Emergent behavior of Cucker–Smale model with time-varying topological structures and reaction-type delays

Qin Xu; Xiao Wang; Yicheng Liu; Qin Xu; Xiao Wang; Yicheng Liu

doi:10.3934/mmc.2022020

Mathematical Modelling and Control

2022, Volume 2, Issue 4: 200-218. doi: 10.3934/mmc.2022020

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Research article

Emergent behavior of Cucker–Smale model with time-varying topological structures and reaction-type delays

Department of Mathematics, National University of Defense Technology, Changsha 410073, China

Received: 19 October 2022 Revised: 08 November 2022 Accepted: 21 November 2022 Published: 19 December 2022

This paper studies the continuous Cucker–Smale model with time-varying topological structures and reaction-type delay. The goal of this paper is to establish a sufficient framework for flocking behaviors. Our method combines strict Lyapunov design with the derivation of an appropriate persistence condition for multi-agent systems. First, to prove that position fluctuations are uniformly bounded, a strict and trajectory-dependent Lyapunov functional is constructed via reparametrization of the time variable. Then, by constructing a global Lyapunov functional and using a novel backward-forward estimate, it is deduced that velocity fluctuations converge to zero. Finally, flocking behaviors are analyzed separately in terms of time delays and communication failures.
- Cucker–Smale model,
- time-varying topological structures,
- time delay,
- flocking,
- persistence of excitation
Citation: Qin Xu, Xiao Wang, Yicheng Liu. Emergent behavior of Cucker–Smale model with time-varying topological structures and reaction-type delays[J]. Mathematical Modelling and Control, 2022, 2(4): 200-218. doi: 10.3934/mmc.2022020

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Abstract

This paper studies the continuous Cucker–Smale model with time-varying topological structures and reaction-type delay. The goal of this paper is to establish a sufficient framework for flocking behaviors. Our method combines strict Lyapunov design with the derivation of an appropriate persistence condition for multi-agent systems. First, to prove that position fluctuations are uniformly bounded, a strict and trajectory-dependent Lyapunov functional is constructed via reparametrization of the time variable. Then, by constructing a global Lyapunov functional and using a novel backward-forward estimate, it is deduced that velocity fluctuations converge to zero. Finally, flocking behaviors are analyzed separately in terms of time delays and communication failures.

References

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