American Institute of Mathematical Sciences

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August  2020, 13(4): 653-676. doi: 10.3934/krm.2020022

On the Cucker-Smale ensemble with $ q $-closest neighbors under time-delayed communications

Jiu-Gang Dong 1, , Seung-Yeal Ha 2,3, and  Doheon Kim 3,,

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

* Corresponding author: Doheon Kim

Received  May 2019 Revised  March 2020 Published  May 2020

Fund Project: The work of S.-Y. Ha was supported by the National Research Foundation of Korea(NRF-2020R1A2C3A01003881), and the work of D. Kim was supported by a KIAS Individual Grant (MG073901) at Korea Institute for Advanced Study

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.

Keywords: Collective dynamics, Cucker-Smale model, nonlinear functional, mono-cluster flocking, multi-agent system.
Mathematics Subject Classification: Primary: 34D05; Secondary: 39A12, 70F99.
Citation: Jiu-Gang Dong, Seung-Yeal Ha, Doheon Kim. On the Cucker-Smale ensemble with $ q $-closest neighbors under time-delayed communications. Kinetic and Related Models, 2020, 13 (4) : 653-676. doi: 10.3934/krm.2020022
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.

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.

Figure 1.  The initial configuration $ ( {\boldsymbol{x}}_i, {\boldsymbol{v}}_i) $ $ (i = 1.\cdots,20) $
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Figure 2.  The graphs of $ \mathcal D(V(t)) $ and $ \log \mathcal D(V(t)) $ for $ q = 9,10,11,12 $
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Figure 3.  The graphs of $ \mathcal D(X(t)) $ and $ \log \mathcal D(X(t)) $ for $ q = 9,10,11,12 $
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Figure 4.  Emergence of bi-cluster flocking for $ q = 9 $
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