Complete Controllability of Impulsive Stochastic Integrodifferential Systems in Hilbert Space

Abstract

This paper concerns the complete controllability of the impulsive stochastic integrodifferential systems in Hilbert space. Based on the semigroup theory and Burkholder-Davis-Gundy's inequality, sufficient conditions of the complete controllability for impulsive stochastic integro-differential systems are established by using the Banach fixed point theorem. An example for the stochastic wave equation with impulsive effects is presented to illustrate the utility of the proposed result.

1. Introduction

It is well known that controllability is one of the fundamental concepts and plays an important role in control theory and engineering. The problem which is about controllability of linear and nonlinear stochastic systems represented by SODE (stochastic ordinary differential equation) in finite dimensional space has been extensively studied (e.g., [1–4] and references therein). The controllability for infinite dimensional stochastic systems represented by SPDE (stochastic partial differential equation) is natural generalization of stochastic systems in finite dimensional space [5]. According to the literature, at least three types of infinite dimensional stochastic systems have been studied, that is, approximate, complete, and 𝑆-controllability [6], so the controllability research of the infinite dimensional stochastic systems is usually more complicated than that of the finite dimensional. For linear stochastic system, the controllability problem has been studied by some authors [6, 7], which is shown as the following SPDE: (𝑡)=[(𝑡)(𝑡(𝑡)(𝑡)[],𝑥(0)𝑑𝑥𝐴𝑥+𝐵𝑢)]𝑑𝑡+Σ𝑑𝑊,𝑡∈0,𝑇=𝑥₀∈ℒ₂(Ω,𝐻), (1) where 𝑥₀ is ℱ₀-measurable, 𝐻 is separable Hilbert space, 𝐴 is the infinitesimal generator of a strongly continuous semigroup 𝑆(𝑡) on 𝐻, 𝐵∈ℒ(𝑈,𝐻), 𝑢(𝑡) is feedback control, 𝑊(𝑡) is 𝑄-Wiener process, andΣ∈ℒ₂(𝑄^1/2𝐸,𝐻). For nonlinear stochastic systems in infinite dimensional space, there are also many results on the controllability theory (see [8–13]).

On the other hand, the impulsive effects exist widely in many evolution processes in which the states are changed abruptly at certain moments of time, involving fields such as finance, economics, mechanics, electronics, and telecommunications (see [14] and references of therein). Impulsive differential systems have emerged as an important area investigation in applied sciences, and many papers have been published about the controllability of impulsive differential systems both in finite and infinite dimensional space. Sakthivel et al. [15] established the sufficient conditions for approximate controllability of nonlinear impulsive differential systems by Schauder’s fixed point theorem; Li et al. [16] investigated the complete controllability of the first-order impulsive functional differential systems in Banach space using Schaefer’s fixed point theorem; Chang [17] studied the complete controllability of impulsive functional differential systems with infinite delay; Sakthivel et al. [18] discussed complete controllability of second-order nonlinear impulsive differential systems. However, the complete controllability problem of impulsive stochastic integro-differential systems has not been investigated in infinite dimensional space yet, to the best of our knowledge, although [19–22], respectively, investigated the controllability of impulsive stochastic control systems in finite dimensional space by using contraction mapping principle; and Subalakshmi and Balachandran [23] studied the approximate controllability of nonlinear stochastic impulsive systems in Hilbert spaces by using Nussbaum’s fixed point theorem. Based on Banach fixed point theorem, the proposed work in this paper on the complete controllability of the integro-differential stochastic systems with impulsive effects in Hilbert spaces is new in the literature.

In this paper, our main purpose is to show the complete controllability of following impulsive stochastic integro-differential systems in Hilbert space, (𝑡)=(𝑡)(𝑡)(𝑡),𝑑𝑥𝐴𝑥+𝐵𝑢+𝐹𝑡,𝑥^𝑡₀𝑓((𝑠(𝑡),𝑡,𝑠,𝑥))𝑑𝑠𝑑𝑡+𝐺𝑡,𝑥^𝑡₀𝑔(𝑡,𝑠,𝑥(𝑠))𝑑𝑠𝑑𝑤(𝑡),𝑡≠𝑡_𝑘𝑡,𝑡⩾0,Δ𝑥_𝑘=𝐼_𝑘𝑥_𝑡⁻_𝑘,𝑡=𝑡_𝑘𝑥(0),𝑘=1,2,…,𝑚,=𝑥₀∈𝐻, (2) where 𝐹∶[0,𝑇]×𝐻×𝐻→𝐻, 𝐺∶[0,𝑇]×𝐻×𝐻→ℒ₂(𝑄^1/2𝐸,𝐻), 𝑓,𝑔∶[0,𝑇×[0,𝑇]×𝐻→𝐻 are measurable mappings. 𝐼_𝑘(𝑥_𝑡⁻_𝑘)=𝑥(𝑡⁺_𝑘)−𝑥(𝑡⁻_𝑘),  𝑡=𝑡_𝑘,  𝑘=1,2,…,𝜌, where 𝑥(𝑡⁺_𝑘) and 𝑥(𝑡⁻_𝑘) denote the right and left limits of 𝑥(𝑡) at 𝑡=𝑡_𝑘, respectively. Also Δ𝑥(𝑡_𝑘)=𝑥(𝑡⁺_𝑘)−𝑥(𝑡⁻_𝑘) represents the jump in the state 𝑥 at time 𝑡_𝑘 with 𝐼_𝑘 determining the size of the jump. For systems (2), if 𝐼_𝑘=0, the controllability problem was studied by Subalakshmi et al. [11]. If 𝐼_𝑘≠0 and 𝐺=0,  𝑓=0, [15] discussed the approximate controllability problem. When 𝐴, 𝐵 are matrices with appropriate dimensions, 𝐹, 𝐺 are vectors (in fact, matrix is aspecial form of operator), and 𝑓=𝑔=0, Karthikeyan et al. [19] obtained the controllability results, so system (2) is of the more general form and has great diversity.

