Optimal Control Strategies and Continuous Dependence for Stochastic Hilfer Fractional Systems With Delay: A Volterra-Fredholm Integro-Differential Approach

This study investigates the conditions necessary for solving Sobolev Hilfer fractional Volterra-Fredholm integro-differential (SHFVFI) control problems within the range (Formula presented.) on a Banach space. We first establish the existence of mild solutions for such systems by employing tools including the Laplace transform, semigroup theory, Mönch's fixed point theorem, the integro-differential approach of mixed type, and nonlocal conditions. We then extend the analysis to stochastic SHFVFI of order (Formula presented.) including finite delays, along with addressing optimal control problems. Furthermore, the study provides insights into the continuous dependence outcomes for the given problems along with relevant hypotheses. To illustrate the practical applicability of our theoretical results, we present two illustrative examples involving Hilfer fractional partial differential equations with and without delay.