This paper focuses on the approximate observability and observer design for a wave equation with a quadratic-cubic nonlinear boundary condition. First, the well-posedness of the wave equation is analysed by means of the method of characteristics and nonlinear boundary reflections. Subsequently, the approximate observability of the wave equation equipped with a boundary displacement output measurement is established. A state observer based on the same boundary displacement output measurement is then designed, and a sufficient condition that ensures the exponential stability of the estimation error system is provided. Rigorous theoretical proofs are presented for the well-posedness, approximate observability, and exponential stability of the estimation error system. Finally, numerical simulations are carried out to verify the effectiveness of the proposed observer.
Zhu et al. (Mon,) studied this question.