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Stabilization of a wave equation and a scalar ODE by a single input

In this paper, we consider a stabilization problem for a system consisting of a wave equation in a one-dimensional spatial domain and a scalar ordinary differential equation (ODE). The wave equation does not depend on the state variable of the ODE-subsystem, but the ODE is driven by the boundary values of the state variable of the wave equation in a nonlinear manner. A scalar control input affects the ODE-subsystem as well as the wave equation. To design a stabilizing control law, the forwarding approach is adapted. We first design a control law for the wave equation. Then, the control law is augmented so that the resulting control law stabilizes the ODE-subsystem. The asymptotic stability of the closed-loop system with the derived control law is proved based on the Lyapunov analysis.