Controllability of stochastic multi-term fractional-order impulsive systems involving state delay

This study investigates the damping behavior of impulsive fractional-order stochastic systems with state delays, which are essential for modeling dynamical processes exhibiting both memory and stochastic effects. The system is formulated using Caputo fractional derivatives and Mittag-Leffler functions, allowing for analytical expressions that accurately capture the hereditary properties of fractional dynamics. The main contribution of this work is the establishment of sufficient conditions for the controllability of both linear and nonlinear systems, achieved through a combination of stochastic analysis and fixed-point techniques, explicitly accounting for the influences of delays, damping, and impulses. The proposed methodology extends existing controllability results to a broader class of fractional stochastic systems and provides a systematic approach for analyzing their damping mechanisms. The theoretical findings are illustrated through a numerical example, confirming the accuracy and effectiveness of the developed methodology.