Exponential Decay of Solutions to Linear Evolution Equations with Time-Dependent Time Delay

In this chapter, we analyze an abstract evolution equation with time-dependent time delay and time-dependent delay feedback coefficient. We assume that the operator corresponding to the nondelayed part of the model generates an exponentially stable semigroup. Under an appropriate assumption on the delay feedback, we prove the well-posedness and an exponential stability estimate for our model. Our arguments combine the use of the Duhamel formula with a Lyapunov functional approach. Applications of our abstract results to concrete models are also illustrated.