Trajectory tracking in rectangular billiards by unfolding the billiard table

In this paper, a position feedback controller is proposed to solve the tracking problem in rectangular billiards. Such a controller is obtained by transforming the billiard table into a surface on which the ball moves without experiencing any impact, i.e., by reflecting the billiard table rather than the ball trajectory, and designing a position feedback controller based on such an unfolded billiard table. Furthermore, it is shown how such an infinite billiard table can be mapped to the surface of a torus, thus leading to bounded trajectories of the ball.