Communications in Mathematical Sciences

Volume 17 (2019)

Number 8

Nonlinear functional approach for mono-cluster flocking to the discrete Cucker–Smale model

Pages: 2239 – 2256



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

Seung-Yeal Ha (Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul, South Korea; and Korea Institute for Advanced Study, Seoul, South Korea)

Doheon Kim (School of Mathematics, Korea Institute for Advanced Study, Seoul, South Korea)


We present a nonlinear functional approach for the discrete Cucker–Smale (C–S) model with a general communication weight, and using the monotonicity of this functional, we provide a simple sufficient condition for the emergence of a mono-cluster flocking in terms of initial data, communication weight function and system parameters. Our proposed nonlinear functionals are the discrete analogues of the continuous nonlinear functionals introduced in [S.-Y. Ha and J.-G. Liu, Commun. Math. Sci., 7:297–325, 2009].


collective dynamics, Cucker–Smale model, nonlinear functional, mono-cluster flocking, multi-agent system

2010 Mathematics Subject Classification

74A25, 76N10, 92D25

Received 18 August 2018

Accepted 2 September 2019

Published 3 February 2020