Trajectory controllability of semilinear dynamic systems on time scales

This paper explores the trajectory controllability of semilinear dynamic systems defined over time scales, which is an important aspect in understanding and manipulating the behavior of such systems across discrete and continuous domains. We address the controllability of these systems under the assumption that the nonlinearities satisfy a Lipschitz-type condition. Our approach involves a detailed analysis of how these conditions impact the ability to steer the system’s state along a desired trajectory within a finite-time horizon. We establish sufficient conditions for trajectory controllability (T-controllability), providing a theoretical framework that extends classical results from differential and difference equations to the broader context of time-scale calculus. To illustrate the practical implications of our theoretical findings, we include several numerical examples that demonstrate the application of our results to specific semilinear dynamic systems, highlighting the versatility and effectiveness of our approach.