Global-In-Time Discrete Approximation of the Cucker–Smale Model with a Unit Speed Constraint

Abstract

In this paper, we study the discrete Cucker–Smale model with a unit-speed constraint. For this, we first propose a discrete-time approximation of the Cucker–Smale model with a unit speed constraint (Choi and Ha, in: Commun Math Sci 14:953–972, 2016) using an exponential map in the state space \(\mathbb {R}^d\times \mathbb {S}^{d-1}\). Then, we present several sufficient frameworks to guarantee its asymptotic flocking. Moreover, we prove the finite-in-time transition from the discrete system to its continuous counterpart under generic initial data and system parameters. With the help of this result and the asymptotic flocking of the discrete and continuous systems, we also demonstrate the uniform-in-time transition between them.