Communications in Mathematical Sciences

Volume 14 (2016)

Number 4

Emergence of flocking for a multi-agent system moving with constant speed

Pages: 953 – 972



Sun-Ho Choi (Department of Applied Mathematics and the Institute of Natural Sciences, Kyung Hee University, Yongin, South Korea)

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)


We present a Cucker–Smale-type flocking model for interacting multi-agents(or particles) moving with constant speed in arbitrary dimensions, and derive a sufficient condition for the asymptotic flocking in terms of spatial and velocity diameters, coupling strength and a communication weight. In literature, several Vicsek-type models with a unit speed constraint have been proposed in the modeling of self-organization and planar models were extensively studied via the dynamics of the heading angle. Our proposed model has a velocity coupling that is orthogonal to the velocity of the test agent to ensure the constancy of speed of the test agent along the dynamic process. For a flocking estimate, we derive a system of dissipative differential inequalities for spatial and velocity diameters, and we also employ a robust Lyapunov functional approach.


Cucker–Smale model, flocking, Lyapunov functional, unit speed constraint, Vicsek model

2010 Mathematics Subject Classification

Primary 70K20. Secondary 34D05.

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Published 6 April 2016