Relaxation of a Bolza Problem Governed by a Time-Delay Sweeping Process

We study in an infinite dimensional Hilbert space a Bolza problem in which the dynamics are given by a time-delay perturbed sweeping process. This is a differential inclusion whose right-hand side involves a normal cone to a moving set, along with a time-delay perturbation. A relaxation result is established from which we deduce a sufficient condition ensuring the existence of an optimal solution.