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On the Cucker-Smale ensemble with $ q $-closest neighbors under time-delayed communications
1. | School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China |
2. | Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826 |
3. | School of Mathematics, Korea Institute for Advanced Study, Seoul 02455, South Korea |
We study time-asymptotic interplay between time-delayed communication and Cucker-Smale (C-S) velocity alignment. For this, we present two sufficient frameworks for the asymptotic flocking to the continuous and discrete C-S models with $ q $-closest neighbors in the presence of time-delayed communications. Communication time-delays result from the finite-propagation speed of information and they are often ignored in the first place modeling of collective dynamics. In the absence of time-delays in communication, Cucker and Dong showed that the C-S model with $ q $-closest neighbors can exhibit a phase-transition like phenomenon for unconditional and conditional flockings depending on the size $ q $ relative to system size. In this paper, we investigate whether Cucker and Dong's result is robust with respect to the time-delayed communications or not. In fact, our flocking estimates show that the critical number of $ q $ for unconditional flocking is the same as in the case for zero time-delay, which shows the robustness of the Cucker and Dong's result with respect to small time-delay.
References:
[1] |
S. M. Ahn and S.-Y. Ha, Stochastic flocking dynamics of the Cucker-Smale model with multiplicative white noises, J. Math. Phys., 51 (2010), 103301, 17 pp.
doi: 10.1063/1.3496895. |
[2] |
S. M. Ahn, H. Choi, S.-Y. Ha and H. Lee,
On the collision avoiding initial-configurations to the Cucker-Smale type flocking models, Commun. Math. Sci., 10 (2012), 625-643.
doi: 10.4310/CMS.2012.v10.n2.a10. |
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M. Aouchiche, O. Favaron and P. Hansen,
Variable neighborhood search for extremal graphs. XXII. Extending bounds for independence to upper irredundance, Discrete Appl. Math., 157 (2009), 3497-3510.
doi: 10.1016/j.dam.2009.04.004. |
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M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini, M. Viale and V. Zdravkovic,
Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study, Proc. Natl. Acad. Sci. U. S. A., 105 (2008), 1232-1237.
doi: 10.1073/pnas.0711437105. |
[5] |
N. Bellomo and S.-Y. Ha,
A quest toward a mathematical theory of the dynamics of swarms, Math. Models Methods Appl. Sci., 27 (2017), 745-770.
doi: 10.1142/S0218202517500154. |
[6] |
A. Blanchet and P. Degond,
Topological interactions in a Boltzmann-type framework, J. Stat. Phys., 163 (2016), 41-60.
doi: 10.1007/s10955-016-1471-6. |
[7] |
F. Bolley, J. A. Cañizo and J. A. Carrillo,
Stochastic mean-field limit: Non-Lipschitz forces and swarming, Math. Models Methods Appl. Sci., 21 (2011), 2179-2210.
doi: 10.1142/S0218202511005702. |
[8] |
M. Camperi, A. Cavaga, I. Giardina, G. Parisi and E. Silvestri,
Spatially balanced topological interaction grants optimal cohesion in flocking models, Interface Focus, 2 (2012), 715-725.
doi: 10.1098/rsfs.2012.0026. |
[9] |
J. A. Cañizo, J. 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. |
[10] |
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. |
[11] |
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. |
[12] |
Y.-P. Choi and Z. 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. |
[13] |
F. Cucker and J.-G. Dong,
Avoiding collisions in flocks, IEEE Trans. Automat. Control, 55 (2010), 1238-1243.
doi: 10.1109/TAC.2010.2042355. |
[14] |
F. Cucker and J.-G. Dong,
On flocks influenced by closest neighbors, Math. Models Methods Appl. Sci., 26 (2016), 2685-2708.
doi: 10.1142/S0218202516500639. |
[15] |
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. |
[16] |
F. Cucker and S. Smale,
Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.
doi: 10.1109/TAC.2007.895842. |
[17] |
J.-G. Dong, S.-Y Ha and 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.
doi: 10.3934/dcdsb.2019072. |
[18] |
J.-G. Dong and L. Qiu,
Flocking of the Cucker-Smale model on general digraphs, IEEE Trans. Automat. Control, 62 (2017), 5234-5239.
