Trajectory tracking for a particle in elliptical billiards

Abstract

An infinitely rigid unitary mass (particle) is considered, moving on a planar region delimited by a rigid elliptical barrier (elliptical billiards) under the action of proper control forces. A class of periodic trajectories, involving an infinite sequence of non-smooth impacts between the mass and the barrier at fixed times, is found by using an LMIs based procedure. The jumps in the velocities at the impact times render difficult (if not impossible) to obtain the classical stability and attractivity properties for the dynamic system describing the tracking error behaviour. Hence, the tracking control problem is properly stated using notions similar to the quasi stability concept in V. Lakshmikantham, D.D. Bainov and P.S. Simeonov, Theory of Impulsive Differential Equations, 6, World Scientific, 1989. A controller (whose state is subject to discontinuities) based on the internal model principle is shown to solve the proposed tracking problem, giving rise to control forces that are piecewise continuous function of time, with discontinuities at the desired impact times and at the impact times of the particle with the barrier.