December  2019, 14(4): 789-804. doi: 10.3934/nhm.2019032

Emergent behavior of Cucker-Smale model with normalized weights and distributed time delays

1. 

Department of Mathematics, Yonsei University, 50 Yonsei-Ro, Seodaemun-Gu, Seoul 03722, Republic of Korea

2. 

Dipartimento di Ingegneria e Scienze dell'Informazione e Matematica, Università di L'Aquila, Via Vetoio, 67010 L'Aquila, Italy

* Corresponding author: Cristina Pignotti

Received  February 2019 Revised  June 2019 Published  October 2019

Fund Project: The first author was supported by National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIP) (No. 2017R1C1B2012918 and 2017R1A4A1014735) and POSCO Science Fellowship of POSCO TJ Park Foundation. The research of the second author was partially supported by the GNAMPA 2018 project Analisi e controllo di modelli differenziali non lineari (INdAM).

We study a Cucker-Smale-type flocking model with distributed time delay where individuals interact with each other through normalized communication weights. Based on a Lyapunov functional approach, we provide sufficient conditions for the velocity alignment behavior. We then show that as the number of individuals $ N $ tends to infinity, the $ N $-particle system can be well approximated by a delayed Vlasov alignment equation. Furthermore, we also establish the global existence of measure-valued solutions for the delayed Vlasov alignment equation and its large-time asymptotic behavior.

Citation: Young-Pil Choi, Cristina Pignotti. Emergent behavior of Cucker-Smale model with normalized weights and distributed time delays. Networks and Heterogeneous Media, 2019, 14 (4) : 789-804. doi: 10.3934/nhm.2019032
References:
[1]

G. AlbiM. Herty and L. Pareschi, Kinetic description of optimal control problems and applications to opinion consensus, Commun. Math. Sci., 13 (2015), 1407-1429.  doi: 10.4310/CMS.2015.v13.n6.a3.

[2]

N. BellomoM. A. Herrero and A. Tosin, On the dynamics of social conflict: Looking for the black swan, Kinet. Relat. Models, 6 (2013), 459-479.  doi: 10.3934/krm.2013.6.459.

[3]

A. Borzì and S. Wongkaew, Modeling and control through leadership of a refined flocking system, Math. Models Methods Appl. Sci., 25 (2015), 255-282.  doi: 10.1142/S0218202515500098.

[4]

J. A. CañizoJ. A. Carrillo and J. A. 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.

[5]

M. CaponigroM. FornasierB. Piccoli and E. Trélat, Sparse stabilization and control of alignment models, Math. Models Methods Appl. Sci., 25 (2015), 521-564.  doi: 10.1142/S0218202515400059.

[6]

J. A. CarrilloY.-P. Choi and S. Perez, A review an attractive-repulsive hydrodynamics for consensus in collective behavior, Active Particles Vol.I: Advances in Theory, Models, Applications, Model. Simul. Sci. Eng. Technol., Birkhäuser/Springer, Cham, 1 (2017), 259-298. 

[7]

J. A. CarrilloY.-P. Choi and M. Hauray, The derivation of swarming models: Mean-field limit and Wasserstein distances, Collective Dynamics from Bacteria to Crowds, CISM Courses and Lect., Springer, Vienna, 533 (2014), 1-46.  doi: 10.1007/978-3-7091-1785-9_1.

[8]

J. A. CarrilloY.-P. Choi and M. Hauray, Local well-posedness of the generalized Cucker-Smale model with singular kernels, MMCS, Mathematical modelling of complex systems, ESAIM Proc. Surveys, EDP Sci., Les Ulis, 47 (2014), 17-35.  doi: 10.1051/proc/201447002.

[9]

J. A. CarrilloY.-P. ChoiM. Hauray and S. Salem, Mean-field limit for collective behavior models with sharp sensitivity regions, J. Eur. Math. Soc. (JEMS), 21 (2019), 121-161.  doi: 10.4171/JEMS/832.

[10]

J. A. CarrilloY.-P. ChoiP. B. Mucha and J. Peszek, Sharp conditions to avoid collisions in singular Cucker-Smale interactions, Nonlinear Anal.-Real World Appl., 37 (2017), 317-328.  doi: 10.1016/j.nonrwa.2017.02.017.

