  The paper deals with initial-boundary value problems for the linear wave
equation whose solutions stabilize to zero in a finite time. We prove that
problems in this class remain exponentially stable in $L^2$ as well as in $C^2$
under small bounded perturbations of the wave operator. To show this for $C^2$,
we prove a smoothing result implying that the solutions to the perturbed
problems become eventually $C^2$-smooth for any $H^1\times L^2$-initial data.
