  This paper addresses the derivation of sufficient linear matrix inequality
conditions ensuring the stability of a coupled system composed of a
reaction-diffusion partial differential equation (PDE), with possibly
spatially-varying coefficients, and a finite-dimensional linear time invariant
ordinary differential equation (ODE). The coupling of the PDE with the ODE is
located at both left and right boundary conditions of the reaction-diffusion
equation and takes the form of the input and output of the ODE. We investigate
the four possible sets of left/right boundary couplings among Dirichlet and
Neumann traces. 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. 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.