doi: 10.1109/TAC.2016.2631608. |
[19] |
R. Erban, J. Haskovec and Y. Sun,
On Cucker-Smale model with noise and delay, SIAM. J. Appl. Math., 76 (2016), 1535-1557.
doi: 10.1137/15M1030467. |
[20] |
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. |
[21] |
J. Haskovec,
Flocking dynamics and mean-field limit in the Cucker-Smale-type model with topological interactions, Phys. D, 261 (2013), 42-51.
doi: 10.1016/j.physd.2013.06.006. |
[22] |
Z. Li and X. Xue,
Cucker-Smale flocking under rooted leadership with fixed and switching topologies, SIAM J. Appl. Math., 70 (2010), 3156-3174.
doi: 10.1137/100791774. |
[23] |
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. |
[24] |
S. Martin,
Multi-agent flocking under topological interactions, Systems Control Lett., 69 (2014), 53-61.
doi: 10.1016/j.sysconle.2014.04.004. |
[25] |
R. Olfati-Saber and R. M. Murray,
Consensus problems in networks of agents with switching topology and time-delays, IEEE Trans. Automat. Control, 49 (2004), 1520-1533.
doi: 10.1109/TAC.2004.834113. |
[26] |
L. Perea, P. Elosegui and G. Gómez,
Extension of the Cucker-Smale control law to space fight formations, J. Guidance Contr. Dyn., 32 (2009), 526-536.
doi: 10.2514/1.36269. |
[27] |
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. |
[28] |
J. Shen,
Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math., 68 (2007/08), 694-719.
doi: 10.1137/060673254. |
[29] |
C. Virágh, G. Vásárhelyi, N. Tarcai, T. Szörényi, G. Somorjai, T. Nepusz, and T. Vicsek, Flocking algorithm for autonomous flying robots, Bioinspir. Biomim., 9 (2014), 025012.
doi: 10.1088/1748-3182/9/2/025012. |
[30] |
G. Vásárhelyi, C. Virágh, G. Somorjai, N. Tarcai, T. Szörényi, T. Nepusz, and T. Vicsek, Outdoor flocking and formation flight with autonomous aerial robots, in Proc. IEEE/RSJ Int. Conf. Intell. Robots Syst., Chicago, IL, 2014, 3866–3873.
doi: 10.1109/IROS.2014.6943105. |
[31] |
T. Vicsek, A. Czirók, 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. |
[32] |
T. Vicsek and A. Zefeiris,
Collective motion, Phys. Rep., 517 (2012), 71-140.
doi: 10.1016/j.physrep.2012.03.004. |
show all references
References:
[1] |
S. M. Ahn and S.-Y. Ha, Stochastic flocking dynamics of the Cucker-Smale model with multiplicative white noises, J. Math. Phys., 51 (2010), 103301, 17 pp.
doi: 10.1063/1.3496895. |
[2] |
S. M. Ahn, H. Choi, S.-Y. Ha and H. Lee,
On the collision avoiding initial-configurations to the Cucker-Smale type flocking models, Commun. Math. Sci., 10 (2012), 625-643.
doi: 10.4310/CMS.2012.v10.n2.a10. |
[3] |
M. Aouchiche, O. Favaron and P. Hansen,
Variable neighborhood search for extremal graphs. XXII. Extending bounds for independence to upper irredundance, Discrete Appl. Math., 157 (2009), 3497-3510.
doi: 10.1016/j.dam.2009.04.004. |
[4] |
M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini, M. Viale and V. Zdravkovic,
Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study, Proc. Natl. Acad. Sci. U. S. A., 105 (2008), 1232-1237.
doi: 10.1073/pnas.0711437105. |
[5] |
N. Bellomo and S.-Y. Ha,
A quest toward a mathematical theory of the dynamics of swarms, Math. Models Methods Appl. Sci., 27 (2017), 745-770.
doi: 10.1142/S0218202517500154. |
[6] |
A. Blanchet and P. Degond,
Topological interactions in a Boltzmann-type framework, J. Stat. Phys., 163 (2016), 41-60.
