  We present a methodology for stabilization of general nonlinear systems with
actuator dynamics governed by general, quasilinear, first-order hyperbolic
PDEs. Since for such PDE-ODE cascades the speed of propagation depends on the
PDE state itself (which implies that the prediction horizon cannot be a priori
known analytically), the key design challenge is the determination of the
predictor state. We resolve this challenge and introduce a PDE
predictor-feedback control law that compensates the transport actuator
dynamics. Due to the potential formation of shock waves in the solutions of
quasilinear, first-order hyperbolic PDEs (which is related to the fundamental
restriction for systems with time-varying delays that the delay rate is bounded
by unity), we limit ourselves to a certain feasibility region around the origin
and we show that the PDE predictor-feedback law achieves asymptotic stability
of the closed-loop system, providing an estimate of its region of attraction.
Our analysis combines Lyapunov-like arguments and ISS estimates. Since it may
be intriguing as to what is the exact relation of the cascade to a system with
input delay, we highlight the fact that the considered PDE-ODE cascade gives
rise to a system with input delay, with a delay that depends on past input
values (defined implicitly via a nonlinear equation).
