Optimal Control Problem for Fractional Stochastic Nonlocal Analytic Resolvent Semilinear Integro-Differential System

This research explores the existence of mild solutions and optimal control strategies for fractional stochastic nonlocal semi linear integro-differential systems using resolvent operators. Utilizing advanced mathematical techniques, including the Banach Fixed Point Theorem and Gronwall's Inequality, we establish conditions ensuring the uniqueness and existence of mild solutions. These findings provide insights into the complex memory and hereditary effects inherent in fractional systems, enhancing control strategies within infinite-dimensional spaces. Additionally, we develop optimal control strategies and prove the existence of time-optimal controls under specific hypotheses. The integration of resolvent operators presents a structured approach to managing fractional-order systems, with potential applications in various scientific and engineering domains. This study contributes to the theoretical foundation and practical implementation of fractional calculus in control theory.