doi: 10.1007/s10955-016-1471-6. |
[7] |
F. Bolley, J. A. Cañizo and J. A. Carrillo,
Stochastic mean-field limit: Non-Lipschitz forces and swarming, Math. Models Methods Appl. Sci., 21 (2011), 2179-2210.
doi: 10.1142/S0218202511005702. |
[8] |
M. Camperi, A. Cavaga, I. Giardina, G. Parisi and E. Silvestri,
Spatially balanced topological interaction grants optimal cohesion in flocking models, Interface Focus, 2 (2012), 715-725.
doi: 10.1098/rsfs.2012.0026. |
[9] |
J. A. Cañizo, J. 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. |
[10] |
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. |
[11] |
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. |
[12] |
Y.-P. Choi and Z. 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. |
[13] |
F. Cucker and J.-G. Dong,
Avoiding collisions in flocks, IEEE Trans. Automat. Control, 55 (2010), 1238-1243.
doi: 10.1109/TAC.2010.2042355. |
[14] |
F. Cucker and J.-G. Dong,
On flocks influenced by closest neighbors, Math. Models Methods Appl. Sci., 26 (2016), 2685-2708.
doi: 10.1142/S0218202516500639. |
[15] |
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. |
[16] |
F. Cucker and S. Smale,
Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.
doi: 10.1109/TAC.2007.895842. |
[17] |
J.-G. Dong, S.-Y Ha and 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.
doi: 10.3934/dcdsb.2019072. |
[18] |
J.-G. Dong and L. Qiu,
Flocking of the Cucker-Smale model on general digraphs, IEEE Trans. Automat. Control, 62 (2017), 5234-5239.
doi: 10.1109/TAC.2016.2631608. |
[19] |
R. Erban, J. Haskovec and Y. Sun,
On Cucker-Smale model with noise and delay, SIAM. J. Appl. Math., 76 (2016), 1535-1557.
doi: 10.1137/15M1030467. |
[20] |
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. |
[21] |
J. Haskovec,
Flocking dynamics and mean-field limit in the Cucker-Smale-type model with topological interactions, Phys. D, 261 (2013), 42-51.
doi: 10.1016/j.physd.2013.06.006. |
[22] |
Z. Li and X. Xue,
Cucker-Smale flocking under rooted leadership with fixed and switching topologies, SIAM J. Appl. Math., 70 (2010), 3156-3174.
doi: 10.1137/100791774. |
[23] |
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. |
[24] |
S. Martin,
Multi-agent flocking under topological interactions, Systems Control Lett., 69 (2014), 53-61.
doi: 10.1016/j.sysconle.2014.04.004. |
[25] |
R. Olfati-Saber and R. M. Murray,
Consensus problems in networks of agents with switching topology and time-delays, IEEE Trans. Automat. Control, 49 (2004), 1520-1533.
doi: 10.1109/TAC.2004.834113. |
[26] |
L. Perea, P. Elosegui and G. Gómez,
Extension of the Cucker-Smale control law to space fight formations, J. Guidance Contr. Dyn., 32 (2009), 526-536.
doi: 10.2514/1.36269. |
[27] |
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. |
[28] |
J. Shen,
Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math., 68 (2007/08), 694-719.
doi: 10.1137/060673254. |
[29] |
C. Virágh, G. Vásárhelyi, N. Tarcai, T. Szörényi, G. Somorjai, T. Nepusz, and T. Vicsek, Flocking algorithm for autonomous flying robots, Bioinspir. Biomim., 9 (2014), 025012.
doi: 10.1088/1748-3182/9/2/025012. |
[30] |
G. Vásárhelyi, C. Virágh, G. Somorjai, N. Tarcai, T. Szörényi, T. Nepusz, and T. Vicsek, Outdoor flocking and formation flight with autonomous aerial robots, in Proc. IEEE/RSJ Int. Conf. Intell. Robots Syst., Chicago, IL, 2014, 3866–3873.
doi: 10.1109/IROS.2014.6943105. |
[31] |
T. Vicsek, A. Czirók, 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. |
[32] |
T. Vicsek and A. Zefeiris,
Collective motion, Phys. Rep., 517 (2012), 71-140.
doi: 10.1016/j.physrep.2012.03.004. |


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