The outline of this paper is as follows: Section 2 contains basic notations, lemmas, and preliminary facts. The controllability results are given in Section 3 by fixed point methods. In Section 4, we provide an example to demonstrate the effectiveness of our method. Finally, conclusions are given in Section 5.

2. Preliminaries

Let (Ω,ℱ,ℙ) be a complete probability space with a filtration {ℱ_𝑡}_𝑡⩾0 satisfying the usual conditions (i.e., it is right continuous and ℱ₀ contains all ℙ-null sets). We consider three Hilbert spaces 𝐸,  𝐻, and 𝑈, and a 𝑄-Wiener process on (Ω,ℱ,ℙ) with the covariance operator 𝑄∈ℒ(𝐸) such that tr𝑄<∞. Let ⟨⋅⟩ and ‖⋅‖ denote inner product and norm of 𝐻, respectively. ℒ(𝑋,𝑌) is the space of all linear bounded operator from a Hilbert space 𝑋 to a Hilbert space 𝑌. We also employ the same notation ‖⋅‖ for the norm of ℒ(𝑋,𝑌). We assume that there exists a complete orthonormal {𝑒_𝑘} in 𝐸, a bounded sequence of nonnegative real numbers 𝜆_𝑘 such that 𝑄𝑒_𝑘=𝜆_𝑘𝑒_𝑘,  𝑘=1,2,…, and a sequence {𝛽_𝑘} of independent Brownian motions such that ⟨𝑤(𝑡),𝑒⟩=^∞_𝑘=1𝜆_𝑘𝑒_𝑘,𝑒𝛽_𝑘(𝑡),𝑒∈𝐸,𝑡∈[0,𝑇], (3) and ℱ_𝑡=ℱ^𝛽_𝑡, where ℱ^𝛽_𝑡 is the 𝜎-algebra generated by {𝛽(𝑠)∶0⩽𝑠⩽𝑡}. Let ℒ⁰₂=ℒ₂(𝑄^1/2𝐸,𝐻) be the space of all Hilbert-Schmidt operator from 𝑄^1/2𝐸 to 𝐻 with the inner product ⟨Ψ,Φ⟩_ℒ⁰₂=tr[Ψ𝑄Φ^∗] and the norm ‖⋅‖_ℒ⁰₂. 𝐿₂(ℱ_𝑇,𝐻) is the Hilbert space of all ℱ_𝑇-measurable square integrable random variables with values in Hilbert space 𝐻. 𝐿^ℱ𝑇₂([0,𝑇],𝐻) is the Hilbert space of square integrable and ℱ_𝑇-adapted processes with values in 𝐻.

Let PC([0,𝑇],𝐿₂(Ω,ℱ,ℙ;𝐻))={𝜙∶𝜙 is a function from [0,𝑇] into 𝐿₂(Ω,ℱ,ℙ;𝐻) such that 𝜙(𝑡) is continuous at 𝑡≠𝑡_𝑘, left continuous at 𝑡=𝑡_𝑘, and the right limit 𝜙(𝑡⁺_𝑘) exists for 𝑘=1,2,…,𝜌}. ℋ₂(𝑈₂) is the closed subspace of PC([0,𝑇],𝐿₂(Ω,ℱ,ℙ;𝐻)) consisting of measurable and ℱ_𝑡-adapted 𝐻-valued(𝑈-valued) process 𝜙(⋅)∈PC([0,𝑇],𝐿₂(Ω,ℱ,ℙ;𝐻))(𝜙(⋅)∈PC([0,𝑇],𝐿₂(Ω,ℱ,ℙ;𝑈))) endowed with the norm ‖𝜙‖²_ℋ2=sup_{0⩽𝑡⩽𝑇}𝔼‖𝜙(𝑡)‖².

