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

• Published in: Physica. D Vol. 241; no. 22; pp. 2029 - 2041
• Main Authors: Leine, R.I., Heimsch, T.F.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 15.11.2012
• Subjects: Attraction; Bouncing; Finite-time attractivity; Gravitational fields; Impact; Lyapunov functions; Mathematical analysis; Measure differential inclusions; Oscillating; Stability; Stability theory; Tables (data); Impact; Finite-time attractivity; Measure differential inclusions; Stability theory
• ISSN: 0167-2789; 1872-8022
• DOI: 10.1016/j.physd.2011.04.013

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. ► Sufficient conditions for global uniform attractive stability of the equilibrium. ► Generalization of Lyapunov’s direct method to temporarily increasing Lyapunov functions. ► Upper-bound for the attraction time of the bouncing ball system.