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arXiv · 2103.04775

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

Hugo Lhachemi·Christophe Prieur
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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.

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Hugo Lhachemi, Christophe Prieur. 2021-03-08. Stability analysis of reaction-diffusion PDEs coupled at the boundaries with an ODE. https://doi.org/10.1016/j.automatica.2022.110465

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