By a solution of system (2), we mean a mild solution of the following nonlinear integral equation: 𝑥(𝑡)=𝑆(𝑡)𝑥₀+^𝑡₀𝑆()(𝑠)+𝑡−𝑠𝐵𝑢𝑑𝑠_𝑡0𝑆(𝑡−𝑠)𝐹𝑠,𝑥(𝑠),^𝑠₀𝑓((𝜏+𝑠,𝜏,𝑥))𝑑𝜏𝑑𝑠_𝑡0𝑆(𝑡−𝑠)𝐺𝑠,𝑥(𝑠),^𝑠₀𝑔((𝜏(𝑠)+𝑠,𝜏,𝑥))𝑑𝜏𝑑𝑤_𝜌𝑘=1𝑆𝑡−𝑡_𝑘𝐼_𝑘𝑥𝑡⁻_𝑘, (4) where 𝑢∈𝑈_𝑎𝑑∶=𝑈₂, 𝑆(𝑡)_𝑡⩾0 denotes the strongly continuous semigroup generated by the operator 𝐴.

Now let us introduce the controllability operator Π^𝑇_𝑠 associated with (4) (see[8]), Π^𝑇_𝑠{⋅}=^𝑇_𝑠𝑆(𝑇−𝑡)𝐵𝐵^∗𝑆^∗(𝑇−𝑡)𝑑𝑡, (5) which belongs to ℒ(𝐻,𝐻); 𝐵^∗ is the adjoint operator of 𝐵.

Definition 1. System (2) is completely controllable on [0,𝑇] if ℛ_𝑇𝑥₀=𝐿^ℱ𝑇₂([0,𝑇],𝐻). (6) That is, all the points in 𝐿^ℱ𝑇₂([0,𝑇],𝐻) can be reached from the point 𝑥₀ at time 𝑇, where ℛ_𝑡(𝑥₀)={𝑥(𝑡;𝑥₀,𝑢)∶𝑢∈𝐿^ℱ₂([0,𝑇],𝐻)}.

Lemma 2 (Burkholder-Davis-Gundy’s inequality [23]). For any 𝑟⩾1 and for arbitrary ℒ⁰₂-valued predictable process Ψ(𝑡),  𝑡∈[0,𝑇], one has 𝔼sup_{0⩽𝑡⩽𝑇}^𝑡₀Ψ(𝑠)𝑑𝑤(𝑠)^2𝑟⩽𝐶_𝑟𝔼^𝑡₀‖Ψ(𝑠)‖²_ℒ02𝑑𝑠^𝑟, (7) where 𝐶_𝑟=(𝑟(2𝑟−1))^𝑟2𝑟2𝑟−1^2𝑟2. (8)

Lemma 3 (Mahmudov [6]). The following linear system 𝑑𝑥(𝑡)=[𝐴𝑥(𝑡)+𝐵𝑢(𝑡)]𝑑𝑡+𝐷(𝑡)𝑑𝑤(𝑡),𝑥(0)=𝑥₀ (9) is completely controllable if and only if Π^𝑇₀⩾𝛾𝐼, where 𝛾 is constant and 𝐼 is unit operator.

Lemma 4. Assume that the operator Π^𝑇_𝑠 is invertible. Then for arbitrary target 𝑥_𝑇∈𝐿₂(ℱ_𝑇,𝐻), the control 𝑢(𝑡)=𝐵^∗𝑆^∗()𝔼×Π𝑇−𝑡_𝑇0−1𝑥_𝑇−𝑆(𝑇)𝑥₀−^𝑇₀𝑆(𝑇−𝑠)𝐹(𝑠)−𝑑𝑠_𝑇0𝑆(𝑇−𝑠)𝐺(𝑠)(𝑠)−𝑑𝑤_𝜌𝑘=1𝑆𝑇−𝑡_𝑘𝐼_𝑘𝑥𝑡⁻_𝑘∣ℱ_𝑡 (10) transfers the systems (4) from 𝑥₀ to 𝑥_𝑇 at time 𝑇, where 𝐹(𝑠)=𝐹(𝑠,𝑥(𝑠),∫^𝑠₀𝑓(𝑠,𝜏,𝑥(𝜏))𝑑𝜏), 𝐺(𝑠)=𝐺(𝑠,𝑥(𝑠),∫^𝑠₀𝑔(𝑠,𝜏,𝑥(𝜏))𝑑𝜏)).

