Results on approximate controllability for fractional stochastic delay differential systems of order r ∈ (1,2)

Abstract

In this paper, we deal with the approximate controllability of fractional stochastic delay differential inclusions of order [Formula: see text]. By using fractional calculus, stochastic analysis, the theory of cosine family and Dhage fixed point techniques, a new set of necessary and sufficient conditions are formulated which guarantees the approximate controllability of the nonlinear fractional stochastic system. In particular, the results are established with the assumption that the associated linear part of the system is approximately controllable. Further, the result is extended to obtain the conditions for the solvability of controllability results for fractional inclusions with nonlocal conditions. Finally, an example is presented to illustrate the theory of the obtained result.