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Cucker-Smale model with normalized communication weights and time delay

  • * Corresponding author: Jan Haskovec

    * Corresponding author: Jan Haskovec

YPC was supported by Engineering and Physical Sciences Research Council (EP/K00804/1) and ERC-Starting grant HDSPCONTR "High-Dimensional Sparse 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

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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.

    Mathematics Subject Classification: Primary: 34A12, 34D05; Secondary: 35Q83.


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  • Figure 1.  The system with two particles: Particle velocities $v_1(t)$, $v_2(t)$ as solututions of (30) (left panels) and velocity diameters $d_V(t)$ (right panels, logarithmic scale) for $\tau=0.25$ (first row) and $\tau=1$ (second row). The initial condition is constant, $v_1(t)\equiv 1$, $v_2(t)\equiv -1$ for $t\in[-\tau,0]$

    Figure 2.  The system with three particles: particle velocities $v_1(t)$, $v_2(t), v_3(t)$ as solutions of (5)–(7) (left panels) and velocity diameters $d_V(t)$ (right panels, logarithmic scale) for $\tau=0.25$ (first row) and $\tau=1$ (second row), with exponentially decaying influence function $\psi(s) = e^{-s}$. The initial condition is in both cases given by (33)–(34)

    Figure 3.  The system with four particles: particle velocities as solutions of (5)–(7) (left panels) and velocity diameters $d_V(t)$ (right panels, logarithmic scale) for $\tau=0.25$ (first row) and $\tau=1$ (second row), with the influence function $\psi(s) = {(1+s^2)^{-4}}$. The initial condition is in both cases given by (35)–(36)

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