Proof. Substituting (10) into (4), we can obtain that 𝑥(𝑡)=𝑆(𝑡)𝑥₀+^𝑡₀𝑆(𝑡−𝑠)×𝐵𝐵^∗𝑆^∗()𝔼×Π𝑇−𝑠_𝑇0−1𝑥_𝑇−𝑆(𝑇)𝑥₀−^𝑇₀𝑆(𝑇−𝑠)𝐹(𝑠)−𝑑𝑠_𝑇0𝑆(𝑇−𝑠)𝐺(𝑠)(𝑠)−𝑑𝑤_𝜌𝑘=1𝑆𝑇−𝑡_𝑘𝐼_𝑘𝑥𝑡⁻_𝑘∣ℱ_𝑡+^𝑡₀𝑆(𝑇−𝑠)𝐹(𝑠)+𝑑𝑠_𝑡0𝑆(𝑇−𝑠)𝐺(𝑠)𝑑𝑤(𝑠)+^𝜌_𝑘=1𝑆𝑇−𝑡_𝑘𝐼_𝑘𝑥𝑡⁻_𝑘=𝑆(𝑡)𝑥₀+Π^𝑡₀𝑆^∗(𝑇−𝑡)Π^𝑇₀⁻¹×𝑥_𝑇−𝑆(𝑇)𝑥₀−^𝑇₀𝑆(𝑇−𝑠)𝐹(𝑠)𝑑𝑠−^𝑡₀𝑆(𝑡−𝑠)𝐺(𝑠)(𝑠)−𝑑𝑤_𝜌𝑘=1𝑆𝑇−𝑡_𝑘𝐼_𝑘𝑥𝑡⁻_𝑘+^𝑡₀𝑆(𝑇−𝑠)𝐹(𝑠)𝑑𝑠+^𝑡₀𝑆(𝑇−𝑠)𝐺(𝑠)(𝑠)+𝑑𝑤_𝜌𝑘=1𝑆𝑇−𝑡_𝑘𝐼_𝑘𝑥𝑡⁻_𝑘. (11) The proof is completed by letting 𝑡=𝑇 in (11).

3. Main Results

In this section, by using contraction mapping principle in Banach space we discuss the complete controllability criteria of semilinear impulsive stochastic systems (2). For the proof of the main result we impose the following assumptions on data of the problem.

Assumption A. The functions 𝐹, 𝐺, and 𝐼 are continuous and satisfy the usual linear growth condition; that is, there exist positive real constants 𝐿₁, 𝛼_𝑘 for arbitrary 𝑥∈𝐻, and 0⩽𝑡⩽𝑇 such that 𝐹(𝑡,𝑥,𝑦)²+𝐺(𝑡,𝑥,𝑦)²_ℒ02⩽𝐿₁1+‖𝑥‖²+𝑦²,𝐼_𝑘(𝑥)²⩽𝛼_𝑘1+‖𝑥‖²,𝑘=1,2,…,𝜌,_𝑡0𝑓(𝑡,𝑠,𝑥(𝑠))𝑑𝑠²+^𝑡₀𝑔(𝑡,𝑠,𝑥(𝑠))𝑑𝑠²⩽𝑘₁‖𝑥‖². (12)

Assumption B. The functions 𝐹, 𝐺, and 𝐼 satisfy the following Lipschitz condition and for every 𝑡⩾0 and 𝑥,𝑦∈𝐻 there exist positive real constants 𝐿₂, 𝛽_𝑘, 𝑘₂ such that 𝐹𝑡,𝑥₁,𝑦₁−𝐹𝑡,𝑥₂,𝑦₂²+𝐺𝑡,𝑥₁,𝑦₁−𝐺𝑡,𝑥₂,𝑦₂²_ℒ02⩽𝐿₂𝑥₁−𝑥₂²+𝑦₁−𝑦₂²,𝐼_𝑘(𝑥)−𝐼_𝑘𝑦⩽𝛽_𝑘𝑥−𝑦,𝑘=1,2,…,𝜌,_𝑡0𝑓(𝑡,𝑠,𝑥(𝑠))−𝑓𝑡,𝑠,𝑦(𝑠)²+^𝑡₀𝑔(𝑡,𝑠,𝑥(𝑠))−𝑔𝑡,𝑠,𝑦(𝑠)²𝑑𝑠⩽𝑘₂𝑥−𝑦². (13)

Assumption C. The linear system (9) is completely controllable. By Lemma 3, for some 𝛾>0, 𝔼⟨Π^𝑇₀𝑧,𝑧⟩⩾𝛾𝔼‖𝑧‖², for all 𝑧∈𝐿₂(ℱ_𝑇,𝐻). Consequently, ‖(Π^𝑇₀)⁻¹‖⩽1/𝛾=𝑙₂.

Assumption D. Let 𝑝=[6𝑇𝑙₁𝐿₂(𝑀𝑙₁𝑙₂+1)(𝑇+4)(1+𝑘₂𝑇)+6𝑙₁𝜌(𝑀𝑙₂+1)∑^𝜌_𝑘=1𝛽_𝑘] be such that 0⩽𝑝<1.

