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This paper addresses the derivation of generic and tractable sufficientconditions ensuring the stability of a coupled system composed of areaction-diffusion partial differential equation (PDE) and a finite-dimensionallinear time invariant ordinary differential equation (ODE). The coupling of thePDE with the ODE is located either at the boundaries or in the domain of thereaction-diffusion equation and takes the form of the input and output of theODE. We investigate boundary Dirichlet/Neumann/Robin couplings, as well asin-domain Dirichlet/Neumann couplings. The adopted approach relies on thespectral reduction of the problem by projecting the trajectory of the PDE intoa Hilbert basis composed of the eigenvectors of the underlying Sturm-Liouvilleoperator and yields a set of sufficient stability conditions taking the form ofLMIs. We propose numerical examples, consisting of an unstablereaction-diffusion equation and an unstable ODE, such that the application ofthe derived stability conditions ensure the stability of the resulting coupledPDE-ODE system.

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