Cucker-Smale model with finite speed of information propagation: well-posedness, flocking and mean-field limit

TL;DR

This paper extends the Cucker-Smale flocking model to include finite information propagation speed, proving well-posedness, flocking behavior, and deriving a mean-field limit that differs from classical Fokker-Planck descriptions.

Contribution

It introduces a finite speed of information propagation into the Cucker-Smale model, analyzes well-posedness, flocking conditions, and develops a novel mean-field limit formulation.

Findings

Unique global solutions exist if agents initially travel slower than the speed c.

A critical propagation speed c* guarantees flocking behavior.

The mean-field limit is well-posed and not described by classical Fokker-Planck equations.

Abstract

We study a variant of the Cucker-Smale model where information between agents propagates with a finite speed $\mathfrak{c}>0$. This leads to a system of functional differential equations with state-dependent delay. We prove that, if initially the agents travel slower than $c$, then the discrete model admits unique global solutions. Moreover, under a generic assumption on the influence function, we show that there exists a critical information propagation speed $\mathfrak{c}^\ast>0$ such that if $\mathfrak{c}\geq\mathfrak{c}^\ast$, the system exhibits asymptotic flocking in the sense of the classical definition of Cucker and Smale. For constant initial datum the value of $\mathfrak{c}^\ast$ is explicitly calculable. Finally, we derive a mean-field limit of the discrete system, which is formulated in terms of probability measures on the space of time-dependent trajectories. We show global well-posedness of the mean-field problem and argue that it does not admit a description in terms of the classical Fokker-Planck equation.