Journal
KINETIC AND RELATED MODELS
Volume 10, Issue 4, Pages 1011-1033Publisher
AMER INST MATHEMATICAL SCIENCES-AIMS
DOI: 10.3934/krm.2017040
Keywords
Cucker-Smale model; Vlasov equarion; normalized communication weights; time delay; asymptotic behavior; flocking
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Funding
- Engineering and Physical Sciences Research Council [EP/K00804/1]
- ERC-Starting grant HDSPCONTR High -Dimensional Sparse Optimal Control
- Alexander Humboldt Foundation through the Humboldt Research Fellowship
- KAUST baseline funds and KAUST [1000000193]
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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.
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