Observer design for wave equations with van der Pol type boundary conditions

In this paper, we study the observer design problem for one-dimensional wave equation with van der Pol type boundary condition, whose dynamics presents spatiotemporal chaotic behaviors. By introducing a linear error feedback on the boundary, we construct an observer via method of characteristic. The main approach is to construct two one-dimensional mappings which can characterize the evolutionary dynamics of the system as well as the observer, and the convergence of error dynamics is obtained in terms of these two mappings. The range of the feedback gain is identified. Numerical simulation is provided to illustrate the theoretical outcomes.