Backstepping-Forwarding Designs for Doubly- Distributed Delay-PDE-ODE Systems

A cascade connection comprising a delay, a partial differential equation (PDE), and an ordinary differential equation (ODE) is referred to as “distributed” if an integral operator of the functional state of the delay or the PDE enters, as a scalar, the next system in the cascade, or if the ODE state acts as an input to the PDE throughout the PDE's domain. While boundary control of systems involving PDEs or delays typically calls for the PDE backstepping approach, such “distributed” interconnections call for the PDE forwarding approach. The cascaded triple consisting of a delay, a parabolic PDE, and an ODE can constitute nine (3 × 3) distinct interconnections and each of the nine structures is “doubly distributed” if both of its two connections are distributed. We focus on the most interesting three of the nine possible structures and present mixed backstepping-forwarding designs of controllers and observers. We first propose controllers to exponentially stabilize delay-PDE-ODE and PDE-delay-ODE cascades. Then, we introduce an observer to estimate the states of PDE-ODE systems with sensor delays. Besides advancing the backstepping-forwarding design for “doubly-distributed” systems, we do so with the aid of the input-to-state stability approach for parabolic PDEs. This enables proving stability without constructing Lyapunov functions, as well as proving stability in both square-integral and supremum spatial norms. Our results are novel even for the particular cases of delay-free cascades. Simulations illustrate our theory.