0<𝑡≤𝑇 be fixed and 0<𝜀<𝑡. For 𝑥∈𝐵_𝑁, define 𝜋^𝜀2𝑥(𝑡)==𝐶𝐶((𝑡𝑡))𝜙𝜙((00))++𝑆(𝜀^0𝑡−𝜀)𝑆(𝑡_0𝑡−𝜀−𝑠)𝑔𝑠,𝑥_𝑠𝑑𝑊(𝑠)+^0𝑡−𝜀𝑆(𝑡−𝑠)𝐵𝑢(𝑠)𝑑𝑠𝑆(𝑡−𝜀−𝑠)𝑔𝑠,𝑥_𝑠𝑑𝑊(𝑠)+𝑆(𝜀)^0𝑡−𝜀𝑆(𝑡−𝜀−𝑠)𝐵𝑢(𝑠)𝑑𝑠.(3.25) Since 𝐶(𝑡),𝑆(𝑡),𝑡>0, are compact, it follows that 𝑉_𝜀(𝑡)={(𝜋^𝜀2𝑥)(𝑡)∶𝑥∈𝐵_𝑁} is precompact in H for every 0<𝜀<𝑡. Moreover, for each 𝑥∈𝐵_𝑁, we have 𝐸‖‖𝜋₂𝑥(𝑡)−𝜋^𝜀2𝑥(𝑡)‖‖²≤2tr(𝑄)𝑀²^{𝑡𝑡−𝜀}𝐸‖‖𝑔𝑠,𝑥_𝑠‖‖²𝑑𝑠+2𝑀²^{𝑡𝑡−𝜀}≤𝜀2𝑀₂𝐸‖𝐵𝑢(𝑠)‖²𝑑𝑠tr(𝑄)𝐸‖𝜑‖²+1+𝑈→0𝑎𝑠𝜀→0⁺(3.26) which means that there are precompact sets arbitrary close to the set 𝑉(𝑡). Thus, 𝑉(𝑡) is precompact in 𝐵_𝑁.
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 𝐵_𝑁.

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)=𝜕𝜕𝑧²²𝑥(𝑡,0)=𝑥(𝑡,𝜋)=0,𝑡∈[𝑥0,(1𝑡,]𝑧,)𝑑𝜕𝑡𝑥+(𝜎0,(𝑧𝑡,)𝑥(𝑡,𝑧))𝑑𝑊(𝑡),𝑡∈[0,1]𝜕𝑡=𝑥¹Δ𝑥𝑡_𝑘(𝑧),𝑧∈[0,𝜋](𝑧)=𝐼_𝑘𝑥𝑡_𝑘(𝑧),Δ𝑥^𝑡_𝑘(𝑧)=𝐼_𝑘𝑥𝑡_𝑘(𝑧),𝑡=𝑡_𝑘,(4.1) to rewrite (4.1) into the abstract form of (1.1), let 𝐻=𝐿²[0,𝜋], 𝐴∶𝐻→𝐻 be an operator by 𝐴𝑥=𝑥^ with domain 𝐷(𝐴)=𝑥∈𝐻∶𝑥,𝑥^areabsolutelycontinuous,𝑥^∈𝐻,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)∶𝑡∈𝑅} in 𝐻 and is given by 𝐶(𝑡)𝑥=^∞_𝑛=1cos(𝑛𝑡)⟨𝑥,𝑒_𝑛⟩𝑒_𝑛,𝑥∈𝐻,(4.3) where 𝑒_𝑛(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,𝑒^𝑛⟩𝑒_𝑛,𝑥∈𝐻.(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).