Anti-collocated observer-based output feedback control of wave equation with cubic velocity nonlinear boundary and Dirichlet control

This study focuses on the observer and observer-based output feedback control design for a one-dimensional wave equation with van der Pol type nonlinear boundary condition and Dirichlet boundary control. The contribution of this work is divided into two parts. Firstly, the anti-collocated backstepping observer design for the nonlinear wave system is investigated, and the well-posedness and the asymptotical stability analysis of the observer error are studied. Secondly, an output feedback observer-based controller is set up by using boundary displacement measurement only to stabilise the nonlinear wave system; also the asymptotical stability of the closed-loop system is shown. The validity of the theoretical results is demonstrated by numerical simulations.