Polynomial stabilization of the wave equation with a time varying delay term in the dynamical control

We consider the one-dimensional wave equation with a time-varying delay term in the dynamical control. Under suitable assumptions, we show the well posedness of the problem. These results are obtained by using semi-group theory. Combining the multiplier method with a non linear integral inequality, a rational energy decay result of the system is established. A fundamental aspect of this paper is that our involved operator is time dependent, therefore the standard frequential method cannot be invoked.