Now for convenience, let us introduce the following notations: 𝑙₁=max_{0⩽𝑡⩽𝑇}‖𝑆(𝑡)‖²,𝑀=max_{𝑠∈[0,𝑇]}Π^𝑇_𝑠². (14)

Theorem 5. Suppose that assumptions A, B, C, and D are satisfied. Then system (4) is completely controllable on [0,𝑇].

Proof. For arbitrary initial data 𝑥₀∈ℋ₂, we can define a nonlinear operator Φ from 𝐻₂ to 𝐻₂ as the following: (Φ𝑥)(𝑡)=𝑆(𝑡)𝑥₀+^𝑡₀𝑆()(𝑠)+𝑡−𝑠𝐵𝑢𝑑𝑠_𝑡0𝑆(𝑡−𝑠)𝐹()+𝑠,𝑥𝑑𝑠_𝑡0𝑆(𝑡−𝑠)𝐺()(𝑠)+𝑠,𝑥𝑑𝑤_𝜌𝑘=1𝑆𝑡−𝑡_𝑘𝐼_𝑘𝑥𝑡⁻_𝑘, (15) where 𝑢(𝑡) is defined by (10).
By Lemma 4, the control (10) transfers system (4) from the initial state 𝑥₀ to the final state 𝑥_𝑇 provided that the operators Φ has a fixed point in ℋ₂. So, if the operator Φ has a fixed point then system (2) is completely controllable. As mentioned before, to prove the complete controllability of the system (2), it is enough to show that Φ has a fixed point in ℋ₂. To do this, we can employ the contraction mapping principle. In the following, we will divide the proof into two steps.
Firstly, we show that ℋ₂ maps ℋ₂ into itself. From (15) we have sup_{0⩽𝑡⩽𝑇}𝔼‖(Φ𝑥)(𝑡)‖²=sup_{0⩽𝑡⩽𝑇}𝔼𝑆(𝑡)𝑥₀+^𝑡₀𝑆(𝑡−𝑠)𝐵𝑢(𝑠)𝑑𝑠+^𝑡₀𝑆(𝑡−𝑠)𝐹()+𝑠,𝑥𝑑𝑠_𝑡0𝑆(𝑡−𝑠)𝐺()(𝑠)+𝑠,𝑥𝑑𝑤_𝜌𝑘=1𝑆𝑡−𝑡_𝑘𝐼_𝑘𝑥𝑡⁻_𝑘²⩽5sup_{0⩽𝑡⩽𝑇}𝔼𝑆(𝑡)𝑥₀²+5sup_{0⩽𝑡⩽𝑇}𝔼^𝑡₀𝑆(𝑡−𝑠)𝐵𝑢(𝑠)𝑑𝑠²+5sup_{0⩽𝑡⩽𝑇}𝔼^𝑡₀𝑆(𝑡−𝑠)𝐹(𝑠,𝑥)𝑑𝑠²+5sup_{0⩽𝑡⩽𝑇}𝔼^𝑡₀𝑆(𝑡−𝑠)𝐺(𝑠,𝑥)𝑑𝑤(𝑠)²+5sup_{0⩽𝑡⩽𝑇}𝔼^𝜌_𝑘=1𝑆𝑡−𝑡_𝑘𝐼_𝑘𝑥𝑡⁻_𝑘²≜⁵_𝑖=1𝐴_𝑖. (16)
