<p>This research focuses on achieving precise fractional controllability for linear hyperbolic systems within a finite time frame. Control is applied at internal points along a specified curve within the spatial domain of the system. We use an extension of (HUM) to design a minimal-energy control, enabling the desired fractional state to be reached at time T. Hence, we consider a numerical approach that enables us to determine the explicit formula of the optimal control. Furthermore, we implement an algorithm on a one-dimensional wave equation with a mobile zone actuator to validate the practical applicability of the obtained results through computer simulations.</p>