[11]

J. A. CarrilloM. FornasierJ. Rosado and G. Toscani, Asymptotic flocking dynamics for the kinetic Cucker-Smale model, SIAM J. Math. Anal., 42 (2010), 218-236.  doi: 10.1137/090757290.

[12]

Y.-P. ChoiS.-Y. Ha and Z. C. Li, Emergent dynamics of the Cucker-Smale flocking model and its variants, Active particles, Advances in theory, models, and applications, Model. Simul. Sci. Eng. Technol., Birkhäuser/Springer, Cham, 1 (2017), 299-331. 

[13]

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.

[14]

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.

[15]

Y.-P. Choi and Z. C. Li, Emergent behavior of Cucker-Smale flocking particles with heterogeneous time delays, Appl. Math. Lett., 86 (2018), 49-56.  doi: 10.1016/j.aml.2018.06.018.

[16]

Y.-P. Choi, D. Kalise, J. Peszek and A. A. Peters, A collisionless singular Cucker-Smale model with decentralized formation control, SIAM J. Appl. Dyn. Syst., to appear.

[17]

I. D. CouzinJ. KrauseN. R. Franks and S. A. Levin, Effective leadership and decision making in animal groups on the move, Nature, 433 (2005), 513-516.  doi: 10.1038/nature03236.

[18]

E. CristianiB. Piccoli and A. Tosin, Multiscale modeling of granular flows with application to crowd dynamics, Multiscale Model. Simul., 9 (2011), 155-182.  doi: 10.1137/100797515.

[19]

F. Cucker and J.-G. Dong, A general collision-avoiding flocking framework, IEEE Trans. Automat. Cont., 56 (2011), 1124-1129.  doi: 10.1109/TAC.2011.2107113.

[20]

F. Cucker and E. Mordecki, Flocking in noisy environments, J. Math. Pures Appl., 89 (2008), 278-296.  doi: 10.1016/j.matpur.2007.12.002.

[21]

F. Cucker and S. Smale, Emergent behaviour in flocks, IEEE Trans. Automat. Contr., 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842.

[22]

F. Cucker and S. Smale, On the mathematics of emergence, Japanese Journal of Mathematics, 2 (2007), 197-227.  doi: 10.1007/s11537-007-0647-x.

[23]

R. ErbanJ. Haškovec and Y. Z. Sun, A Cucker-Smale model with noise and delay, SIAM J. Appl. Math., 76 (2016), 1535-1557.  doi: 10.1137/15M1030467.

[24] A. Halanay, Differential Equations: Stability, Oscillations, Time Lags, Academic Press, New York and London, 1966. 
[25]

S.-Y. HaK. Lee and D. Levy, Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system, Commun. Math. Sci., 7 (2009), 453-469.  doi: 10.4310/CMS.2009.v7.n2.a9.

[26]

S.-Y. Ha and J.-G. Liu, A simple proof of the Cucker-Smale flocking dynamics and mean-field limit, Commun. Math. Sci., 7 (2009), 297-325.  doi: 10.4310/CMS.2009.v7.n2.a2.

[27]

S.-Y. Ha and M. Slemrod, Flocking dynamics of singularly perturbed oscillator chain and the Cucker-Smale system, J. Dyn. Diff. Equat., 22 (2010), 325-330.  doi: 10.1007/s10884-009-9142-9.

[28]

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.

[29]

S. LemercierA. JelicR. KulpaJ. HuaJ. FehrenbachP. DegondC. Appert- RollandS. Donikian and J. Pettré, Realistic following behaviors for crowd simulation, Comput. Graph. Forum, 31 (2012), 489-498.  doi: 10.1111/j.1467-8659.2012.03028.x.

[30]

N. A. MecholskyE. Ott and T. M. Antonsen, Obstacle and predator avoidance in a model for flocking, Physica D, 239 (2010), 988-996.  doi: 10.1016/j.physd.2010.02.007.

[31]

S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behaviour, J. Stat. Phys., 144 (2011), 923-947.  doi: 10.1007/s10955-011-0285-9.

[32]

J. Peszek, Existence of piecewise weak solutions of a discrete Cucker-Smale's flocking model with a singular communication weight, J. Differential Equations, 257 (2014), 2900-2925.  doi: 10.1016/j.jde.2014.06.003.

[33]

B. PiccoliF. Rossi and E. Trélat, Control to flocking of the kinetic Cucker-Smale model, SIAM J. Math. Anal., 47 (2015), 4685-4719.  doi: 10.1137/140996501.

