  This paper is concerned with the analysis of a one dimensional wave equation
$z_{tt}-z_{xx}=0$ on $[0,1]$ with a Dirichlet condition at $x=0$ and a damping
acting at $x=1$ which takes the form $(z_t(t,1),-z_x(t,1))\in\Sigma$ for every
$t\geq 0$, 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 aim at
determining conditions on $\Sigma$ ensuring existence and uniqueness of
solutions of that wave equation as well as strong stability and uniform global
asymptotic stability of its solutions. In the latter case, we also study the
decay rates of the solutions and their optimality. We first establish a
one-to-one 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 simple
necessary and sufficient condition on $\Sigma$ ensuring existence and
uniqueness of solutions of the wave equation as well as an efficient strategy
for determining optimal decay rates when $\Sigma$ verifies a generalized sector
condition. As an application, we solve two conjectures stated in the
literature, the first one seeking a specific optimal decay rate and the second
one associated with a saturation type of damping. In case the boundary damping
is subject to perturbations, we derive sharp results regarding asymptotic
perturbation rejection and input-to-state issues.
