Stability analysis of reaction-diffusion PDEs coupled at the boundaries with an ODE

TL;DR

This paper develops generic stability conditions for coupled reaction-diffusion PDEs and ODEs using spectral reduction and LMIs, applicable to various boundary and in-domain couplings.

Contribution

It introduces a systematic approach to derive tractable stability conditions for PDE-ODE coupled systems with boundary and in-domain interactions.

Findings

Derived LMI-based stability conditions for coupled PDE-ODE systems.

Validated conditions with numerical examples ensuring stability.

Applicable to various boundary and in-domain coupling scenarios.

Abstract

This paper addresses the derivation of generic and tractable sufficient conditions ensuring the stability of a coupled system composed of a reaction-diffusion partial differential equation (PDE) and a finite-dimensional linear time invariant ordinary differential equation (ODE). The coupling of the PDE with the ODE is located either at the boundaries or in the domain of the reaction-diffusion equation and takes the form of the input and output of the ODE. We investigate boundary Dirichlet/Neumann/Robin couplings, as well as in-domain Dirichlet/Neumann couplings. The adopted approach relies on the spectral reduction of the problem by projecting the trajectory of the PDE into a Hilbert basis composed of the eigenvectors of the underlying Sturm-Liouville operator and yields a set of sufficient stability conditions taking the form of LMIs. We propose numerical examples, consisting of an unstable reaction-diffusion equation and an unstable ODE, such that the application of the derived stability conditions ensure the stability of the resulting coupled PDE-ODE system.