Nonlinear Control Through Wave PDE with State-Dependent Propagation Speed

We consider predictor control of a nonlinear ODE/wave PDE cascaded system with state-dependent propagation speed. First, the original cascaded system is transferred into a coupled hyperbolic PDE cascading with a nonlinear ODE by the first-step backstepping transformation. Further, this cascaded system is transferred into a pair of transport PDEs with state-dependent coefficients cascading into a nonlinear ODE by the second-step backstepping transformation. The key challenge is to prove the well-posedness and uniqueness of kernel equations, and design a compensator for the resulting cascaded system. Our design provides asymptotic stability and attraction region estimation for the closed-loop system since the propagation speed of wave depends on the ODE's state. An example is given to show the validity of the proposed design.