On a new method for the state and output feedback exponential stabilization of a cascaded wave PDE-ODE system with nonlinear boundary condition

Abstract This study presents a novel approach for the state and output feedback exponential control of cascaded unstable ODE-wave equations with nonlinear boundary condition. The new approach can be also used to investigate the state and output feedback stabilization of other cascaded ODE-partial differential equation (PDE) systems with nonlinear boundary condition. The interconnection between the ODE and wave PDE with the nonlinear boundary is bi-directional at two points. First, an exponential controller is planned and the well-posed and exponential stability of the closed-loop system is derived. Next, an exponential observer is designed, whereby the exponential stability of the overall error system is shown. Then, an observer-based output feedback control that causes the overall real system to be exponentially stable is built. The existence and exponential stability of the solution to the overall closed-loop system are further deduced. Finally, simulation results are given to verify the effectiveness, feasibility and validity of the proposed technique.