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-eld 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 rst 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.
Bibliographical noteFunding Information:
YPC was supported by Engineering and Physical Sciences Research Council (EP/K00804/1) and ERC-Starting grant HDSPCONTR \High-Dimensional parse Optimal Control". He was also supported by the Alexander Humboldt Foundation through the Humboldt Research Fellowship for Postdoctoral Researchers. JH was supported by KAUST baseline funds and KAUST grant no. 1000000193.
All Science Journal Classification (ASJC) codes
- Numerical Analysis
- Modelling and Simulation