具有时滞的Hegselmann-Krause意见形成模型的一致性
Consensus of the Hegselmann-Krause opinion formation model with time delay
摘要：
本文研究了具有时变时滞的Hegselmann-Krause模型。在适当的假设下，当时滞满足适当的小性假设时，我们证明了指数渐近一致性。我们的主要策略，这是基于李雅普诺夫函数的方法和仔细估计的轨迹。然后研究了当个体数N趋于无穷大时，从多个体Hegselmann-Krause方程到连续型偏微分方程的平均场极限.对于极限方程，我们证明了测度值解的时间整体存在唯一性。我们还利用粒子系统的一致性估计中出现的常数与N无关的事实，将指数一致性结果扩展到连续模型。最后给出了数值试验结果。
英文摘要：
In this paper, we study Hegselmann-Krause models with a time-variable time delay. Under appropriate assumptions, we show the exponential asymptotic consensus when the time delay satisfies a suitable smallness assumption. Our main strategies for this are based on Lyapunov functional approach and careful estimates on the trajectories. We then study the mean-field limit from the many-individual Hegselmann-Krause equation to the continuity-type partial differential equation as the number N of individuals goes to infinity. For the limiting equation, we prove global-in-time existence and uniqueness of measure-valued solutions. We also use the fact that constants appearing in the consensus estimates for the particle system are independent of N to extend the exponential consensus result to the continuum model. Finally, some numerical tests are illustrated.
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