An Investigation on the Approximate Controllability of Non-instantaneous Impulsive Hilfer Sobolev-Type Fractional Stochastic System Driven by the Rosenblatt Process and Poisson Jumps

In this paper, we aim to establish a set of sufficient conditions for the existence of an integral form mild solution and approximate controllability for a class of Sobolev-type Hilfer fractional stochastic differential systems driven by the Rosenblatt process and Poisson jumps. In the proposed control problem, we deal with a system that is under non-instantaneous impulsive effect. The sufficient condition for the existence of a mild solution for the proposed nonlinear control system has been established by using Schauder's fixed point theorem. The approximate controllability results for the proposed control problem have been established under the consideration that the corresponding linear system is approximate controllable. By utilising stochastic analysis, the theory of resolvent operator, fractional calculus, and the fixed point technique, sufficient conditions have been established. At the end, an example is given to illustrate the abstract results.