Stability analysis of a system coupled to a heat equation

TL;DR

This paper introduces a novel Lyapunov-based method using polynomial approximation and integral inequalities to analyze the stability of coupled finite-infinite dimensional systems, specifically those involving heat equations.

Contribution

It presents a new stability analysis methodology for coupled systems using polynomial approximation and Bessel inequality, inspired by time delay system techniques.

Findings

Stability conditions expressed as linear matrix inequalities.

Effective polynomial approximation of the heat equation state.

Validated approach on academic examples.

Abstract

As a first approach to the study of systems coupling finite and infinite dimensional natures, this article addresses the stability of a system of ordinary differential equations coupled with a classic heat equation using a Lyapunov functional technique. Inspired from recent developments in the area of time delay systems, a new methodology to study the stability of such a class of distributed parameter systems is presented here. The idea is to use a polynomial approximation of the infinite dimensional state of the heat equation in order to build an enriched energy functional. A well known efficient integral inequality (Bessel inequality) will allow to obtain stability conditions expressed in terms of linear matrix inequalities. We will eventually test our approach on academic examples in order to illustrate the efficiency of our theoretical results.