Controllability Result for Nonlocal Nonlinear Impulsive Fuzzy Stochastic Differential Systems

This paper is concerned with sufficient conditions for the controllability for nonlocal nonlinear impulsivefuzzy stochastic systems in Banach space by using the concept of Mild solution. And the conditions are drive by using Sadovskii fixed point theorem, Hausdorff measure of noncompactness and operator semigroup. 1. Introduction The problem of controllability is to show the existence ofa control function, which steers the solution of the system from its initial state to afinal state, where the initial and final states may vary over the entire space. A largeclass of scientific and engineering problems is modeled by partial differential equations, integral equations or coupled ordinary and partial differential, integrodifferential equations. So it becomes importantto study the controllability results of such systems using available techniques. Several authorshave studied the problem of controllability of semilinear and nonlinear systems represented by differential and integrodifferential equations in finite or infinite dimensional Banach spaces [7, 9]. A standard approach is to transform the controllabilityproblem into a fixed point problem for an appropriate operator in a functionalspace.In this paper we studied the controllability of impulsive nonlinear nonlocal fuzzy stochastic differential equation described by dxሺtሻ = Axሺtሻ + Buሺtሻdt + f൫t, xሺtሻ൯dt + g൫t, xሺtሻ൯dwሺtሻ, t ∈ J ≔ ሾ0, aሿ, t ≠ t ୧ (1)