具有重整化和时滞的Hegselmann-Krause和Cucker-Smale模型渐近一致性的简单证明
A simple proof of asymptotic consensus in the Hegselmann-Krause and Cucker-Smale models with renormalization and delay
摘要：
给出了离散Hegselmann-Krause模型的渐近一致性和离散Cucker-Smale模型的群集性的一个简单证明。它基于重正化通信权的凸性和Gronwall-Halanay型不等式。我们的方法的主要优点，相比以前的方法的延迟Hegselmann-Krause模型，是它不需要任何限制的最大时间延迟，或初始数据，或衰减率的影响函数。从这个角度来看，结果是最佳的。对于Cucker-Smale模型，它提供了一个类似的结果，在制度的无条件群集充分缓慢衰减的通信速率，但仍然没有任何限制的最大时间延迟的长度。此外，我们证明了该方法可以很容易地扩展到Hegselmann-Krause和Cucker-Smale系统的平均场极限，使用适当的稳定性结果的测量值的解决方案。
英文摘要：
We present a simple proof of asymptotic consensus in the discrete Hegselmann-Krause model and flocking in the discrete Cucker-Smale model with renormalization and variable delay. It is based on convexity of the renormalized communication weights and a Gronwall-Halanay-type inequality. The main advantage of our method, compared to previous approaches to the delay Hegselmann-Krause model, is that it does not require any restriction on the maximal time delay, or the initial data, or decay rate of the influence function. From this point of view the result is optimal. For the Cucker-Smale model it provides an analogous result in the regime of unconditonal flocking with sufficiently slowly decaying communication rate, but still without any restriction on the length of the maximal time delay. Moreover, we demonstrate that the method can be easily extended to the mean-field limits of both the Hegselmann-Krause and Cucker-Smale systems, using appropriate stability results on the measure-valued solutions.
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