Stabilization of a Coupled ODE-wave System with Disturbances

This paper is devoted to stabilizing an ODE-PDE coupled system with boundary disturbance, where the PDE is chosen to be a wave equation with viscous damping. Applying the active disturbance rejection control (ADRC) technic, the unknown disturbance is estimated by the extended state observer (ESO). Assuming the absence of disturbance, the dynamic boundary connection is designed. The methods of spectral analysis and separation of operator are applied, which ensures the Riesz basis property as well as the spectrum determined growth condition. The exponential stability of the coupled system without disturbance is then concluded. Finally, combining the disturbance estimation and the dynamic boundary connection, a feedback controller designed to stabilize the coupled system, and the closed-loop system is proved to be asymptotically stable.