# Lyapunov stability analysis of a string equation coupled with an ordinary differential system

## TL;DR

This paper develops a Lyapunov functional approach to analyze the stability of a coupled system involving a string PDE and an ODE, providing tractable conditions via linear matrix inequalities.

## Contribution

It introduces a new Lyapunov functional based on augmented states, extending classical methods for coupled PDE-ODE systems with a novel stability analysis framework.

## Key findings

- Effective stability conditions derived as LMIs.
- Numerical examples demonstrate stabilization of various coupled systems.
- Method applicable to both stable and unstable ODEs coupled with PDEs.

## Abstract

This paper considers the stability problem of a linear time invariant system in feedback with a string equation. A new Lyapunov functional candidate is proposed based on the use of augmented states which enriches and encompasses the classical Lyapunov functional proposed in the literature. It results in tractable stability conditions expressed in terms of linear matrix inequalities. This methodology follows from the application of the Bessel inequality together with Legendre polynomials. Numerical examples illustrate the potential of our approach through three scenari: a stable ODE perturbed by the PDE, an unstable open-loop ODE stabilized by the PDE and an unstable closed-loop ODE stabilized by the PDE.