Exponential asymptotic flocking in the Cucker-Smale model with distributed reaction delays

Jan Haskovec; Ioannis Markou; Jan Haskovec; Ioannis Markou

doi:10.3934/mbe.2020304

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 5: 5651-5671. doi: 10.3934/mbe.2020304

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Exponential asymptotic flocking in the Cucker-Smale model with distributed reaction delays

Jan Haskovec ^{1
,
,},
Ioannis Markou ²

1.
Computer, Electrical and Mathematical Sciences & Engineering, King Abdullah University of Science and Technology, 23955 Thuwal, KSA
2.
Institute of Applied and Computational Mathematics (IACM-FORTH), N. Plastira 100, Vassilika Vouton GR - 700 13, Heraklion, Crete, Greece

Received: 13 May 2020 Accepted: 13 August 2020 Published: 25 August 2020

We study a variant of the Cucker-Smale system with distributed reaction delays. Using backward-forward and stability estimates on the quadratic velocity fluctuations we derive sufficient conditions for asymptotic flocking of the solutions. The conditions are formulated in terms of moments of the delay distribution and they guarantee exponential decay of velocity fluctuations towards zero for large times. We demonstrate the applicability of our theory to particular delay distributions - exponential, uniform and linear. For the exponential distribution, the flocking condition can be resolved analytically, leading to an explicit formula. For the other two distributions, the satisfiability of the assumptions is investigated numerically.
- Cucker-Smale system,
- flocking,
- distributed time delay,
- velocity fluctuation
Citation: Jan Haskovec, Ioannis Markou. Exponential asymptotic flocking in the Cucker-Smale model with distributed reaction delays[J]. Mathematical Biosciences and Engineering, 2020, 17(5): 5651-5671. doi: 10.3934/mbe.2020304

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Abstract

We study a variant of the Cucker-Smale system with distributed reaction delays. Using backward-forward and stability estimates on the quadratic velocity fluctuations we derive sufficient conditions for asymptotic flocking of the solutions. The conditions are formulated in terms of moments of the delay distribution and they guarantee exponential decay of velocity fluctuations towards zero for large times. We demonstrate the applicability of our theory to particular delay distributions - exponential, uniform and linear. For the exponential distribution, the flocking condition can be resolved analytically, leading to an explicit formula. For the other two distributions, the satisfiability of the assumptions is investigated numerically.

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