[34]

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.

[35]

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 (2019), 233–253. doi: 10.1007/978-3-030-17949-6_12.

[36]

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.

[37]

J. Shen, Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math., 68 (2007/08), 694-719.  doi: 10.1137/060673254.

[38]

C. H. Tan, A discontinuous Galerkin method on kinetic flocking models, Math. Models Methods Appl. Sci., 27 (2017), 1199-1221.  doi: 10.1142/S0218202517400139.

[39]

G. Toscani, Kinetic models of opinion formation, Commun. Math. Sci., 4 (2006), 481-496.  doi: 10.4310/CMS.2006.v4.n3.a1.

[40]

T. VicsekA. CzirokE. Ben-JacobI. Cohen and O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229.  doi: 10.1103/PhysRevLett.75.1226.

show all references

References:
[1]

G. AlbiM. Herty and L. Pareschi, Kinetic description of optimal control problems and applications to opinion consensus, Commun. Math. Sci., 13 (2015), 1407-1429.  doi: 10.4310/CMS.2015.v13.n6.a3.

[2]

N. BellomoM. A. Herrero and A. Tosin, On the dynamics of social conflict: Looking for the black swan, Kinet. Relat. Models, 6 (2013), 459-479.  doi: 10.3934/krm.2013.6.459.

[3]

A. Borzì and S. Wongkaew, Modeling and control through leadership of a refined flocking system, Math. Models Methods Appl. Sci., 25 (2015), 255-282.  doi: 10.1142/S0218202515500098.

[4]

J. A. CañizoJ. A. Carrillo and J. A. 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.

[5]

M. CaponigroM. FornasierB. Piccoli and E. Trélat, Sparse stabilization and control of alignment models, Math. Models Methods Appl. Sci., 25 (2015), 521-564.  doi: 10.1142/S0218202515400059.

[6]

J. A. CarrilloY.-P. Choi and S. Perez, A review an attractive-repulsive hydrodynamics for consensus in collective behavior, Active Particles Vol.I: Advances in Theory, Models, Applications, Model. Simul. Sci. Eng. Technol., Birkhäuser/Springer, Cham, 1 (2017), 259-298. 

[7]

J. A. CarrilloY.-P. Choi and M. Hauray, The derivation of swarming models: Mean-field limit and Wasserstein distances, Collective Dynamics from Bacteria to Crowds, CISM Courses and Lect., Springer, Vienna, 533 (2014), 1-46.  doi: 10.1007/978-3-7091-1785-9_1.

[8]

J. A. CarrilloY.-P. Choi and M. Hauray, Local well-posedness of the generalized Cucker-Smale model with singular kernels, MMCS, Mathematical modelling of complex systems, ESAIM Proc. Surveys, EDP Sci., Les Ulis, 47 (2014), 17-35.  doi: 10.1051/proc/201447002.

[9]

J. A. CarrilloY.-P. ChoiM. Hauray and S. Salem, Mean-field limit for collective behavior models with sharp sensitivity regions, J. Eur. Math. Soc. (JEMS), 21 (2019), 121-161.  doi: 10.4171/JEMS/832.

[10]

J. A. CarrilloY.-P. ChoiP. B. Mucha and J. Peszek, Sharp conditions to avoid collisions in singular Cucker-Smale interactions, Nonlinear Anal.-Real World Appl., 37 (2017), 317-328.  doi: 10.1016/j.nonrwa.2017.02.017.

[11]

J. A. CarrilloM. FornasierJ. Rosado and G. Toscani, Asymptotic flocking dynamics for the kinetic Cucker-Smale model, SIAM J. Math. Anal., 42 (2010), 218-236.  doi: 10.1137/090757290.

[12]

Y.-P. ChoiS.-Y. Ha and Z. C. Li, Emergent dynamics of the Cucker-Smale flocking model and its variants, Active particles, Advances in theory, models, and applications, Model. Simul. Sci. Eng. Technol., Birkhäuser/Springer, Cham, 1 (2017), 299-331. 

[13]

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.

[14]

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.

[15]

Y.-P. Choi and Z. C. Li, Emergent behavior of Cucker-Smale flocking particles with heterogeneous time delays, Appl. Math. Lett., 86 (2018), 49-56.  doi: 10.1016/j.aml.2018.06.018.

