Trajectory controllability of multi-term fractional stochastic differential equations with Poisson jumps

This paper demonstrates the solvability and trajectory controllability of a multi-term fractional stochastic differential equation driven by Poisson jumps in the pth-moment. In general, an integer order differential equation with n derivatives can be turned into an abstract form of the first order differential system, but fractional differential equation does not have this property. This study introduces a method to investigate fractional differential equation with more than one fractional derivatives. The key findings of this work include the existence and trajectory control of the proposed model without imposing the compactness condition on the resolvent operators. The solvability of the proposed system is addressed by employing fractional calculus, semigroup theory, and the stochastic version of Banach fixed point theorem. The existence of trajectory controllability is established using Gronwall's inequality. Finally, a multi-term time-fractional wave equation with stochastic diffusion is provided to illustrate and justify the importance of the proposed study.