# American Institute of Mathematical Sciences

February  2021, 26(2): 1223-1241. doi: 10.3934/dcdsb.2020251

## 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

Received  February 2020 Revised  July 2020 Published  February 2021 Early access  August 2020

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

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.

Citation: Maoli Chen, Xiao Wang, Yicheng Liu. Collision-free flocking for a time-delay system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1223-1241. doi: 10.3934/dcdsb.2020251
##### References:

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##### References:
$\beta$ = 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
$\beta$ = 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
$\beta$ = 0.6, $\tau$ = 0.1, System (2)-(3) fail to form a flock
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