On the well-posedness and stability of retarded impulsive wave equations with a frictional damping

Of concern is, first, a wave equation subject to a discrete time delay in the state itself (and not in its time derivative). It is also under the action of impulses and a control given in the form of a frictional damping. We prove the well-posedness and exponential stability of our system. Second, we give an extension to the case where the time delay occurs in both, the state and its time derivative, in addition to impulsive boundary conditions in the delays. Our stability results show that the dissipation generated by the frictional damping is not destroyed neither by the impulses nor by the delays even when acting simultaneously