Trajectory controllability of second order dynamic equations on time scales

This paper is concerned with the trajectory controllability (T-controllability) of second-order nonlinear dynamic equations defined on time scales. We first study linear systems, providing detailed proof of T-controllability by employing the Gronwall inequality. In the case of nonlinear equations, we assume the Lipschitz continuity on the nonlinearity to ensure the applicability of our results. We include several numerical examples to demonstrate the practical application of the derived results. The formulation of the problem using the theory of time scales ensures that the obtained results hold true in both the continuous and discrete time domains and any combination of these domains, demonstrating the non-trivial nature of the results and their generalisation to existing ones.