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We consider the problem of stabilizing PDE-ODE cascade systems in which theinput is applied to the PDE system whose output drives the ODE system. We alsoconsider the dual problem of constructing an observer for ODE-PDE cascadesystems in which the output of the ODE system drives the PDE system, whoseoutput is measured. The PDE in these problems is stable and the ODE isunstable. While the ODE system models the plant in both the problems, the PDEsystem models the actuator in the stabilization problem and the sensor in thedual problem. In the literature, these problems have been solved for specificPDE models using the backstepping approach. In contrast, in the present work weconsider these problems in an abstract framework by letting the PDE system beany regular linear system. Using a state transformation obtained by solving aSylvester equation with unbounded operators, we first diagonalize the stateoperator corresponding to the cascade systems. We then solve the stabilizationproblem and the dual estimation problem, provided they are solvable, by solvingcertain finite-dimensional counterparts. We also derive necessary andsufficient conditions for verifying the solvability of these problems. We showthat the controller which solves the stabilization problem is robust to certainunbounded perturbations. We illustrate our theory by designing a stabilizingcontroller for a PDE-ODE cascade in which the PDE is a 1D diffusion equationand an observer for a ODE-PDE cascade in which the PDE is a 1D wave equation.

Compensating PDE actuator and sensor dynamics using Sylvester equation