# Lyapunov Differential Equation Hierarchy and Polynomial Lyapunov Functions for Switched Linear Systems

## TL;DR

This paper introduces a hierarchy of Lyapunov differential equations that simplifies the search for polynomial Lyapunov functions in stable switched linear systems, offering a new computational approach and comparison with traditional methods.

## Contribution

It establishes an equivalence between polynomial and quadratic Lyapunov functions via a hierarchy of differential equations, enabling efficient stability analysis and high-order polynomial function generation.

## Key findings

- The proposed method effectively generates high-order polynomial Lyapunov functions.
- It provides a computationally competitive alternative to sum-of-squares approaches.
- The approach offers an intuitive procedure for stability verification of switched linear systems.

## Abstract

This work studies the problem of searching for homogeneous polynomial Lyapunov functions for stable switched linear systems. Specifically, we show an equivalence between polynomial Lyapunov functions for systems of this class and quadratic Lyapunov functions for a related hierarchy of Lyapunov differential equations. This creates an intuitive procedure for checking the stability properties of switched linear systems and a computationally competitive algorithm is presented for generating high-order homogeneous polynomial Lyapunov functions in this manner. Additionally, we provide a comparison between polynomial Lyapunov functions generated with our proposed approach and polynomial Lyapunov functions generated with a more traditional sum-of-squares based approach.