Collision-avoidance and flocking in the Cucker–Smale-type model with a discontinuous controller

Jianfei Cheng; Xiao Wang; Yicheng Liu

doi:10.3934/dcdss.2021169

Article Contents

2022, Volume 15, Issue 7: 1733-1748. Doi: 10.3934/dcdss.2021169

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Collision-avoidance and flocking in the Cucker–Smale-type model with a discontinuous controller

Department of Mathematics, College of Liberal Arts and Sciences, National University of Defense Technology, Changsha 410073, China

^* Corresponding author: Xiao Wang
^* Corresponding author: Xiao Wang

Received: August 31, 2021

Revised: October 31, 2021

Published: July 2022

The third author is supported by National Natural Science Foundation of China (11671011)

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Abstract

The collision-avoidance and flocking of the Cucker–Smale-type model with a discontinuous controller are studied. The controller considered in this paper provides a force between agents that switches between the attractive force and the repulsive force according to the movement tendency between agents. The results of collision-avoidance are closely related to the weight function $ f(r) = (r-d_0)^{-\theta } $. For $ \theta \ge 1 $, collision will not appear in the system if agents' initial positions are different. For the case $ \theta \in [0,1) $ that not considered in previous work, the limits of initial configurations to guarantee collision-avoidance are given. Moreover, on the basis of collision-avoidance, we point out the impacts of $ \psi (r) = (1+r^2)^{-\beta } $ and $ f(r) $ on the flocking behaviour and give the decay rate of relative velocity. We also estimate the lower and upper bound of distance between agents. Finally, for the special case that agents moving on the 1-D space, we give sufficient conditions for the finite-time flocking.

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Mathematics Subject Classification: Primary: 93A16; Secondary: 93C10, 93D20.

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Figure 1. Flocking: the evolution of agents' position (left) and velocity (right)

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Figure 2. Flocking: the lower and upper bound of the distance between agents (left) and the logarithm of the maximum velocity difference (right)

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Figure 3. Swarming: the evolution of agents' position (left) and velocity (right)

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Figure 4. Swarming: the lower and upper bound of the distance between agents (left) and the maximum velocity difference (right)

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Figure 5. Finite-time flocking: the evolution of agents' position (left) and velocity (right)

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Figure 6. Finite-time flocking: the lower and upper bound of the distance between agents (left) and the logarithm of the maximum velocity difference (right)

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Table 1. Initial configurations of system (3)

$ i $	1	2	3	4
$ (x_i(0)) $	(1, 1)	(2, 2)	(3, 3)	(4, 4)
$ (v_i(0)) $	(-3.25, -1.25)	(0.75, 3.75)	(-1.25, 0.75)	(3.75, -3.25)

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Table 2. Initial configurations of system (3)

$ i $	1	2	3	4
$ (x_i(0)) $	(1, 1)	(2, 2)	(3, 3)	(4, 4)
$ (v_i(0)) $	(-0.325, -0.125)	(0.075, 0.375)	(-0.125, 0.075)	(0.375, -0.325)

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Table 3. Initial configurations of system (3) on the real line

$ i $	1	2	3	4
$ (x_i(0),v_i(0)) $	(1, -3.25)	(2, 0.75)	(3, -1.25)	(4, 3.75)

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