Fractional stochastic systems with memory: Existence, Ulam–Hyers stability, and local approximate controllability

This paper studies a class of fractional stochastic integro-differential systems with memory effects and control inputs. The model involves a Caputo fractional derivative of order $\alpha \in (1/2,1)$, a Volterra-type memory kernel, and stochastic perturbations driven by a Wiener process. Under standard Lipschitz and boundedness assumptions, we establish the existence and uniqueness of mild solutions in the space of mean-square continuous processes via the Banach contraction principle, together with an explicit contraction condition. We further prove Ulam–Hyers stability, providing quantitative bounds that characterize the sensitivity of solutions to perturbations. In addition, we investigate local approximate controllability through a scaled bounded-control approximation framework. We show that, for sufficiently small terminal times and for targets approaching the initial state at the fractional scaling rate $\Vert x_T-x_0\Vert \le \rho T^{\alpha -1/2}$ the controllability of the nonlinear stochastic system can be inferred from that of an associated reduced linear system using controls whose $L^2$-norms remain uniformly bounded as $T\to 0^{+}$. A finite-dimensional example is provided to demonstrate that the scaled bounded-control approximation hypothesis can be verified explicitly in a concrete setting and to illustrate the applicability of the controllability framework. The results provide a unified analytical framework for studying well-posedness, stability, and controllability in fractional stochastic systems with memory.