Integral contractor and conformable fractional differential transform approach to study optimal control, T-controllability and Ulam-Hyer’s-Rassia’s stability for higher-order stochastic system

Abstract

The increasing complexity of dynamical systems in science and engineering, particularly those influenced by memory and uncertainty, motivates the study of conformable fractional stochastic differential equations (CFSDEs). These models capture both fractional-order dynamics and stochastic behavior, offering a richer framework than classical approaches. This paper investigates the optimal control, uniqueness, Ulam–Hyers–Rassias stability (UHRS), and trajectory (T)-controllability of mild solutions to CFSDEs with fractional-order initial conditions. The analysis is further extended to coupled CFSDEs and higher-order Sobolev-type systems, where both the governing equations and boundary conditions involve fractional-order terms. Methodologically, stochastic analysis, sequencing techniques, and bounded integral contractors are employed to establish the main results. Unlike earlier studies, the proposed framework avoids reliance on the induced inverse of the controllability operator and does not impose Lipschitz restrictions on nonlinear functions. Grownwall’s inequality is used to derive T-controllability, while Balder’s theorem ensures optimal controllability. In addition, dynamic constraints are expressed in terms of conformable fractional derivatives, and variational methods yield optimality conditions. To support practical implementation, the conformable fractional differential transform (CFDT) method is applied for the numerical modeling of fractional optimal control problems (FOCPs), with illustrative applications demonstrating the effectiveness of the proposed approach.