Skip to main content
SpringerLink
Log in
Book cover

Trends in Control Theory and Partial Differential Equations pp 233–253Cite as

Asymptotic Analysis of a Cucker–Smale System with Leadership and Distributed Delay

  • Chapter
  • First Online:

Part of the book series: Springer INdAM Series ((SINDAMS,volume 32))

Abstract

We extend the analysis developed in Pignotti and Reche Vallejo (J Math Anal Appl 464:1313–1332, 2018) [34] in order to prove convergence to consensus results for a Cucker–Smale type model with hierarchical leadership and distributed delay. Flocking estimates are obtained for a general interaction potential with divergent tail. We analyze also the model when the ultimate leader can change its velocity. In this case we give a flocking result under suitable conditions on the leader’s acceleration.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Albi, G., Herty, M., Pareschi, L.: Kinetic description of optimal control problems and applications to opinion consensus. Commun. Math. Sci. 13, 1407–1429 (2015)

    Article  MathSciNet  Google Scholar 

  2. Bellomo, N., Herrero, M.A., Tosin, A.: On the dynamics of social conflict: Looking for the Black Swan. Kinet. Relat. Models 6, 459–479 (2013)

    Article  MathSciNet  Google Scholar 

  3. Borzì, A., Wongkaew, S.: Modeling and control through leadership of a refined flocking system. Math. Models Methods Appl. Sci. 25, 255–282 (2015)

    Article  MathSciNet  Google Scholar 

  4. Canuto, C., Fagnani, F., Tilli, P.: An Eulerian approach to the analysis of Krause’s consensus models. SIAM J. Control Optim. 50, 243–265 (2012)

    Article  MathSciNet  Google Scholar 

  5. Caponigro, M., Fornasier, M., Piccoli, B., Trélat, E.: Sparse stabilization and optimal control of the Cucker-Smale model. Math. Cont. Relat. Fields 3, 447–466 (2013)

    Article  MathSciNet  Google Scholar 

  6. Caponigro, M., Fornasier, M., Piccoli, B., Trélat, E.: Sparse stabilization and control of alignment models. Math. Models Methods Appl. Sci. 25, 521–564 (2015)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  10. Couzin, I., Krause, J., Franks, N., Levin, S.: Effective leadership and decision making in animal groups on the move. Nature 433, 513–516 (2005)

    Article  Google Scholar 

  11. Cristiani, E., Piccoli, B., Tosin, A.: Multiscale modeling of granular flows with application to crowd dynamics. Multiscale Model. Simul. 9, 155–182 (2011)

    Article  MathSciNet  Google Scholar 

  12. Cucker, F., Dong, J.G.: A general collision-avoiding flocking framework. IEEE Trans. Automat. Cont. 56, 1124–1129 (2011)

    Article  MathSciNet  Google Scholar 

  13. Cucker, F., Mordecki, E.: Flocking in noisy environments. J. Math. Pures Appl. 89, 278–296 (2008)

    Article  MathSciNet  Google Scholar 

  14. Cucker, F., Smale, S.: Emergent behaviour in flocks. IEEE Trans. Autom. Control 52, 852–862 (2007)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  16. Dalmao, F., Mordecki, E.: Cucker-smale flocking under hierarchical leadership and random interactions. SIAM J. Appl. Math. 71, 1307–1316 (2011)

    Article  MathSciNet  Google Scholar 

  17. Degond, P., Motsch, S.: Continuum limit of self-driven particles with orientation interaction. Math. Models Methods Appl. Sci. 18, 1193–1215 (2008)

    Article  MathSciNet  Google Scholar 

  18. Erban, R., Haskovec, J., Sun, Y.: On cucker-smale model with noise and delay. SIAM J. Appl. Math. 76(4), 1535–1557 (2016)

    Article  MathSciNet  Google Scholar 

  19. Ha, S.Y., Lee, K., Levy, D.: Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system. Commun. Math. Sci. 7, 453–469 (2009)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  21. Ha, S.Y., Slemrod, M.A.: Flocking dynamics of singularly perturbed oscillator chain and the cucker-smale system. J. Dyn. Diff. Equ. 22, 325–330 (2010)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  23. Lemercier, S., Jelic, A., Kulpa, R., Hua, J., Fehrenbach, J., Degond, P., Appert Rolland, C., Donikian, S., Pettré, J.: Realistic following behaviors for crowd simulation. Comput. Graph. Forum 31, 489–498 (2012)

    Article  Google Scholar 

  24. Li, Z.: Effectual leadershipin flocks with hierarchy and individual preference. Discret. Contin. Dyn. Syst. 34, 3683–3702 (2014)

    Article  Google Scholar 

  25. Li, Z., Ha, S.Y., Xue, X.: Emergent phenomena in an ensemble of Cucker-Smale particles under joint rooted leadership. Math. Models Methods Appl. Sci. 24, 1389–1419 (2014)

