Results on approximate controllability of fractional stochastic differential equations driven by mixed fractional Brownian motion with infinite delay via resolvent operators

This paper addresses the issue of approximate controllability for a class of control system which is represented by nonlinear factional order stochastic differential equations with infinite delay. By using fractional calculus, semigroup theory, stochastic analysis, and the technique of stochastic control theory, a new set of sufficient conditions for the approximate controllability of a fractional stochastic differential system are formulated and proved. More precisely, the results are established under the assumption that the corresponding deterministic fractional linear system is approximately controllable with using resolvent condition and techniques on fractional power operators. As an illustration of the application of the obtained results, an example is also provided.