Collision-free flocking for a time-delay system

Maoli Chen; Xiao Wang; Yicheng Liu

doi:10.3934/dcdsb.2020251

Article Contents

2021, Volume 26, Issue 2: 1223-1241. Doi: 10.3934/dcdsb.2020251

This issue Previous Article Forced oscillation of viscous Burgers' equation with a time-periodic force Next Article

Collision-free flocking for a time-delay system

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

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

Received: January 31, 2020

Revised: June 30, 2020

Early access: August 2020

Published: February 2021

This work was supported by the NSFC (Grant No.11401577 and No.11671011)

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Abstract

The co-existence of collision avoidance and time-asymptotic flocking of multi-particle systems with measurement delay is considered. Based on Lyapunov stability theory and some auxiliary differential inequalities, a delay-related sufficient condition is established for this system to admit a time-asymptotic flocking and collision avoidance. The estimated range of the delay is given, which may affect the flocking performance of the system. An analytical expression was proposed to quantitatively analyze the upper bound of this delay. Under the flocking conditions, the exponential decay of the relative velocity of any two particles in the system is characterized. Particularly, the collision-free flocking conditions are also given for the case without delay. This work verifies that both collision avoidance and flocking behaviors can be achieved simultaneously in a delay system.

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Mathematics Subject Classification: Primary: 34K20, 37N20; Secondary: 93D20.

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Figure 1. $ \beta $ = 0.5, $ \tau $ = 0.01, the relative position of any two particles in System (2)-(3) is always bounded, and the velocity will asymptotically converge, i.e., the emergence of flocking

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Figure 2. $ \beta $ = 0.6, $ \tau $ = 0.01, the relative position of any two particle in System (2)-(3) is always bounded, and the velocity will asymptotically converge, i.e., the emergence of flocking

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Figure 3. $ \beta $ = 0.6, $ \tau $ = 0.1, System (2)-(3) fail to form a flock

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