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A discrete districting plan
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 |
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.
References:
[1] |
G. Albi, M. 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. Bellomo, M. 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. Cañizo, J. 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. Caponigro, M. Fornasier, B. 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. Carrillo, Y.-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. Carrillo, Y.-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. Carrillo, Y.-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. Carrillo, Y.-P. Choi, M. 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. Carrillo, Y.-P. Choi, P. 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. Carrillo, M. Fornasier, J. 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. Choi, S.-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. Couzin, J. Krause, N. 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. Cristiani, B. 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. Erban, J. 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. Ha, K. 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. Lemercier, A. Jelic, R. Kulpa, J. Hua, J. Fehrenbach, P. Degond, C. Appert- Rolland, S. 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. Mecholsky, E. 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. Piccoli, F. 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. Vicsek, A. Czirok, E. Ben-Jacob, I. 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. Albi, M. 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. Bellomo, M. 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. Cañizo, J. 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. Caponigro, M. Fornasier, B. 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. Carrillo, Y.-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. Carrillo, Y.-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. Carrillo, Y.-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. Carrillo, Y.-P. Choi, M. 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. Carrillo, Y.-P. Choi, P. 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. Carrillo, M. Fornasier, J. 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. Choi, S.-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. Couzin, J. Krause, N. 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. Cristiani, B. 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. Erban, J. 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. Ha, K. 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. Lemercier, A. Jelic, R. Kulpa, J. Hua, J. Fehrenbach, P. Degond, C. Appert- Rolland, S. 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. Mecholsky, E. 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. Piccoli, F. 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. Vicsek, A. Czirok, E. Ben-Jacob, I. 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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