Using Holder inequality, B-D-G inequality (here 𝐶₁=4), and Assumption C, we have the following estimates: 𝐴₁⩽5𝑙₁𝔼𝑥₀²,𝐴₃⩽5𝑇sup_{0⩽𝑡⩽𝑇}𝔼^𝑡₀𝑆(𝑡−𝑠)𝐹(𝑠,𝑥)²𝑑𝑠⩽5𝑇𝑙₁𝐿₁^𝑇₀1+1+𝑘₁sup_{0⩽𝑠⩽𝑇}𝔼‖𝑥(𝑠)‖²𝐴𝑑𝑠,₄⩽20𝑙₁𝐿₁^𝑇₀1+1+𝑘₁sup_{0⩽𝑠⩽𝑇}𝔼‖𝑥(𝑠)‖²𝐴𝑑𝑠,₅⩽5𝜌sup_{0⩽𝑡⩽𝑇}𝔼^𝜌_𝑘=1𝑆𝑡−𝑡_𝑘𝐼_𝑘𝑥𝑡⁻_𝑘²⩽5𝜌𝑙₁^𝜌_𝑖−1𝛼_𝑘1+sup_{0⩽𝑡⩽𝑇}𝔼‖𝑥(𝑠)‖². (17) Meanwhile by control function (15), we have 𝐴₂=5sup_{0⩽𝑡⩽𝑇}𝔼×^𝑡₀𝑆(𝑡−𝑟)𝐵𝐵^∗𝑆^∗()𝔼×Π𝑇−𝑟_𝑇0−1×𝑥(𝑇)−𝑆(𝑇)𝑥₀−^𝑇₀𝑆(𝑇−𝑠)𝐹()−𝑠,𝑥𝑑𝑠_𝑇0𝑆(𝑇−𝑠)𝐺()(𝑠)−𝑠,𝑥𝑑𝑤_𝜌𝑘=1𝑆𝑇−𝑡_𝑘𝐼_𝑘𝑥𝑡⁻_𝑘𝑑𝑟²=5sup_{0⩽𝑡⩽𝑇}𝔼^𝑡₀𝑆(𝑡−𝑟)𝐵𝐵^∗𝑆^∗(𝑡−𝑟)𝑆^∗(𝑇−𝑡)Π^𝑇₀⁻¹×𝑥(𝑇)−𝑆(𝑇)𝑥₀−^𝑇₀𝑆(𝑇−𝑠)𝐹()−𝑠,𝑥𝑑𝑠_𝑇0𝑆(𝑇−𝑠)𝐺()(𝑠)−𝑠,𝑥𝑑𝑤_𝜌𝑘=1𝑆𝑇−𝑡_𝑘𝐼_𝑘𝑥𝑡⁻_𝑘𝑑𝑟²=5sup_{0⩽𝑡⩽𝑇}𝔼Π^𝑡₀𝑆^∗(𝑇−𝑡)Π^𝑇₀⁻¹×𝑥_𝑇−𝑆(𝑇)𝑥₀−^𝑇₀𝑆(𝑇−𝑠)𝐹(𝑠)𝑑𝑠−^𝑡₀𝑆(𝑡−𝑠)𝐺()(𝑠)−𝑠,𝑥𝑑𝑤_𝜌𝑘=1𝑆𝑇−𝑡_𝑘𝐼_𝑘𝑥𝑡⁻_𝑘². (18) So similar as in (17), we get 𝐴₂⩽25𝑀𝑙₁𝑙₂𝔼𝑥_𝑇²+𝑙₁𝑥₀²+𝑇𝑙₁𝐿₁^𝑇₀1+1+𝑘₁sup_{0⩽𝑠⩽𝑇}‖𝑥(𝑠)‖²𝑑𝑠+4𝑙₁𝐿₁^𝑇₀1+1+𝑘₁sup_{0⩽𝑠⩽𝑇}‖𝑥(𝑠)‖²𝑑𝑠+𝜌^𝜌_𝑖=1𝛼_𝑘1+sup_{0⩽𝑡⩽𝑇}‖𝑥(𝑡)‖². (19) From (17)–(19), we have sup_{0⩽𝑡⩽𝑇}𝔼‖Φ𝑥(𝑡)‖²⩽𝐶1+^𝑇₀1+1+𝑘₁sup_{0⩽𝑠⩽𝑇}𝔼‖𝑥(𝑠)‖²𝑑𝑠⩽𝐶1+1+1+𝑘₁𝑇sup_{0⩽𝑠⩽𝑇}𝔼‖𝑥(𝑠)‖² (20) for all 𝑡∈[0,𝑇], where 𝐶 is constant. This implies that Φ maps ℋ₂ into itself.
