Well-Posedness and Trajectory Controllability of Coupled Fractional Stochastic Delay Differential Equations with Periodic Motion

Abstract This paper investigates the well-posedness and exponential stability of a class of coupled fractional stochastic differential equations (FSDEs) with periodic motion. Using the Banach fixed point theorem (FPT), we establish existence and uniqueness of mild solutions under relaxed and nonstandard conditions, thereby extending classical results that typically rely on strong Lipschitz continuity. Furthermore, we derive weaker sufficient conditions for exponential stability of nonlinear functionals associated with the coupled fractional stochastic system. The novelty of this work lies in the integration of fractional stochastic dynamics with periodic behavior under relaxed stability constraints and in the use of an integral inequality framework to analyze exponential stability in coupled FSDEs. This approach not only generalizes previous studies on deterministic or single fractional systems but also provides a unified treatment of coupling, randomness, and memory effects within the same analytical framework. The solution structure is characterized via the two-parameter Mittag-Leffler (M-L) function, which elegantly captures the fractional-order memory and decay properties of the system. Finally, a numerical illustration is provided to validate the theoretical findings and demonstrate the practical applicability of the proposed results.