Stabilization of chains of linear parabolic PDE–ODE cascades

Over the past decade, stabilization problems have been solved for various cascade and “sandwich” configurations involving linear ODEs and PDEs of both hyperbolic and parabolic types. In this paper, we consider systems in which the output of the (i+1)th ODE subsystem is the control input of the ith PDE subsystem, and in which the state of the ith PDE subsystem enters as control into the ith ODE subsystems. We extend the existing results, among which a representative one is for the case where the ODEs in the chain are scalar and the PDEs are pure delays, in two major directions. First, we allow for the virtual inputs to be affected by PDE dynamics different from pure delays: we allow the PDEs to include diffusion, i.e., to be parabolic, and to even have counter-convection, and, in addition, for the PDE dynamics to enter the ODEs not only with the PDE's boundary value but also in a spatially-distributed (integrated) fashion. Second, we allow the ODEs in the chain to be not just scalar ODEs in a strict-feedback configuration but general LTI systems. We develop an n-step backstepping procedure and prove that the resulting closed-loop system is exponentially stable. A simulation example is provided to illustrate the effectiveness of our controllers.