Advanced Search
Article Contents
Article Contents

Cucker-Smale model with finite speed of information propagation: Well-posedness, flocking and mean-field limit

  • *Corresponding author: Jan Haskovec

    *Corresponding author: Jan Haskovec
Abstract Full Text(HTML) Related Papers Cited by
  • We study a variant of the Cucker-Smale model where information between agents propagates with a finite speed $ {{\mathfrak{c}}}>0 $. This leads to a system of functional differential equations with state-dependent delay. We prove that, if initially the agents travel slower than $ {{\mathfrak{c}}} $, then the discrete model admits unique global solutions. Moreover, under a generic assumption on the influence function, we show that there exists a critical information propagation speed $ {{{{\mathfrak{c}}}^\ast}}>0 $ such that if $ {{\mathfrak{c}}}\geq{{{{\mathfrak{c}}}^\ast}} $, the system exhibits asymptotic flocking in the sense of the classical definition of Cucker and Smale. For constant initial datum the value of $ {{{{\mathfrak{c}}}^\ast}} $ is explicitly calculable. Finally, we derive a mean-field limit of the discrete system, which is formulated in terms of probability measures on the space of time-dependent trajectories. We show global well-posedness of the mean-field problem and argue that it does not admit a description in terms of the classical Fokker-Planck equation.

    Mathematics Subject Classification: Primary: 34K43, 34K25, 35Q84; Secondary: 82C22, 93D50.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] S. CamazineJ.-L. DeneubourgN. R. FranksJ. SneydG. Theraulaz and  E. BonabeauSelf-Organization in Biological Systems, Princeton University Press, Princeton, NJ, 2003. 
    [2] J. A. CañizoJ. A. Carrillo and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion, Math. Models Methods Appl. Sci., 21 (2011), 515-539.  doi: 10.1142/S0218202511005131.
    [3] Y.-P. Choi and J. Haskovec, Cucker-Smale model with normalized communication weights and time delay, Kinet. Relat. Models, 10 (2017), 1011-1033.  doi: 10.3934/krm.2017040.
    [4] Y.-P. Choi and J. Haskovec, Hydrodynamic Cucker-Smale model with normalized communication weights and time delay, SIAM J. Math. Anal., 51 (2019), 2660-2685.  doi: 10.1137/17M1139151.
    [5] Y.-P. Choi and C. Pignotti, Emergent behavior of Cucker-Smale model with normalized weights and distributed time delays, Netw. Heterog. Media, 14 (2019), 789-804.  doi: 10.3934/nhm.2019032.
    [6] F. Cucker and S. Smale, Emergent behaviour in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842.
    [7] F. Cucker and S. Smale, On the mathematics of emergence, Jpn. J. Math., 2 (2007), 197-227.  doi: 10.1007/s11537-007-0647-x.
    [8] R. ErbanJ. Haškovec and Y. Sun, A Cucker-Smale model with noise and delay, SIAM J. Appl. Math., 76 (2016), 1535-1557.  doi: 10.1137/15M1030467.
    [9] S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinet. Relat. Models, 1 (2008), 415-435.  doi: 10.3934/krm.2008.1.415.
    [10] H. Hamman, Swarm Robotics: A Formal Approach, Springer, Cham, 2018. doi: 10.1007/978-3-319-74528-2.
    [11] J. Haskovec, A simple proof of asymptotic consensus in the Hegselmann-Krause and Cucker-Smale models with normalization and delay, SIAM J. Appl. Dyn. Syst., 20 (2021), 130-148.  doi: 10.1137/20M1341350.
    [12] J. Haskovec, Well posedness and asymptotic consensus in the Hegselmann-Krause model with finite speed of information propagation, Proc. Amer. Math. Soc., 149 (2021), 3425-3437.  doi: 10.1090/proc/15522.
    [13] J. Haskovec and 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.  doi: 10.3934/krm.2020027.
    [14] J. Haskovec and I. Markou, Exponential asymptotic flocking in the Cucker-Smale model with distributed reaction delays, Math. Biosci. Eng., 17 (2020), 5651-5671.  doi: 10.3934/mbe.2020304.
    [15] R. Hegselmann and U. Krause, Opinion dynamics and bounded confidence models, analysis, and simulation, J. Artif. Soc. Soc. Simul., 5 (2002), 1-33. Available from: https://www.jasss.org/5/3/2/2.pdf.
    [16] A. JadbabaieJ. Lin and A. S. Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Trans. Automat. Control, 48 (2003), 988-1001.  doi: 10.1109/TAC.2003.812781.
    [17] P. Krugman, The Self Organizing Economy, Blackwell Publishers, 1995.
    [18] Y. Liu and J. Wu, Flocking and asymptotic velocity of the Cucker-Smale model with processing delay, J. Math. Anal. Appl., 415 (2014), 53-61.  doi: 10.1016/j.jmaa.2014.01.036.
    [19] G. Naldi, L. Pareschi and G. Toscani (eds.), Mathematical Modeling of Collective Behaviour in Socio-Economic and Life Sciences, Modelling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Ltd., Boston, MA, 2010. doi: 10.1007/978-0-8176-4946-3.
    [20] C. Pignotti and I. Reche Vallejo, Asymptotic analysis of a Cucker-Smale system with leadership and distributed delay, in Trends in Control Theory and Partial Differential Equations, Springer INdAM Ser., 32, Springer, Cham, 2019,233-253.
    [21] C. Pignotti and I. Reche Vallejo, Flocking estimates for the Cucker-Smale model with time lag and hierarchical leadership, J. Math. Anal. Appl., 464 (2018), 1313-1332.  doi: 10.1016/j.jmaa.2018.04.070.
    [22] C. Pignotti and E. Trélat, Convergence to consensus of the general finite-dimensional Cucker-Smale model with time-varying delays, Commun. Math. Sci., 16 (2018), 2053-2076.  doi: 10.4310/CMS.2018.v16.n8.a1.
    [23] M. Rodriguez Cartabia, Cucker-Smale model with time delay, Discrete Contin. Dyn. Syst., 42 (2022), 2409-2432.  doi: 10.3934/dcds.2021195.
    [24] H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Texts in Applied Mathematics, 57, Springer, New York, 2011. doi: 10.1007/978-1-4419-7646-8.
    [25] C. Somarakis and J. S. Baras, Delay-independent stability of consensus networks with application to flocking, IFAC, 48 (2015), 159-164.  doi: 10.1016/j.ifacol.2015.09.370.
    [26] K. Szwaykowska, I. B. Schwartz, L. Mier-y-Teran Romero, C. R. Heckman, D. Mox and M. A. Hsieh, Collective motion patterns of swarms with delay coupling: Theory and experiment, Phys. Rev. E, 93 (2016), 11 pp. doi: 10.1103/physreve.93.032307.
    [27] G. Valentini, Achieving Consensus in Robot Swarms. Design and Analysis of Strategies for the best-of-n Problem, Studies in Computational Intelligence, 706, Springer, Cham, 2017. doi: 10.1007/978-3-319-53609-5.
    [28] T. Vicsek and A. Zafeiris, Collective motion, Phys. Rep., 517 (2012), 71-140.  doi: 10.1016/j.physrep.2012.03.004.
    [29] C. Villani, Optimal Transport. Old and New, Grundlehren der mathematischen Wissenschaften, 338, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.
  • 加载中

Article Metrics

HTML views(296) PDF downloads(161) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint