Adaptive Boundary Observer Design for a Class of Nonlinear Wave PDEs with Uncertain Domain and Boundary Parameters

We consider the problem of state observer design for wave PDEs containing Lipschitz nonlinearities in the domain and parameter uncertainties in the domain and at the boundaries. Using the decoupling-transformation design approach, we develop an adaptive boundary observer consisting of a state observer with boundary output error injection, a least-squares type parameter adaptive law, and a hyperbolic auxiliary filter. Using Lyapunov stability analysis, we show that the observer is exponentially convergent under a persistent excitation condition. The novelty is twofold: (i) the class of systems is much wider than those studied in previous works, it particularly accounts for structured disturbances acting on the domain and all boundaries; (ii) the proposed adaptive observer is quite different from existing ones for wave-type PDEs.