Boundary stabilization of a vibrating string with variable length

We study small vibrations of a string with time-dependent length $\ell(t)$ and boundary damping. The vibrations are described by a 1-d wave equation in an interval with one moving endpoint at a speed $\ell'(t)$ slower than the speed of propagation of the wave c=1. With no damping, the energy of the solution decays if the interval is expanding and increases if the interval is shrinking. The energy decays faster when the interval is expanding and a constant damping is applied at the moving end. However, to ensure the energy decay in a shrinking interval, the damping factor $\eta$ must be close enough to the optimal value $\eta=1$, corresponding to the transparent condition. In all cases, we establish lower and upper estimates for the energy with explicit constants.