Lyapunov functions obtained from first order approximations

TL;DR

This paper explores how to construct Lyapunov functions from first order approximations to analyze local and global stability of dynamical systems, providing a systematic approach for stability verification.

Contribution

It introduces a method to derive Lyapunov functions from first order approximations for both local exponential stability and global stability analysis.

Findings

Lyapunov functions can be constructed from first order approximations for local stability.

First order approximations enable Lyapunov function construction for global stability.

The approach simplifies stability analysis by leveraging linearized system properties.

Abstract

In this paper, we study the construction of Lyapunov functions based on first order approximations. In a first part, the study of local exponential stability property of a transverse invariant manifold is considered. This part is mainly a rephrasing of the result of [3]. It is shown with this framework how to construct a Lyapunov function which characterizes this local stability property. In a second part, when considering the global stability property of an equilibrium point it is shown that the study of first order approximation along solutions of the system allows to construct a Lyapunov function.