Wellposedness and trajectory controllability of random impulsive stochastic differential systems driven by fBm

We present a framework of studying the mild solution to random impulsive stochastic differential equations (RISDEs) with fractional Brownian motion (fBm) and Poisson jumps in Hilbert spaces. The existence results of the proposed stochastic system is obtained using stochastic analysis and Kransnoselskii’s fixed point theorem. Then the stability results including continuous dependence of solutions on initial conditions, Hyer’s-Ulam stability and mean square exponential stability of the mild solution are studied. Furthermore, under some reasonable assumptions, the Trajectory (T-)controllability of the investigated system is established using extended Gronwall’s inequality. The obtained results will generalize the existing work of Shu et al.