Asymptotic tracking of periodic trajectories for a simple mechanical system subject to non-smooth impacts

An infinitely rigid, fully-actuated mass is considered, which moves on a plane within a closed region delimited by an infinitely massive and rigid circular barrier. The tracking problem of a class of periodic trajectories involving an infinite number of impacts is considered. Since the jumps in the velocities at the impact times render difficult (if not impossible) to obtain the classical stability and attractivity properties for the tracking error, such properties are properly amended for the case of interest. A simple PD-like control law is proposed, giving rise to control forces that are piecewise continuous functions of time.