A discussion on approximate controllability of Sobolev-type Hilfer neutral fractional stochastic differential inclusions

In this paper, the approximate controllability of Sobolev-type Hilfer neutral fractional stochastic differential inclusions in Hilbert spaces is considered. By using the stochastic analysis theory, fractional calculus, multivalued maps, and Karlin's fixed point technique, a new set of necessary and sufficient conditions, are formulated which guarantees the approximate controllability of the nonlinear Hilfer fractional stochastic system. The results are obtained under the assumption that the associated linear system is approximately controllable. Finally, two examples are presented to illustrate the theory of the obtained results.