Strong decay for one-dimensional wave equations with nonmonotone boundary damping

This paper is a contribution to the following question : consider the classical wave equation damped by a nonlinear feedback control which is only assumed to decrease the energy. Then, do solutions to the perturbed system still exist for all time? Does strong stability occur in the sense that the energy tends to zero as time tends to infinity? We prove here that the answer to both questions is positive in the specific case of the one-dimensional wave equation damped by boundary controls which are functions of the observed velocity. The main point is that no monotonicity assumption is made on the damping term