On the Existence and Stability of a Hybrid Delay Itô-Differential Equation with Stochastic Feedback Control

In this paper, the main objective is to prove the existence of solutions U∈C([0, ϖ], L2(Q)) for a hybrid delay Itô-differential equation with stochastic feedback control, a problem that incorporates delay effects introducing memory-like behavior into the system and leading to intricate dynamics. Additionally, the uniqueness of the solutions with sufficient conditions are provided. Furthermore, the solutions with the continuous dependence on initial data and certain functions, as well as the concept of Hyers-Ulam stability, are analyzed. These findings serve as a foundational framework from which well-established results in the literature naturally emerge. The concluding section is dedicated to applying these results to specific examples, illustrating the uniqueness and existence of solutions for hybrid delay Itô-differential equations with stochastic feedback control. These examples not only validate the theoretical findings but also enhance understanding by offering practical insights into the study of such equations.