Delay-robust stabilization of a hyperbolic PDE-ODE system

We detail in this article the development of a delay-robust stabilizing feedback control law for a linear ordinary differential equation coupled with two linear first order hyperbolic equations in the actuation path. The proposed method combines the use of a backstepping approach, required to construct a canceling feedback for the in-domain coupling terms of the PDEs, with a second change of variables that reduces the stabilization problem of the PDE-ODE system to that of a time-delay system for which a predictor can be constructed. The proposed controller can be tuned, with some restrictions imposed by the system structure, either by adjusting a reflection coefficient left on the PDE after the backstepping transformation, or by choosing the pole placement on the ODE when constructing the predictor, enabling a trade-off between convergence rate and delay-robustness. The proposed feedback law is finally proved to be robust to small delays in the actuation.