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.

I. Introduction

In control engineering, the study of coupled systems has been a popular subject in recent years, which has rich physical backgrounds such as coupled mechanical, coupled chemical reactions. Meanwhile, one could point to its many applications in fluid dynamics, plasma system, and other engineering problems [1]. Thus, the stability issue of coupled system is significant both theoretically and practically. Recently, some works have been addressed on control design of cascaded PDE-ODE systems through making use of decoupling and PDE backstepping method such as in [2]-[5].