Output feedback stabilization of an ODE-Reaction–Diffusion PDE cascade with a long interconnection delay

Abstract

This paper addresses the output feedback stabilization of an ODE–PDE cascade. The PDE takes the form of a 1-D reaction–diffusion equation. The output of the ODE enters into the PDE via a Dirichlet/Neumann/Robin boundary condition. It is assumed that both the output of the ODE and the output of the PDE, selected as a boundary Dirichlet trace, are available for feedback control. The proposed control strategy takes the form of a finite-dimensional observer-based controller. Under a suitable structural controllability property, we show that the reported control strategy achieves the exponential stabilization of the plant when the order of the observer is selected large enough. We then demonstrate how such a control strategy can be adapted and augmented with a predictor component in order to achieve the stabilization of the above mentioned PDE–ODE cascade when the output of the ODE enters into the PDE with an arbitrarily long delay.