Instability conditions for reaction-diffusion-ODE systems

TL;DR

This paper develops spectral and Lyapunov-based methods to identify instability conditions in reaction-diffusion-ODE systems with boundary control, providing practical criteria and numerical validation.

Contribution

It introduces novel analytical and numerical instability certificates for reaction-diffusion-ODE systems with boundary coupling, combining spectral root locus analysis and Lyapunov inequalities.

Findings

Spectral root locus analysis identifies instability regions.

Lyapunov inequalities provide sufficient instability conditions.

Numerical results suggest the criteria are necessary and sufficient.

Abstract

This paper analyzes the stability of a reactiondiffusion equation coupled with a finite-dimensional controller through Dirichlet boundary input and Neumann boundary output. Going against the flow, we intend to propose numerical certificates of instability for such interconnections. From one side, using spectral methods, an analytical condition based on root locus analysis can determine the instability regions in the parameters space and can sometimes be tested. On the other side, using Lyapunov direct and converse approaches, two sufficient conditions of instability are established in terms of linear matrix inequalities. The novelties lie both in the type of system studied and in the methods used. The numerical results demonstrate the performance of the different criteria set up in this paper and allow us to conjecture that these conditions seem to be necessary and sufficient.