[16]

Y.-P. Choi, D. Kalise, J. Peszek and A. A. Peters, A collisionless singular Cucker-Smale model with decentralized formation control, SIAM J. Appl. Dyn. Syst., to appear.

[17]

I. D. CouzinJ. KrauseN. R. Franks and S. A. Levin, Effective leadership and decision making in animal groups on the move, Nature, 433 (2005), 513-516.  doi: 10.1038/nature03236.

[18]

E. CristianiB. Piccoli and A. Tosin, Multiscale modeling of granular flows with application to crowd dynamics, Multiscale Model. Simul., 9 (2011), 155-182.  doi: 10.1137/100797515.

[19]

F. Cucker and J.-G. Dong, A general collision-avoiding flocking framework, IEEE Trans. Automat. Cont., 56 (2011), 1124-1129.  doi: 10.1109/TAC.2011.2107113.

[20]

F. Cucker and E. Mordecki, Flocking in noisy environments, J. Math. Pures Appl., 89 (2008), 278-296.  doi: 10.1016/j.matpur.2007.12.002.

[21]

F. Cucker and S. Smale, Emergent behaviour in flocks, IEEE Trans. Automat. Contr., 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842.

[22]

F. Cucker and S. Smale, On the mathematics of emergence, Japanese Journal of Mathematics, 2 (2007), 197-227.  doi: 10.1007/s11537-007-0647-x.

[23]

R. ErbanJ. Haškovec and Y. Z. Sun, A Cucker-Smale model with noise and delay, SIAM J. Appl. Math., 76 (2016), 1535-1557.  doi: 10.1137/15M1030467.

[24] A. Halanay, Differential Equations: Stability, Oscillations, Time Lags, Academic Press, New York and London, 1966. 
[25]

S.-Y. HaK. Lee and D. Levy, Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system, Commun. Math. Sci., 7 (2009), 453-469.  doi: 10.4310/CMS.2009.v7.n2.a9.

[26]

S.-Y. Ha and J.-G. Liu, A simple proof of the Cucker-Smale flocking dynamics and mean-field limit, Commun. Math. Sci., 7 (2009), 297-325.  doi: 10.4310/CMS.2009.v7.n2.a2.

[27]

S.-Y. Ha and M. Slemrod, Flocking dynamics of singularly perturbed oscillator chain and the Cucker-Smale system, J. Dyn. Diff. Equat., 22 (2010), 325-330.  doi: 10.1007/s10884-009-9142-9.

[28]

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.

[29]

S. LemercierA. JelicR. KulpaJ. HuaJ. FehrenbachP. DegondC. Appert- RollandS. Donikian and J. Pettré, Realistic following behaviors for crowd simulation, Comput. Graph. Forum, 31 (2012), 489-498.  doi: 10.1111/j.1467-8659.2012.03028.x.

[30]

N. A. MecholskyE. Ott and T. M. Antonsen, Obstacle and predator avoidance in a model for flocking, Physica D, 239 (2010), 988-996.  doi: 10.1016/j.physd.2010.02.007.

[31]

S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behaviour, J. Stat. Phys., 144 (2011), 923-947.  doi: 10.1007/s10955-011-0285-9.

[32]

J. Peszek, Existence of piecewise weak solutions of a discrete Cucker-Smale's flocking model with a singular communication weight, J. Differential Equations, 257 (2014), 2900-2925.  doi: 10.1016/j.jde.2014.06.003.

[33]

B. PiccoliF. Rossi and E. Trélat, Control to flocking of the kinetic Cucker-Smale model, SIAM J. Math. Anal., 47 (2015), 4685-4719.  doi: 10.1137/140996501.

[34]

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.

[35]

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 (2019), 233–253. doi: 10.1007/978-3-030-17949-6_12.

[36]

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.

[37]

J. Shen, Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math., 68 (2007/08), 694-719.  doi: 10.1137/060673254.

[38]

C. H. Tan, A discontinuous Galerkin method on kinetic flocking models, Math. Models Methods Appl. Sci., 27 (2017), 1199-1221.  doi: 10.1142/S0218202517400139.

[39]

G. Toscani, Kinetic models of opinion formation, Commun. Math. Sci., 4 (2006), 481-496.  doi: 10.4310/CMS.2006.v4.n3.a1.

[40]

T. VicsekA. CzirokE. Ben-JacobI. Cohen and O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229.  doi: 10.1103/PhysRevLett.75.1226.

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