The stability of non-smooth dynamical systems is a novel research field which is receiving much attention in the mathematical as well as engineering community. Mechanical systems with impact phenomena and unilateral constraints form an important class of non-smooth systems as they arise in many engineering applications. Despite the huge amount of textbooks and papers on (Lyapunov) stability in various fields of engineering science, Lyapunov stability properties of non-autonomous non-smooth systems described by measure differential inclusions has not been considered so far. In order to study Lyapunov stability criteria of equilibria in non-smooth, explicitly time-dependent (i.e. non-autonomous) mechanical systems with unilateral constraints, we attempt to investigate the stability of the equilibrium of an apparently simple mechanical system meeting the requirements of non-smoothness and explicit time-dependence. A standard problem of chaotic dynamics, which has been extensively studied in the literature, is a ball in a constant gravitational field which bounces inelastically on a flat vibrating table. The governing equations of motion are highly nonlinear due to the unilateral contact and generally do not allow for any closed form solution.

Global attractive stability conditions for the equilibrium of the bouncing ball system are proven in this research project using an extension of Lyapunov’s direct method to non-autonomous systems.Furthermore, it is proven that the attractivity of the equilibrium 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 determined.

Trajectory of the ball bouncing on the table (left) and a Lyapunov candidate function (right).