Approximate controllability of a semilinear impulsive stochastic system with nonlocal conditions and Poisson jumps

The objective of this paper is to investigate the approximate controllability of a semilinear impulsive stochastic system with nonlocal conditions and Poisson jumps in a Hilbert space. Nonlocal initial condition is a generalization of the classical initial condition and is motivated by physical phenomena. The results are obtained by using Sadovskii’s fixed point theorem. Finally, an example is provided to illustrate the effectiveness of the obtained result.


Introduction
The concept of controllability plays a major role in both finite and infinite dimensional spaces for systems represented by ordinary differential equations and partial differential equations. One of the basic qualitative behaviors of a dynamical system is the controllability. The problem of controllability is to show the existence of control function, which steers the solution of the system from its initial state to the final state, where the initial and final states may vary over the entire space. Conceived by Kalman, the controllability concept has been studied extensively in the fields of finite and infinite-dimensional systems. If a system cannot be controlled completely, then different types of controllability, such as approximate, null, local null, and local approximate null controllability, can be defined. For more details, the reader may refer to [1][2][3][4][5][6][7][8][9] and the references therein.
Meanwhile, the theory of impulsive differential equations describes processes which experience a sudden change of their state at certain moments. Dynamics of many evolutionary processes with such a characteristic arise naturally and are often, for example, population dynamics, control theory, physics, biology, medicine, etc. These perturbations can be well-approximated as instantaneous change of state or impulses. These processes are modeled by impulsive differential equations. Few works have been reported in the study of stochastic integrodifferential equations with impulsive effects, refer to [2,[10][11][12][13].
The Poisson jumps have become very popular, they are extensively used to model many of the phenomena arising in areas such as economics, finance, physics, biology, medicine, and other sciences. Moreover, many practical systems (such as sudden price variations [jumps] due to market crashes, earthquakes, epidemics, and so on) may undergo some jump-type stochastic perturbations. The sample paths of such systems are not continuous. Therefore, it is more appropriate to consider stochastic processes with jumps to describe such models. In general, these jump models are derived from Poisson random measure. The sample paths of such systems are right-continuous and have left limits. Recently, an increasing interest in the study of stochastic differential equations with jumps has been observed [14][15][16]. Luo and Liu [17] established the existence and uniqueness theory of mild solutions to stochastic partial functional differential equations with Markovian switching and Poisson jumps. It should be noted that most of the literature in this direction was mainly concerned with results on controllability of stochastic equations without jumps. However, up to now controllability problems for nonlinear stochastic dynamical systems with jumps have not been considered in the literature. In order to fill this gap, this paper considers the approximate controllability of semilinear stochastic control systems with nonlocal conditions using Sadovskii's fixed point theorem.
The goal of the present research work is to focus on studying the approximate controllability of a semilinear impulsive stochastic system with nonlocal conditions and Poisson jumps of the form Also, the fixed moments of time t k satisfy 0 = t 0 < t 1 < · · · < t m < t m+1 = b, z(t + k ) and z(tk ) denotes the right and left limits of z(t) at t = t k , respectively; z(t k ) = z(t + t )z(tk ) represents the jump in the state z at time t k with I k determining the size of the jump.

Preliminaries
Let (Ω, , P) be a complete probability space with a normal filtration t , t ∈ [0, T]. Let H, U, and E be the separable Hilbert spaces. Let w be a Q-Wiener process on (Ω, T , P) with the covariance operator Q such that tr Q < ∞. Let us suppose that there exists a complete orthonormal system e n in E, a bounded sequence of real numbers λ n , where λ n > 0 such that Qe n = λ n e n , n = 1, 2, 3, . . . , and a sequence β n of independent Brownian motions such that λ n β n (t)e n , t ∈ J, Let H 2 be a closed subspace of PC (J 1 , L 2 (Ω, t , H)) that consists of all measurable and t -adapted processes z(·) : t ∈ [-h, T] with the norm topology by where v is the characteristic measure of N P , which is called the compensated Poisson random measure.

Definition 2.2
A stochastic process z ∈ H 2 is a mild solution of (1) if, for each u ∈ L 2 ([0, T], U), it satisfies the integral equation The following are the main assumptions in this paper: (H4) There exist some positive constants M p such that (H5) I k : H → H satisfies is the controllability Gramian. Observe that the linear deterministic system corresponding to (1) is approximately controllable on [t, T] iff the operator ζ (ζ I + Γ T t ) -1 → 0 strongly as ζ → 0 + .

Main result
For any ζ > 0 and z T ∈ L 2 (Ω, T , H), we define the control function T (Tt k )I k z 1 (t k ) .

Lemma 3.2
There exists M u > 0 such that, for all z 1 , z 2 ∈ H 2 , we have Proof Let z 1 , z 2 ∈ H 2 . By hypothesis and Holder's inequality, we obtain When u ζ (t, z 2 ) = 0, the second inequality can be proved in the same approach.
Step 2. Let B n = {z ∈ H 2 : E z(t) 2 H ≤ n}, then the set B n is obviously a bounded, closed, and convex set in H 2 for each integer n > 0.
From Holder's inequality and (H1), we get From (H2), we get Next, from (H4) and (H5), we get Similarly, from assumption (H7), we get Now, we have to prove that there exists a number n > 0 such that Θ ζ (B n ) ⊆ B n .
If not, for each n > 0, there exists a function z n (·) ∈ B n but Θ ζ z n does not belong to B n , that is, E Θ ζ z n (t) 2 H > n for t ∈ J. Also, by Lemma 3.2 and hypotheses (H2), (H3), we get Now dividing each side by n and considering the limit as n → ∞, we get This is a contradiction to condition (5). Thus, Θ ζ B n ⊆ B n for any n > 0.

Conclusion
In this paper we have established the approximate controllability of a semilinear impulsive stochastic system with nonlocal conditions and Poisson jumps in a Hilbert space. The results are obtained by using Sadovskii's fixed point theorem and semigroup theory. Further the results have been verified by a proper example. In future the criteria may be extended to semilinear impulsive stochastic integrodifferential equations driven by a fractional Brownian motion.