This paper investigates the conditions required for approximate controllability of a class of \(\psi \) -Caputo fractional-order semilinear delay systems. By leveraging the contraction mapping principle and Schauder’s fixed-point theorem, we rigorously establish the existence and uniqueness of solutions for these systems. The theoretical results are supported by several illustrative examples, demonstrating their applicability in the broader context of control theory and fractional differential equations. This work provides new insights into the analysis and control of fractional-order systems, offering valuable contributions to the field of dynamic systems and control.
