Delayed output feedback stabilization of ODE-heat cascades

This paper addresses the problem of output feedback stabilization of ODE-heat cascades. The ODE enters into the heat equation through a either Dirichlet or Neumann boundary condition. The system output applies solely to the heat equation and takes the form of either a distributed measurement or boundary/in-domain pointwise Dirichlet/Neumann measurement. The adopted approach relies on spectral reduction methods through a detailed study of the eigenelements of the ODE-heat cascade. A full characterization of the controllability and observability properties of each individual mode of the cascade is reported. Provided the satisfaction of these controllability and observability properties for the unstable modes, we show that a finite-dimensional output feedback control strategy can always be successfully designed for achieving the exponential stabilization of the ODE-heat cascade. We discuss in particular how such a strategy extends to the case of an input delay by designing a predictor feedback.