Export file:

Format

  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text

Content

  • Citation Only
  • Citation and Abstract

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

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

Special Issues: Mathematics of collective dynamics and pattern formation in biological systems

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.
  Figure/Table
  Supplementary
  Article Metrics

References

1. S. Camazine, J. L. Deneubourg, N. R. Franks, J. Sneyd, G. Theraulaz, E. Bonabeau, Self-Organization in Biological Systems, Princeton University Press, Princeton, NJ, 2001.

2. T. Vicsek, A. Zafeiris, Collective motion, Phys. Rep., 517 (2012), 71-140.

3. P. Krugman, The Self Organizing Economy, Blackwell Publishers, 1995.

4. G. Naldi, L. Pareschi, G. Toscani, Mathematical Modeling of Collective behaviour in Socio-Economic and Life Sciences, in Series: Modelling and Simulation in Science and Technology, Birkhäuser, 2010.

5. H. Hamman, Swarm Robotics: A Formal Approach, Springer, 2018.

6. A. Jadbabaie, J. Lin, A. S. Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Trans. Automat. Control, 48 (2003), 988-1001.

7. N. Bellomo, P. Degond, E. Tamdor, Active Particles, Volume I. Advances in Theory, Models, and Applications, Series: Modelling and Simulation in Science, Engineering and Technology, Birkhäuser, 2017.

8. Y.-P. Choi, S.-Y. Ha, Z. Li., Emergent dynamics of the Cucker-Smale flocking model and its variants, in Active Particles, Volume 1. Modeling and Simulation in Science, Engineering and Technology (eds N. Bellomo, P. Degond, E. Tadmor), Birkhäuser, 2017.

9. J.-G. Dong, S.-Y. Ha, D. Kim, Interplay of time-delay and velocity alignment in the Cucker-Smale model on a general digraph, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 5569-5596.

10. S.-Y. Ha, J. Kim, J. Park, X. Zhang, Complete cluster predictability of the Cucker-Smale flocking model on the real line, Arch. Ration. Mech. Anal., 231 (2019), 319-365.

11. D. Kalise, J. Peszek, A. Peters, Y.-P. Choi, A collisionless singular Cucker-Smale model with decentralized formation control, SIAM J. Appl. Dyn. Syst., 18 (2019), 1954-1981.

12. Z. Liu, X. Li, Y. Liu, X. Wang, Asymptotic flocking behavior of the general finite-dimensional Cucker-Smale model with distributed time delays, Bull. Malays. Math. Sci. Soc., 2020. Available from: https://doi.org/10.1007/s40840-020-00917-8.

13. I. Markou, Collision avoiding in the singular Cucker-Smale model with nonlinear velocity couplings, Discrete Contin. Dyn. Syst., 38 (2018), 5245-5260.

14. L. Pareschi, G. Toscani, Interacting Multiagent Systems: Kinetic equations and Monte Carlo methods, Oxford University Press, 2014.

15. C. Pignotti, E. Trelat, Convergence to consensus of the general finite-dimensional Cucker-Smale model with time-varying delays, Commun. Math. Sci., 16 (2018), 2053-2076.

16. C. Pignotti, I. Reche Vallejo, Flocking estimates for the Cucker-Smale model with time lag and hierarchical leadership, J. Math. Anal. Appl., 464 (2018), 1313-1332.

17. C. Pignotti, I. Reche Vallejo, Asymptotic analysis of a Cucker-Smale system with leadership and distributed delay, in Trends in Control Theory and Partial Differential Equations (eds. F. Alabau-Boussouira, F. Ancona, A. Porretta, C. Sinestrari), Springer INdAM Series, vol. 32 (2019).

18. F. Cucker, S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.

19. F. Cucker, S. Smale, On the mathematics of emergence, Jpn. J. Math., 2 (2007), 197-227.

20. S.-Y. Ha, E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinet. Relat. Models, 1 (2008), 415-435.

21. S.-Y. Ha, J.-G. Liu, A simple proof of the Cucker-Smale flocking dynamics and mean-field limit, Commun. Math. Sci., 7 (2009), 297-325.

22. J. A. Carrilo, M. Fornasier, J. Rosado, G. Toscani, Asymptotic flocking dynamics for the kinetic Cucker-Smale model, SIAM J. Math. Anal., 42 (2010), 218-236.

23. I. Gyori, G. Ladas, Oscillation Theory of Delay Differential Equations with Applications, Oxford Science Publications, Clarendon Press, Oxford, 1991.

24. J. Haskovec, I. Markou, Asymptotic flocking in the Cucker-Smale model with reaction-type delays in the non-oscillatory regime, Kinet. Relat. Models, 13 (2020), 795-813.

25. J. Haskovec, Exponential decay for negative feedback loop with distributed delay, Appl. Math. Lett., 107 (2020), 106419.

26. Y. Liu, J. Wu, Flocking and asymptotic velocity of the Cucker-Smale model with processing delay, J. Math. Anal. Appl., 415 (2014), 53-61.

27. Y.-P. Choi, J. Haskovec, Cucker-Smale model with normalized communication weights and time delay, Kinet. Relat. Models, 10 (2017), 1011-1033.

28. Y.-P. Choi, J. Haskovec, Hydrodynamic Cucker-Smale model with normalized communication weights and time delay, SIAM J. Math. Anal., 51 (2019), 2660-2685.

29. R. Erban, J. Haskovec, Y. Sun, A Cucker-Smale model with noise and delay, SIAM J. Appl. Math., 76 (2016), 1535-1557.

30. I. Barbalat, Systèmes d'équations différentielles d'oscillations nonlinéaires, Rev. Math. Pures Appl., 4 (1959), 267-270.

31. Y.-P. Choi, Z. Li, Emergent behavior of Cucker-Smale flocking particles with heterogeneous time delays, Appl. Math. Lett., 86 (2018), 49-56.

32. H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer New York Dordrecht Heidelberg London, 2011.

© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Article outline

Show full outline
Copyright © AIMS Press All Rights Reserved