Piecewise structure of Lyapunov functions and densely checked decrease conditions for hybrid systems

TL;DR

This paper introduces a piecewise structured class of Lyapunov functions for hybrid systems, enabling easier verification of stability conditions by checking inequalities on dense sets rather than at nondifferentiable points.

Contribution

It proposes a new class of piecewise Lipschitz Lyapunov functions and shows how to verify stability conditions on dense sets, simplifying analysis of hybrid systems.

Findings

Lyapunov inequalities can be checked on dense sets, reducing computational complexity.

Connections established between piecewise Lipschitz functions and other regular function classes.

Applications demonstrated for hybrid dynamical systems stability analysis.

Abstract

We propose a class of locally Lipschitz functions with piecewise structure for use as Lyapunov functions for hybrid dynamical systems. Subject to some regularity of the dynamics, we show that Lyapunov inequalities can be checked only on a dense set and thus we avoid checking them at points of nondifferentiability of the Lyapunov function. Connections to other classes of locally Lipschitz or piecewise regular functions are also discussed and applications to hybrid dynamical systems are included.