Approximate controllability of stochastic Hilfer fractional control system with delay

This paper investigates the approximate controllability of stochastic Hilfer fractional control systems with time delay. First, we derive the mild solution of the considered system through rigorous applications of fractional calculus, Laplace transform and its inverse operator. Subsequently, the existence and uniqueness of this solution are systematically established via the Picard iteration method. Furthermore, under the fundamental assumption that the associated linear system is approximately controllable, we obtain sufficient conditions for achieving approximate controllability in the nonlinear case. Finally, an example is provided to validate the theoretical findings. The study contributes to the field by extending controllability analysis to stochastic Hilfer fractional systems with delay, while providing a unified approach integrating operator theory and iterative techniques.