Secondly, we prove that Φ is a contraction mapping on ℋ₂, for any 𝑥,𝑦∈ℋ₂, sup_{0⩽𝑡⩽𝑇}𝔼(Φ𝑥)(𝑡)−Φ𝑦(𝑡)²=sup_{0⩽𝑡⩽𝑇}𝔼Π^𝑡₀𝑆^∗(𝑇−𝑡)Π^𝑇₀⁻¹×^𝑇₀𝑆(𝑇−𝑠)𝐹𝑠,𝑦−𝐹()+𝑠,𝑥𝑑𝑠_𝑇0𝑆(𝑇−𝑠)𝐺𝑠,𝑦−𝐺((𝑠)+𝑠,𝑥)𝑑𝑤_𝜌𝑘=1𝑆𝑇−𝑡_𝑘𝐼_𝑘𝑦𝑡⁻_𝑘−𝐼_𝑘𝑥𝑡⁻_𝑘+^𝑡₀𝑆(𝑡−𝑠)𝐹(𝑠,𝑥)−𝐹+𝑠,𝑦𝑑𝑠_𝑡0𝑆(𝑡−𝑠)𝐺(𝑠,𝑥)−𝐺(𝑠)+𝑠,𝑦𝑑𝑤_𝜌𝑘=1𝑆𝑡−𝑡_𝑘𝐼_𝑘𝑥𝑡⁻_𝑘−𝐼_𝑘𝑦𝑡⁻_𝑘²⩽6𝑀𝑙₁𝑙₂^𝑇₀𝑆(𝑇−𝑠)𝐹(𝑠,𝑥)−𝐹𝑠,𝑦𝑑𝑠²+^𝑇₀𝑆(𝑇−𝑠)𝐺(𝑠,𝑥)−𝐺𝑠,𝑦𝑑𝑠²+^𝜌_𝑘=1𝑆𝑇−𝑡_𝑘𝐼_𝑘𝑥𝑡⁻_𝑘−𝐼_𝑘𝑦𝑡⁻_𝑘²+6sup_{0⩽𝑡⩽𝑇}𝔼^𝑡₀𝑆(𝑡−𝑠)𝐹(𝑠,𝑥)−𝐹𝑠,𝑦𝑑𝑠²+6sup_{0⩽𝑡⩽𝑇}𝔼^𝑡₀𝑆(𝑡−𝑠)𝐺(𝑠,𝑥)−𝐺𝑠,𝑦𝑑𝑤(𝑠)²+6sup_{0⩽𝑡⩽𝑇}𝔼^𝜌_𝑘=1𝑆𝑡−𝑡_𝑘𝐼_𝑘𝑥𝑡⁻_𝑘−𝐼_𝑘𝑦𝑡⁻_𝑘²≜6𝑀𝑙₁𝑙₂³_𝑖=1𝐵_𝑖+⁶_𝑖=4𝐵_𝑖. (21) Using Lipschitz condition, similiar to 𝐴₁--𝐴₅, we have the following estimates: 𝐵₁⩽𝑇𝑙₁𝐿₂1+𝑘₂𝑇^𝑇₀sup_{0⩽𝑠⩽𝑇}𝔼𝑥(𝑠)−𝑦(𝑠)²𝐵𝑑𝑠,₂⩽4𝑙₁𝐿₂1+𝑘₂𝑇^𝑇₀sup_{0⩽𝑠⩽𝑇}𝔼𝑥(𝑠)−𝑦(𝑠)²𝐵𝑑𝑠,₃⩽𝑙₁𝜌^𝜌_𝑘=1𝛽_𝑘sup_{0⩽𝑠⩽𝑇}𝔼𝑥(𝑠)−𝑦(𝑠)²,𝐵₄⩽6𝑇𝑙₁𝐿₂1+𝑘₂𝑇^𝑇₀sup_{0⩽𝑠⩽𝑇}𝔼𝑥(𝑠)−𝑦(𝑠)²𝐵𝑑𝑠,₅⩽24𝑙₁𝐿₂1+𝑘₂𝑇^𝑇₀sup_{0⩽𝑠⩽𝑇}𝔼𝑥(𝑠)−𝑦(𝑠)²𝐵𝑑𝑠,₆⩽6𝑙₁𝜌^𝜌_𝑘=1𝛽_𝑘sup_{0⩽𝑠⩽𝑇}𝔼𝑥(𝑠)−𝑦(𝑠)² (22)(23)(24)(25)(26)(27) together with inequalities (22)–(27): sup_{0⩽𝑡⩽𝑇}𝔼(Φ𝑥)(𝑡)−Φ𝑦(𝑡)²⩽6𝑇𝑀𝑙²₁𝑙₂𝐿₂1+𝑘₂𝑇^𝑇₀sup_{0⩽𝑠⩽𝑇}𝔼𝑥(𝑠)−𝑦(𝑠)²𝑑𝑠+24𝑀𝑙²₁𝑙₂𝐿₂1+𝑘₂𝑇^𝑇₀sup_{0⩽𝑠⩽𝑇}𝔼𝑥(𝑠)−𝑦(𝑠)²+𝑑𝑠6𝑀𝑙₁𝑙₂𝜌^𝜌_𝑘=1𝛽_𝑘+6𝑙₁𝜌^𝜌_𝑘=1𝛽_𝑘sup_{0⩽𝑡⩽𝑇}𝔼𝑥(𝑡)−𝑦(𝑡)²⩽6𝑇𝑙₁𝐿₂𝑀𝑙₁𝑙₂+1(𝑇+4)1+𝑘₂𝑇+6𝑙₁𝜌𝑀𝑙₂+1^𝜌_𝑘=1𝛽_𝑘sup_{0⩽𝑡⩽𝑇}𝔼𝑥(𝑡)−𝑦(𝑡)². (28)
Theorefore, Φ is a contraction mapping from ℋ₂ to ℋ₂, and hence Φ has a unique fixed point. Thus the system (4) is completely controllability on [0,𝑇].

