Results on optimal control of impulsive Hilfer fractional stochastic integro-differential equations

This paper addresses a class of Hilfer fractional stochastic nonlinear integro-differential equations incorporating impulsive effects and optimal control in Hilbert spaces. We first establish the existence of mild solutions, ensuring the solvability of the system through the application of fractional calculus, stochastic analysis, and fixed-point techniques. The analytical framework effectively manages the combined difficulties arising from nonlocal operators, stochastic perturbations, and impulsive dynamics. Subsequently, we formulate the associated optimal control problem and derive the necessary conditions for optimality. An illustrative example is provided to demonstrate the practicality and robustness of the theoretical results.