Nonlinear boundary feedback control of the one-dimensional wave equation

In this paper, we analyze the dynamical behavior of the linear wave equation on an interval, where the right endpoint has a van der Pol type nonlinearity or boundary controller, while the left endpoint has a boundary condition involving displacement. The asymptotic behavior of the system can be classified into two basic types: classical unbounded instability, or spatial pointwise convergence to periodic points of a nonlinear map corresponding to the van der Pol condition.