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

Abstract

We study a variant of the Cucker-Smale model where information between agents propagates with a finite speed \begin{document}$ {{\mathfrak{c}}}>0 $\end{document}. This leads to a system of functional differential equations with state-dependent delay. We prove that, if initially the agents travel slower than \begin{document}$ {{\mathfrak{c}}} $\end{document}, then the discrete model admits unique global solutions. Moreover, under a generic assumption on the influence function, we show that there exists a critical information propagation speed \begin{document}$ {{{{\mathfrak{c}}}^\ast}}>0 $\end{document} such that if \begin{document}$ {{\mathfrak{c}}}\geq{{{{\mathfrak{c}}}^\ast}} $\end{document}, the system exhibits asymptotic flocking in the sense of the classical definition of Cucker and Smale. For constant initial datum the value of \begin{document}$ {{{{\mathfrak{c}}}^\ast}} $\end{document} 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.