Approximate Controllability of Impulsive Stochastic Fractional Differential Equations With Nonlocal Conditions

Abstract

This paper studies the approximate controllability of an impulsive neutral stochastic integro-differential equation with nonlocal conditions and infinite delay involving the Caputo fractional derivative of order q ∈ (1, 2) in separable Hilbert space. The existence of the mild solution to fractional stochastic system with nonlocal and impulsive conditions is first proved utilizing fixed point theorem, stochastic analysis, fractional calculus and solution operator theory. Then, a new set of sufficient conditions proving approximate controllability of nonlocal semilinear fractional stochastic system involving impulsive effects is derived by assuming the associated linear system is approximately controllable. Illustrating the obtained abstract results, an example is considered at the end of the paper.

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

“…In order to study the approximate controllability (see [12,32,38,42,51]) of the fractional control system (1), we need to introduce the relevant operator Note that the assumption (H0) is equivalent to the fact that the fractional linear control system (2) is approximately controllable on '.…”

“…In the infinite dimensional systems, two basic concepts of controllability are exact and approximate controllability. Exact controllability enables to steer the system to arbitrary final state while approximate controllability is weaker concept of controllability and it is possible to steer the system to an arbitrary small neighborhood of the final state (see, for example, [2,9,12,46,50]). Impulsive effects [47] exist widely in many evolution process because, the impulsive effects may bring an abrupt change at a certain moments of time involving such fields as economics, mechanics, electronics, telecommunications, medicine and biology, etc.…”

“…Muthukumar and Thiagu [32] proved the existence of solutions and approximate controllability of fractional nonlocal stochastic differential equations of order ∈ (1,2] with infinite delay and Poisson jumps in Hilbert spaces by using fixed point theory and natural assumption that the corresponding linear system is approximately controllable. Hence problem of existence of approximate controllability for nonlinear fractional impulsive stochastic differential equations with nonlocal conditions and infinite delay has been studied by several authors were received significant attention in modern days (see [10,12,33,40,49] and references therein). Rajivganthi et al [38] studied existence of solutions and approximate controllability of impulsive fractional stochastic differential systems with infinite delay and poisson jumps.…”

An analysis on the approximate controllability of neutral impulsive stochastic integrodifferential inclusions via resolvent operators

“…In the manuscript [36] , the authors discussed the approximate controllability of neutral stochastic integrodifferential systems with impulsive effects. For more details, refer to [37] , [38] , [39] , [40] .…”

Complete controllability for a class of fractional evolution equations with uncertainty

“…It is well-known that many problems in control theory such as stabilization, optimal control or pole assignment can be solved under assumption that the considered system is controllable. After first introducing by Kalman in [20], the controllability of dynamic systems has attracted a lot of interest, see [5,11,16,18,31]. The concept of controllability can be separated into complete controllability and approximate controllability, where the concept of complete controllability is that the dynamical system can be steered exactly from one state to another state while the concept of approximate controllability means that the dynamical system can be steered to a small neighborhood of final state in a given time, that is to say that the complete controllability always implies the approximate controllability, but in general, the converse statement is not true excepting the case of finite dimensional system.…”