  We study a Cucker-Smale-type system with time delay in which agents interact
with each other through normalized communication weights. We construct a
Lyapunov functional for the system and provide sufficient conditions for
asymptotic flocking, i.e., convergence to a common velocity vector. We also
carry out a rigorous limit passage to the mean-field limit of the particle
system as the number of particles tends to infinity.For the resulting
Vlasov-type equation we prove the existence, stability and large-time behavior
of measure-valued solutions. This is, to our best knowledge, the first such
result for a Vlasov-type equation with time delay. We also present numerical
simulations of the discrete system with few particles that provide further
insights into the flocking and oscillatory behaviors of the particle velocities
depending on the size of the time delay.
