Communications in Mathematical Sciences

Volume 20 (2022)

Number 4

Cucker–Smale flocking under hierarchical leadership with time-delay and a free-will leader

Pages: 1103 – 1115

DOI: https://dx.doi.org/10.4310/CMS.2022.v20.n4.a7

Authors

Chen Wu (School of Mathematical Sciences, Dalian University of Technology, Dalian, China)

Jiu-Gang Dong (School of Mathematical Sciences, Dalian University of Technology, Dalian, China)

Abstract

We study the discrete-time Cucker–Smale model under hierarchical leadership with a constant time delay. The overall leader of the flock is assumed to have a free-will acceleration. The strength of the interaction is measured by a parameter $\beta \geq 0$. Under some suitable constraints on the acceleration, we prove that unconditional convergence to flocking can be achieved for arbitrarily large constant delay as long as $\beta \leq \frac{1}{2}$. The convergence rate is also provided depending on the leader’s acceleration.

Keywords

Cucker–Smale model, flocking, hierarchical leadership, time delay, free-will

2010 Mathematics Subject Classification

34K25, 39A12, 93A13

Full Text (PDF format)

This work was supported by the National Natural Science Foundation of China through grant 12171069, and by the Fundamental Research Funds for the Central Universities.

Received 23 March 2021

Received revised 28 October 2021

Accepted 29 October 2021

Published 11 April 2022