Stabilization of a Class of Nonlinear ODE/Wave PDE Cascaded Systems

We investigate stabilization of a class of cascaded systems of nonlinear ordinary differential equation (ODE)/wave partial differential equation (PDE) with time-varying propagation speed based on a two-step PDE backstepping transformation. A time-varying propagation velocity of wave PDE leads to two difficulties. One is how to prove the well-posedness and uniqueness of the time-varying kernel PDEs in the first-step backstepping transformation, the other is how to construct a backstepping transform to map the original system into a suitable target system during the second-step transformation. We prove that there exists a unique continuous $2 \times 2$ matrix-valued solution to the time-varying kernel PDEs, and design a predictor control for the original cascaded system. An example is provided to illustrate the feasibility of the proposed design.