Complete Controllability of Impulsive Stochastic Integrodifferential Systems in Hilbert Space

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.


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][2][3][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 -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:  () = [ () +  ()]  + Σ ()  () ,  ∈ [0, ] , where  0 is F 0 -measurable,  is separable Hilbert space,  is the infinitesimal generator of a strongly continuous semigroup () on ,  ∈ L(, ), () is feedback control, () is -Wiener process, and Σ ∈ L 2 ( 1/2 , ).For nonlinear stochastic systems in infinite dimensional space, there are also many results on the controllability theory (see [8][9][10][11][12][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][20][21][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.
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.

Preliminaries
Let (Ω, F, P) be a complete probability space with a filtration {F  } ⩾0 satisfying the usual conditions (i.e., it is right continuous and F 0 contains all P-null sets).We consider three Hilbert spaces , , and , and a -Wiener process on (Ω, F, P) with the covariance operator  ∈ L() such that tr  < ∞.Let ⟨⋅⟩ and ‖ ⋅ ‖ denote inner product and norm of , respectively.L(, ) is the space of all linear bounded operator from a Hilbert space  to a Hilbert space .We also employ the same notation ‖ ⋅ ‖ for the norm of L(, ).We assume that there exists a complete orthonormal {  } in , a bounded sequence of nonnegative real numbers   such that   =     ,  = 1, 2, . .., and a sequence {  } of independent Brownian motions such that and Let PC([0, ],  2 (Ω, F, P; )) = { :  is a function from [0, ] into  2 (Ω, F, P; ) such that () is continuous at  ̸ =   , left continuous at  =   , and the right limit . By a solution of system (2), we mean a mild solution of the following nonlinear integral equation: where  ∈   :=  2 , () ⩾0 denotes the strongly continuous semigroup generated by the operator .Now let us introduce the controllability operator Π   associated with (4) (see [8]), which belongs to L(, );  * is the adjoint operator of .

Lemma 4. Assume that the operator Π 𝑇
is invertible.Then for arbitrary target   ∈  2 (F  , ), the control Proof.Substituting (10) into (4), we can obtain that The proof is completed by letting  =  in (11).

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 C. The linear system ( 9) is completely controllable.By Lemma 3, for some  > 0, E⟨Π  0 , ⟩ ⩾ E‖‖ 2 , for all  ∈  2 (F  , ).Consequently, Now for convenience, let us introduce the following notations: Theorem 5. Suppose that assumptions A, B, C, and D are satisfied.Then system (4) is completely controllable on [0, ].
Proof.For arbitrary initial data  0 ∈ H 2 , we can define a nonlinear operator Φ from  2 to  2 as the following: where () is defined by (10).By Lemma 4, the control (10) transfers system (4) from the initial state  0 to the final state   provided that the operators Φ has a fixed point in H 2 .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 H 2 .To do this, we can employ the contraction mapping principle.In the following, we will divide the proof into two steps. Firstly Using Holder inequality, B-D-G inequality (here  1 = 4), and Assumption C, we have the following estimates: Meanwhile by control function (15), we have So similar as in (17), we get From ( 17)-( 19), we have for all  ∈ [0, ], where  is constant.This implies that Φ maps H 2 into itself.Secondly, we prove that Φ is a contraction mapping on Using Lipschitz condition, similiar to  1 - 5 , we have the following estimates: 2 ⩽ 4  (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 , , , ,  1  ,  2  satisfy Lipschitz condition and linear growth condition.

Conclusions
The complete controllability of impulsive stochastic integrodifferential 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.