Optimal control of fractional-order stochastic systems under uncertainty with Poisson and V-jump perturbations

Abstract

We address the optimal control problem for a novel class of fractional-order uncertain--stochastic dynamical systems perturbed simultaneously by stochastic and epistemic jump disturbances. The system dynamics are governed by Caputo fractional derivatives and driven by a multi-noise framework comprising Brownian motion, Poisson random measures, canonical Liu processes, and finite-variation uncertain V-jump processes, thereby establishing a hybrid fractional system with double-jump features. The primary novelty is a unified analytical framework that combines memory effects with dual-source jump discontinuities under probabilistic randomness and epistemic uncertainty. We prove the existence, uniqueness, and continuous dependence of mild solutions in a hybrid probability--belief L2 framework under standard Lipschitz and growth conditions. We then define an optimal control problem with a combined probabilistic--uncertain performance criterion, verify the existence of optimal controls, and derive a Pontryagin-type maximum principle using a backward fractional adjoint system. Finally, numerical simulations for a fractional portfolio optimisation problem demonstrate the practical implications of memory, control, and multiple-jump disruptions.