On Approximate Controllability of Second Order Fractional Impulsive Stochastic Differential System with Nonlocal, State-dependent Delay and Poisson Jumps

Krishnan Thiagu

doi:10.11648/j.ajam.20210902.13

Abstract

The main scope of this paper is to focus the approximate controllability of second order (q∈(1,2]) fractional impulsive stochastic differential system with nonlocal, state-dependent delay and Poisson umps in Hilbert spaces. The existence of mild solutions is derived by using Schauder fixed point theorem. Sufficient conditions for the approximate controllability are established by under the assumptions that the corresponding linear system is approximately controllable and it is checked by using Lebesgue dominated convergence theorem. The main results are completly based on the results that the existence and approximate controllability of the fractional stochastic system of order 1<q≤2 and are derived by using stochastic analysis theory, fixed point technique, q-order cosine family {Ca(t)}t≥0, new set of novel sufficient conditions and methods adopted directly from deterministic fractional equations for the second order nonlinear impulsive fractional nonlocal stochastic differential systems with state-dependent delay and Poisson jumps in Hildert space H. Finally an example is added to illustrate the main results.

Highlights

The concept of semigroups of bounded linear operators is taken as an important concept to dealing with differential and integro-differential equations in Banach spaces [8, 14, 15, 17, 36]
The main purpose of this paper is to obtain the sufficient conditions of approximate controllability results for fractional impulsive stochastic differential system of order ∈
We can apply the previous results to study the approximate controllability of fractional impulsive stochastic differential system of order 1 < ≤ 2 with nonlocal, state-dependent delay and Poisson jumps of the form

Summary

Introduction

The concept of semigroups of bounded linear operators is taken as an important concept to dealing with differential and integro-differential equations in Banach spaces [8, 14, 15, 17, 36]. Many authors (see [13, 19, 26, 48, 51] and references therein) studied the existence and approximate controllability as well as stability of different types of fractional stochastic differential equations with state-dependent delay in Hilbert spaces under different suitable aspects. Approximate controllability results for fractional impulsive stochastic differential system of order ∈ The main purpose of this paper is to obtain the sufficient conditions of approximate controllability results for fractional impulsive stochastic differential system of order ∈ (1,2] with nonlocal, state-dependent delay and Poisson jumps in Hilbert spaces of the form ( ) = ( ) + ( ) + ( , ( , )). Further =( ) is an U -measurable D-valued random variable independent of !( ) and Poisson point process T with finite second moment

Preliminaries

Approximate Controllability

Application

Conclusion