A note on norm-based Lyapunov functions via contraction analysis

TL;DR

This paper introduces an alternative Lyapunov function for globally contractive autonomous systems by demonstrating that the velocity magnitude along trajectories decreases exponentially, complementing existing contraction analysis methods.

Contribution

It presents a novel approach to Lyapunov functions based on contraction analysis, focusing on velocity decay in autonomous systems.

Findings

Velocity magnitude decreases exponentially along trajectories.

Provides an alternative Lyapunov function for contractive systems.

Enhances understanding of stability via contraction analysis.

Abstract

It is well know that for globally contractive autonomous systems, there exists a unique equilibrium and the distance to the equilibrium evaluated along any trajectory decreases exponentially with time. We show that, additionally, the magnitude of the velocity evaluated along any trajectory decreases exponentially, thus giving an alternative choice of Lyapunov function.