On the Cucker-Smale ensemble with <inline-formula><tex-math id="M1">\begin{document}$ q $\end{document}</tex-math></inline-formula>-closest neighbors under time-delayed communications

Abstract

We study time-asymptotic interplay between time-delayed communication and Cucker-Smale (C-S) velocity alignment. For this, we present two sufficient frameworks for the asymptotic flocking to the continuous and discrete C-S models with \begin{document}$ q $\end{document} -closest neighbors in the presence of time-delayed communications. Communication time-delays result from the finite-propagation speed of information and they are often ignored in the first place modeling of collective dynamics. In the absence of time-delays in communication, Cucker and Dong showed that the C-S model with \begin{document}$ q $\end{document} -closest neighbors can exhibit a phase-transition like phenomenon for unconditional and conditional flockings depending on the size \begin{document}$ q $\end{document} relative to system size. In this paper, we investigate whether Cucker and Dong's result is robust with respect to the time-delayed communications or not. In fact, our flocking estimates show that the critical number of \begin{document}$ q $\end{document} for unconditional flocking is the same as in the case for zero time-delay, which shows the robustness of the Cucker and Dong's result with respect to small time-delay.