Stabilization of PDE-ODE cascade systems using Sylvester equations

We consider the stabilization problem for PDEODE cascade interconnections in which the input is applied to the PDE system, whose output drives the ODE system. The PDE system is stable, while the ODE system is unstable. In the literature, this problem has been solved for specific nontrivial examples of such interconnections using the backstepping approach. In contrast, in the present work we consider this problem in a unified abstract setting for all PDEs that are regular linear systems. In our approach, using a state transformation obtained by solving a Sylvester equation with unbounded operators, we first diagonalize the interconnection. Then by solving a finite-dimensional stabilization problem, we get a stabilizing controller for the interconnection. We illustrate our approach using an example in which the ODE is an unstable scalar system and the PDE is the 1D diffusion equation.