This paper is concerned with the analysis of a 1D wave equation $z_{tt}-z_{xx}=0$ on $[0,1]$ with a Dirichlet condition at $x=0$ and a damping acting at $x=1$ of the form $(z_t(t,1),-z_x(t,1))\in\Sigma$ for $t\geq 0$, where $\Sigma$ is a given subset of $\mathbb R^2$. The study is performed within an $L^p$ functional framework, $p\in [1, +\infty]$. We determine conditions on $\Sigma$ ensuring existence and uniqueness of solutions of that wave equation, its strong stability and uniform global asymptotic stability of the solutions. In the latter case, we study the corresponding decay rates and their optimality. We first establish a correspondence between the solutions of that wave equation and the iterated sequences of a discrete-time dynamical system in terms of which we investigate the above mentioned issues. This enables us to provide a necessary and sufficient condition on $\Sigma$ ensuring existence and uniqueness of solutions of the wave equation and an efficient strategy for determining optimal decay rates when $\Sigma$ verifies a generalized sector condition. In case the boundary damping is subject to perturbations, we derive sharp results regarding asymptotic perturbation rejection and input-to-state issues.
One-dimensional wave equation with set-valued boundary damping: well-posedness, asymptotic stability, and decay rates
