Abstract <p>This paper addresses the trajectory controllability of time-varying fractional dynamical systems, covering both linear and nonlinear cases in the Caputo type fractional derivative framework. The study employs advanced mathematical tools, including the state-transition matrix and the Grönwall–Bellman inequality, to establish theoretical results for these systems. For the non-linear case, the analysis incorporates assumptions of Lipschitz-type conditions on the nonlinear terms, ensuring the mathematical rigor and feasibility of the approach. Additionally, detailed examples are presented to illustrate the theoretical outcomes, providing insight into their practical implications, and demonstrating the applicability of the proposed methods.</p>