  We study a wave equation in one space dimension with a general diffusion
coefficient which degenerates on part of the boundary. Degeneracy is measured
by a real parameter $\mu_a>0$. We establish observability inequalities for
weakly (when $\mu_a \in [0,1[$) as well as strongly (when $\mu_a \in [1,2[$)
degenerate equations. We also prove a negative result when the diffusion
coefficient degenerates too violently (i.e. when $\mu_a>2$) and the blow-up of
the observability time when $\mu_a$ converges to $2$ from below. Thus, using
the HUM method we deduce the exact controllability of the corresponding
degenerate control problem when $\mu_a \in [0,2[$. We conclude the paper by
studying the boundary stabilization of the degenerate linearly damped wave
equation and show that a suitable boundary feedback stabilizes the system
exponentially. We extend this stability analysis to the degenerate nonlinearly
boundary damped wave equation, for an arbitrarily growing nonlinear feedback
close to the origin. This analysis proves that the degeneracy does not affect
the optimal energy decay rates at large time. We apply the optimal-weight
convexity method of \cite{alaamo2005, alajde2010} together with the results of
the previous section, to perform this stability analysis.
