Controllability study on fractional order impulsive stochastic differential equation∗

In this paper, we consider a fractional order stochastic differential system with impulsive conditions. The necessary and sufficient conditions for the controllability of associated linear stochastic system is studied by using the controllability Grammian matrix defined by Mittag-Leffler function. The sufficient condition for controllability of the proposed nonlinear system is proved by using Banach fixed point theorem. An example is provided to illustrate the theory using the numerical integration by Haar wavelet approximation method.