Effects of Time Delay on Cucker-Smale Flocking Under Hierarchical Leadership

We study the effects of time delay on the Cucker-Smale (CS) model under hierarchical leadership (HL). It is well known that time delays are ubiquitous in both physical phenomena and engineering applications. In this article, we establish an easily verified flocking condition for the CS model under HL subject to both transmission and reaction delays. Precisely, for the long-range weight function, we show that the flock converges to flocking exponentially fast when the size of the reaction delay is relatively small. Our condition is independent of the initial data of the flock. In contrast to the known flocking conditions for all-to-all interactions, our result may show the distinctive effectiveness of the HL structure. Finally, we perform some numerical simulations to illustrate our theoretical result.