    Article  MathSciNet  Google Scholar 

  26. Li, Z., Xue, X.: Cucker-Smale flocking under rooted leadership with fixed and switching topologies. SIAM J. Appl. Math. 70, 3156–3174 (2010)

    Article  MathSciNet  Google Scholar 

  27. Li, C.H., Yang, S.Y.: A new discrete Cucker-Smale flocking model under hierarchical leadership. Discret. Contin. Dyn. Syst. Ser. B 21, 2587–2599 (2016)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  29. Mecholsky, N., Ott, E., Antonsen, T.M.: Obstacle and predator avoidance in a model for flocking. Phys. D 239, 988–996 (2010)

    Article  MathSciNet  Google Scholar 

  30. Motsch, S., Tadmor, E.: A new model for self-organized dynamics and its flocking behavior. J. Stat. Phys. 144, 923–947 (2011)

    Article  MathSciNet  Google Scholar 

  31. Motsch, S., Tadmor, E.: Heterophilious dynamics enhances consensus. SIAM Rev. 56, 577–621 (2014)

    Article  MathSciNet  Google Scholar 

  32. Nicaise, S., Pignotti, C.: Stabilization of the wave equation with boundary or internal distributed delay. Differ. Integral Equ. 21, 935–958 (2008)

    MathSciNet  MATH  Google Scholar 

  33. Piccoli, B., Rossi, F., Trélat, E.: Control to flocking of the kinetic Cucker-Smale model. SIAM J. Math. Anal. 47, 4685–4719 (2015)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  35. Pignotti, C., Trélat, E.: Convergence to consensus of the general finite-dimensional Cucker-Smale model with time-varying delays. Commun. Math. Sci. (to appear). arXiv:1707.05020

  36. Shen, J.: Cucker-Smale flocking under hierarchical leadership. SIAM J. Appl. Math. 68, 694–719 (2007/08)

    Article  MathSciNet  Google Scholar 

  37. Toscani, G.: Kinetic models of opinion formation. Commun. Math. Sci. 4, 481–496 (2006)

    Article  MathSciNet  Google Scholar 

  38. Vicsek, T., Czirok, A., Ben Jacob, E., Cohen, I., Shochet, O.: Novel type of phase transition in a system of self-driven particles. Phys. Rev. Lett. 75, 1226–1229 (1995)

    Article  MathSciNet  Google Scholar 

  39. Wongkaew, S., Caponigro, M., Borzì, A.: On the control through leadership of the Hegselmann-Krause opinion formation model. Math. Models Methods Appl. Sci. 25, 565–585 (2015)

    Article  MathSciNet  Google Scholar 

  40. Yates, C., Erban, R., Escudero, C., Couzin, L., Buhl, J., Kevrekidis, L., Maini, P., Sumpter, D.: Inherent noise can facilitate coherence in collective swarm motion. Proc. Natl. Acad. Sci. USA 106, 5464–5469 (2009)

    Article  Google Scholar 

Download references

Acknowledgements

The research of the first author was partially supported by the GNAMPA 2017 project Comportamento asintotico e controllo di equazioni di evoluzione non lineari and the GNAMPA 2018 project Analisi e controllo di modelli differenziali non lineari (INdAM).

Author information

Authors and Affiliations

  1. Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica, Università di L’Aquila, Via Vetoio, Loc. Coppito, 67010, L’Aquila, Italy

    Cristina Pignotti & Irene Reche Vallejo

Authors
  1. Cristina Pignotti
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Irene Reche Vallejo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Cristina Pignotti .

Editor information

Editors and Affiliations

  1. Laboratoire Jacques-Louis Lions UMR 7598, Sorbonne Université, Paris, France

    Fatiha Alabau-Boussouira

  2. Dipartimento di Matematica “Tullio Levi-Civita”, Università degli Studi di Padova, Padua, Italy

    Fabio Ancona

  3. Dipartimento di Matematica, Università di Roma “Tor Vergata”, Rome, Italy

    Alessio Porretta

  4. Dipartimento di Ingegneria Civile e Ingegneria Informatica, Università di Roma “Tor Vergata”, Rome, Italy

    Carlo Sinestrari

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Pignotti, C., Reche Vallejo, I. (2019). Asymptotic Analysis of a Cucker–Smale System with Leadership and Distributed Delay. In: Alabau-Boussouira, F., Ancona, F., Porretta, A., Sinestrari, C. (eds) Trends in Control Theory and Partial Differential Equations. Springer INdAM Series, vol 32. Springer, Cham. https://doi.org/10.1007/978-3-030-17949-6_12

Download citation

Publish with us

Policies and ethics

Search

Navigation