Finite-time flocking of Cucker-Smale model with unknown intrinsic dynamics

Lining Ru; Xiaoyu Li; Yicheng Liu; Xiao Wang

doi:10.3934/dcdsb.2022237

Article Contents

2023, Volume 28, Issue 6: 3680-3696. Doi: 10.3934/dcdsb.2022237

This issue Previous Article Monotonicity results for fractional parabolic equations in the whole space Next Article An overview on deep learning-based approximation methods for partial differential equations

Finite-time flocking of Cucker-Smale model with unknown intrinsic dynamics

1.
Department of Mathematics, National University of Defense Technology, Changsha 410073, China
2.
School of Mathematical Sciences, Suzhou University of Science and Technology, Suzhou 215009, China
3.
School of Science, Nanjing University of Posts and Telecommunications, Nanjing 210003, China

^*Corresponding author: Yicheng Liu
^*Corresponding author: Yicheng Liu

Received: February 2021

Revised: October 2022

Early access: December 2022

Published: June 2023

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Abstract

In this paper, we investigate the finite-time flocking problem for a modified Cucker-Smale model with unknown Hölder continuous intrinsic dynamics. Firstly, we use the energy method to show conditional finite-time flocking would occur under some conditions depending on initial data, exponent $ \beta $ and Hölder constant $ L $. Then we give an example consisting of two agents to show that for any $ \beta>0 $ or large enough $ L $, there is no unconditional finite-time flocking occurrence. Finally, we present some numerical simulations to show our theoretical results are valid.

Keywords:

Mathematics Subject Classification: Primary: 93A16, 92D25; Secondary: 92D50, 91D30.

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Figure 1. Trajectories of velocity $ (V_1, \; V_2) $ with ten agents for $ t\in [0, \; 1] $

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Figure 2. Evolutions of $ \|x(t)\| $ and $ \|v(t)\| $ with ten agents for $ t\in [0, \; 0.4] $

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Figure 3. Evolutions of $ \|x(t)\| $ and $ \|v(t)\| $ with different $ \beta $ for $ t\in [0, \; 0.25] $

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Figure 4. Evolutions of $ \|x(t)\| $ and $ \|v(t)\| $ with different $ K $ for $ t\in [0, \; 0.25] $

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Figure 5. Evolutions of $ \|x(t)\| $ and $ \|v(t)\| $ with different $ N $ for $ t\in [0, \; 0.25] $

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