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

Muath Awadalla; Maryam G. Alshehri; Muath Awadalla; Maryam G. Alshehri

doi:10.3934/math.2026670

AIMS Mathematics

2026, Volume 11, Issue 6: 16305-16333. doi: 10.3934/math.2026670

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

Muath Awadalla ^{1
,
,},
Maryam G. Alshehri ^{2
,
,}

1.
Department of Mathematics, College of Science, King Faisal University, P.O. Box 400, Al-Ahsa, 31982, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box741, Tabuk 71491, Saudi Arabia

Received: 07 April 2026 Revised: 27 May 2026 Accepted: 02 June 2026 Published: 08 June 2026
MSC : 93C10, 93C40, 26A33, 93A16, 93C95, 93B35, 93D23, 68T09

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 $ \|x_T - x_0\| \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 \rightarrow 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.
- fractional-order control,
- variable-order systems,
- power grid resilience,
- data-driven control,
- adaptive control
Citation: Muath Awadalla, Maryam G. Alshehri. Fractional stochastic systems with memory: Existence, Ulam–Hyers stability, and local approximate controllability[J]. AIMS Mathematics, 2026, 11(6): 16305-16333. doi: 10.3934/math.2026670

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Abstract

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 $ \|x_T - x_0\| \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 \rightarrow 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.

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