This research explores the effect of the tempered \(\Psi \) -Caputo fractional derivative on the controllability characteristics of both linear and nonlinear fractional dynamical systems. The research covers systems without delays as well as those featuring multiple delays. For linear systems, controllability is determined by verifying the positive definiteness of the Grammian matrix, which outlines the necessary and sufficient conditions. In the case of nonlinear systems without delays, Schauder’s fixed point theorem is employed to establish sufficient conditions for the existence of a solution. For nonlinear systems involving multiple delays, iterative methods are employed to establish the required conditions. To support the theoretical insights, the study includes illustrative examples accompanied by graphs that demonstrate the derived conclusions.
