Global uniform symptotic attractive stability of the non-autonomous bouncing ball system

The non-autonomous bouncing ball system consists of a point mass in a constant gravitational field, which bounces inelastically on a flat vibrating table. A sufficient condition for the global uniform attractive stability of the equilibrium of the non-autonomous bouncing ball system is proved in this paper by using a Lyapunov-like method which can be regarded as an extension of Lyapunov’s direct method to Lyapunov functions which may also temporarily increase along solution curves. The presented Lyapunov-like method is set up for non-autonomous measure differential inclusions and constructs a decreasing step function above the oscillating Lyapunov function. Furthermore, it is proved that the attractivity of the equilibrium of the bouncing ball system is symptotic, i.e. there exists a finite time for which the solution has converged exactly to the equilibrium. For this attraction time, an upper-bound is given in this paper.