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

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

* 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.

Citation: Jiu-Gang Dong, Seung-Yeal Ha, Doheon Kim. On the Cucker-Smale ensemble with $ q $-closest neighbors under time-delayed communications. Kinetic & 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.  Google Scholar

[2]

S. M. AhnH. ChoiS.-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.  Google Scholar

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M. BalleriniN. CabibboR. CandelierA. CavagnaE. CisbaniI. GiardinaV. LecomteA. OrlandiG. ParisiA. ProcacciniM. 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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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F. BolleyJ. 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.  Google Scholar

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M. CamperiA. CavagaI. GiardinaG. 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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[17]

J.-G. DongS.-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.  Google Scholar

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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.  Google Scholar

[19]

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

[20]

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

S. Martin, Multi-agent flocking under topological interactions, Systems Control Lett., 69 (2014), 53-61.  doi: 10.1016/j.sysconle.2014.04.004.  Google Scholar

[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.  Google Scholar

[26]

L. PereaP. 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.  Google Scholar

[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.  Google Scholar

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[31]

T. VicsekA. CzirókE. 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.  Google Scholar

[32]

T. Vicsek and A. Zefeiris, Collective motion, Phys. Rep., 517 (2012), 71-140.  doi: 10.1016/j.physrep.2012.03.004.  Google Scholar

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.  Google Scholar

[2]

S. M. AhnH. ChoiS.-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.  Google Scholar

[3]

M. AouchicheO. 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.  Google Scholar

[4]

M. BalleriniN. CabibboR. CandelierA. CavagnaE. CisbaniI. GiardinaV. LecomteA. OrlandiG. ParisiA. ProcacciniM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

F. BolleyJ. 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.  Google Scholar

[8]

M. CamperiA. CavagaI. GiardinaG. 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.  Google Scholar

[9]

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[17]

J.-G. DongS.-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.  Google Scholar

[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.  Google Scholar

[19]

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

[20]

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

S. Martin, Multi-agent flocking under topological interactions, Systems Control Lett., 69 (2014), 53-61.  doi: 10.1016/j.sysconle.2014.04.004.  Google Scholar

[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.  Google Scholar

[26]

L. PereaP. 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.  Google Scholar

[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.  Google Scholar

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[31]

T. VicsekA. CzirókE. 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.  Google Scholar

[32]

T. Vicsek and A. Zefeiris, Collective motion, Phys. Rep., 517 (2012), 71-140.  doi: 10.1016/j.physrep.2012.03.004.  Google Scholar

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