Finite‐time flocking problem of a Cucker–smale‐type self‐propelled particle model

Abstract

In this article, we propose a Cucker–Smale‐type self‐propelled particle model with continuous non‐Lipschitz protocol. We show that the flocking can occur in finite time if the communication rate function satisfies a lower bound condition. Both our theoretical and numerical results uncover a power‐law relationship between the convergence time and the number of individuals. Our result implies that the individuals in groups with high density can transit rapidly to ordered collective motion. We also investigate the influence of control parameter on the convergence speed. © 2015 Wiley Periodicals, Inc. Complexity 21: 354–361, 2016

Finite-time flocking control of a swarm of cucker-smale agents with collision avoidance

“…It has been defined a very similar way in [18], where the difference of all velocities tends to zero in finite time and the diameter of the group is bounded. To investigate the control of flock diameter, the flocking would be defined as follows:…”

“…Proof: From [18], [19], proving the finite-time convergence of velocity is the same as proving V (t) tends to zero in finite time. Applying finite-time control (6) to (12), we havė…”

“…T 0 is to achieve flocking behaviour, and it has been proven in [18], [19], where the communication rate ψ from 2 satisfies a lower bound conditions. In other words, there exists ψ * > 0 such that inf s≥0 ψ(s) ≥ ψ * .…”

“…The first one is to achieve finite-time flocking which is similar to [18], [19]. The other two terms are utilized to adjust velocities to ensure collision does not occur.…”

“…In [18], [19], the original Cucker-Smale model has been modified to be non-Lipschitz continuous so that a finite upper bound on the flocking time could be established. An interesting result is that this flocking time bound decreases as the diameter of the flock N increases.…”