Exponential stability estimates for an axially travelling string damped at one end

We study the small vibrations of an axially travelling string with a dashpot damping at one end. The is modelled by wave equation in time-dependent interval two endpoints moving constant speed v. For undamped case, we obtain conserved functional equivalent to energy solution. derive precise upper and lower exponentially decaying estimates for explicit constants. These do not seem be reported literature even non-travelling case v=0.