State Feedback Exponential Stabilization of Coupled ODE-Wave System with Nonlinear Boundary Condition

In this note, we concentrate on the state feedback stabilization for coupled wave PDE-ODE systems associated with cubic nonlinearity term at the uncontrolled end. Due to the energy injection, cubic displacement nonlinearity and the interconnection between the coupled wave PDE-ODE system, the system considered in this note is more complex and interesting. First, two novel transformations are introduced to convert the original system into a desired system. Then a state feedback controller is proposed. Next, the desired system is shown to be exponentially stable by using semigroup theory. At the end, the well-posed and the exponential stability of the closed-loop system is derived. The numerical simulation results are presented to illustrate the effectiveness of the controller.