Approximate controllability of Fractional stochastic systems

   The aim of this book is to study the principal concepts of the approximate controllability for some complicated classes of   stochastic control systems of fractional order driven by mixed fractional Brownian motion in  real separable Hilbert spaces and presented as new classes. The problems of approximate controllability of nonlinear fractional Riemann-Liouville type of order   stochastic integro-differential equations driven by mixed fractional Brownian motion investigated and proved  by using necessary condition of independence the fractional Brownian motion and Wiener process and also, by using  Banach fixed point theorem.    Also, the result is extended to study the approximate controllability for nonlinear fractional Riemann-Liouville type of order  stochastic perturbed control systems with driven by mixed fractional Brownian motion by using Krasnoselskii fixed point theorem.    Finally, the approximate controllability of nonlinear fractional Caputo of order Sobolev type stochastic differential equations driven by mixed fractional Brownian motion   has been studied by using some complicated integrations and the compactness of operators.