A Corollary for Nonsmooth Systems

TL;DR

This paper extends classical stability theorems to nonsmooth, nonautonomous systems with discontinuities, providing new Lyapunov-based tools for analyzing their asymptotic behavior.

Contribution

It introduces two generalized corollaries to the LaSalle-Yoshizawa Theorem for systems with discontinuous dynamics, using differential inclusions.

Findings

Provides conditions for asymptotic convergence in nonsmooth systems

Develops Lyapunov analysis methods for discontinuous differential equations

Extends classical stability results to broader classes of systems

Abstract

In this note, two generalized corollaries to the LaSalle-Yoshizawa Theorem are presented for nonautonomous systems described by nonlinear differential equations with discontinuous right-hand sides. Lyapunov-based analysis methods are developed using differential inclusions to achieve asymptotic convergence when the candidate Lyapunov derivative is upper bounded by a negative semi-definite function.