4. Example

Consider the impulsive stochastic integrao-differential wave equation with control 𝑢(𝑡,𝑧)∈𝐿²[0,1], 𝜕²𝑧𝜕𝑡²𝜕=𝑎²𝑧𝜕𝜉²+𝑏𝑢(𝑡,𝑧)+𝐹𝑡,𝑧(𝑡),^𝑡₀𝑓((𝑠(𝑡)𝑡,𝑠,𝑧))𝑑𝑠+𝐺𝑡,𝑧,^𝑡₀𝑔(𝑡,𝑠,𝑧(𝑠))𝑑𝑠𝜕𝑤,𝜕𝑡𝑡≠𝑡_𝑘,0⩽𝜉⩽1, (29) where 𝜕𝑤/𝜕𝑡 is white noise and initial and boundary conditions are 𝑧(𝑡,0)=𝑧(𝑡,1)=0,𝑡≠𝑡_𝑘,Δ𝑧𝑡_𝑘(𝜉)=𝐼¹_𝑘𝑧_𝑡𝑘,Δ𝑧^′𝑡_𝑘(𝜉)=𝐼²_𝑘𝑧_𝑡𝑘,𝑧(0,𝜉)=𝑧₀(𝜉),𝜕𝑧(0,𝜉)𝜕𝑡=𝑧₁(𝜉). (30) Let 𝐻=𝐿²[0,1]; then 𝐴∶𝐻→𝐻, 𝐴𝑧=𝑧^′′. Domain of operator 𝐴 is 𝐷(𝐴)=𝑧∈𝐻∣𝑧,𝑧^′iscontinuous,𝑧^′′𝑧(0)(1).∈𝐻,=𝑧=0 (31) Let 𝑧𝑍=𝜕𝑧(0)=𝑧𝜕𝑡,𝑍⁰𝑧₁0𝑏,𝐵=,0𝐹(𝑡)𝐹=𝑡,𝑧,∫^𝑡₀𝑓((𝑠,𝑡,𝑠,𝑧))𝑑𝑠0𝐺(𝑡)𝐺=𝑡,𝑧,∫^𝑡₀𝑔((𝑠.𝑡,𝑠,𝑧))𝑑𝑠 (32)(33)

Then the system (29) is 𝑑𝑍=𝒜𝑍+𝐵𝑢+𝐹(𝑡,𝑍)𝑑𝑡+𝐺(𝑡,𝑍)𝑑𝑊,𝑡≠𝑡_𝑘,Δ𝑍_𝑘=𝐼¹_𝑘𝐼²_𝑘,𝑡=𝑡_𝑘(0)=𝑧,𝑘=1,2,…,𝑚,𝑍⁰𝑧₁, (34) where 𝒜=0𝐼−𝐴0, (35) and 𝒜 is the infinitesimal generator of a strongly continuous semigroup ℎ(𝑡),𝑡⩾0, on 𝑋=𝐷(𝐴^1/2)⊕𝐻, for 𝑥×𝑦∈𝑋: 𝑆(𝑡)𝑥𝑦=^∞_𝑛=1cos(𝑛𝜋𝑡)(𝑛𝜋)⁻¹()−()()()sin𝑛𝜋𝑡𝑥𝑛𝜋sin𝑛𝜋𝑡cos𝑛𝜋𝑡𝑦^𝑛_𝑛𝑒_𝑛, (36)𝑒_𝑛(𝜃)=√2sin(𝑛𝜋𝜃),𝜃∈[0,1]. Write ℎ(𝑡) is ℎ(𝑡)=𝐶(𝑡)𝑆(𝑡)𝐴𝑆(𝑡)𝐶(𝑡), (37) where 𝑆(𝑡)𝑥∶=^𝑡₀𝐶(𝑠)𝑥𝑑𝑠,𝑥∈𝐻. (38) The mild solution of system (29) is 𝑧(𝑡)=𝐶(𝑡)𝑧₀+𝑆(𝑡)𝑧₁+^𝑡₀𝑆((𝑠)(+𝑡−𝑠)𝐵𝑢+𝑓𝑠,𝑧)𝑑𝑠_𝑡0𝑆(𝑡−𝑠)𝑔(𝑠,𝑧)𝑑𝑤_𝑠+_{0⩽𝑡𝑘⩽𝑡}𝐶𝑡−𝑡_𝑘𝐼¹_𝑘𝑧_𝑡𝑘+_{0⩽𝑡𝑘⩽𝑡}𝑆𝑡−𝑡_𝑘𝐼²_𝑘𝑧_𝑡𝑘 (39) by [7]; the stochastic linear system of (29) is complete controllable. Then from Theorem 5 one can easily prove system (29) is completely controllable, if the functions 𝐹, 𝐺, 𝑓, 𝑔, 𝐼¹_𝑘, 𝐼²_𝑘 satisfy Lipschitz condition and linear growth condition.

5. Conclusions

The complete controllability of impulsive stochastic integro-differential systems in Hilbert space has been investigated in this paper. Sufficient conditions of complete controllability for impulsive stochastic integro-differential systems are established by using the Banach fixed point theorem. An example illustrates the efficiency of proposed results.

Acknowledgments

This work is supported by National Natural Science Foundation of China under Grant nos. 61273126 and 61174078, PhD Start-up Fundation of Guangxi University of Science and Technology (no. 03081520), and Guangxi Higher Education Science Research Projection (no. 201203YB125).