Algebraic Lyapunov Functions for Homogeneous Dynamic Systems

TL;DR

This paper introduces a new method for constructing homogeneous Lyapunov functions for linear and certain nonlinear systems, enabling more flexible stability analysis beyond symmetric functions.

Contribution

It presents a novel approach to build homogeneous Lyapunov functions from polynomial invariant sets, applicable to a broader class of systems.

Findings

Method successfully constructs Lyapunov functions for various systems

Allows non-symmetric Lyapunov functions, expanding applicability

Illustrated with simple, clear examples

Abstract

A method for constructing homogeneous Lyapunov functions of degree 1 from polynomial invariant sets is presented for linear time varying systems, homogeneous dynamic systems and the class of nonlinear systems that can be represented as such. The method allows the development of Lyapunov functions that are not necessarily symmetric about the origin, unlike the polynomial, homogeneous Lyapunov functions discussed in the prior literature. The work is illustrated by very simple examples.