A differential Lyapunov framework for contraction analysis

TL;DR

This paper introduces a differential Lyapunov framework that uses Finsler structures on the tangent bundle to analyze incremental stability via contraction, extending classical Lyapunov methods.

Contribution

It develops an analog of Lyapunov's second theorem for incremental stability using tangent bundle lifting and Finsler metrics, providing a new approach for contraction analysis.

Findings

Provides a new Lyapunov-based method for incremental stability

Establishes a connection between Finsler geometry and contraction analysis

Enables stability inference through integration along solution curves

Abstract

Lyapunov's second theorem is an essential tool for stability analysis of differential equations. The paper provides an analog theorem for incremental stability analysis by lifting the Lyapunov function to the tangent bundle. The Lyapunov function endows the state-space with a Finsler structure. Incremental stability is inferred from infinitesimal contraction of the Finsler metrics through integration along solutions curves.