New exploration on the existence and null controllability of fractional Hilfer stochastic systems driven by Poisson jumps and fractional Brownian motion with non-instantaneous impulse

This manuscript aims to study a new class of non-instantaneous impulsive Hilfer stochastic differential system of fractional-order 1 2 ≤ β ≤ 1 , driven by fractional Brownian motion and Poisson jumps in a Hilbert space. A new set of sufficient conditions is derived to ensure the existence of the mild solution and exact null controllability of the proposed control system utilizing the stochastic analysis theory, fractional calculus, fixed point technique, and the theory of a resolvent operator. Further, we extend the proposed fractional-order system to the nonlocal differential system in which we consider the nonlocal initial conditions, and briefly discuss the exact null controllability result. At the end, we provide two examples to demonstrate the applicability of the abstract results.