doi: 10.3934/krm.2021030
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The delayed Cucker-Smale model with short range communication weights

Department of Mathematics, School of Science, Nanchang University, Nanchang, Jiangxi 330031, China

* Corresponding author

Received  March 2021 Revised  July 2021 Early access September 2021

Fund Project: The first author is supported by NSFC grant 11961046. The second author is supported by NSFC grant 61963028) and Natural Science Foundation of Jiangxi Province grant 20192BAB207023

Various flocking results have been established for the delayed Cucker-Smale model, especially in the long range communication case. However, the short range communication case is more realistic due to the limited communication ability. In this case, the non-flocking behavior can be frequently observed in numerical simulations. Furthermore, it has potential applications in many practical situations, such as the opinion disagreement in society, fish flock breaking and so on. Therefore, we firstly consider the non-flocking behavior of the delayed Cucker$ - $Smale model. Based on a key inequality of position variance, a simple sufficient condition of the initial data to the non-flocking behavior is established. Then, for general communication weights we obtain a flocking result, which also depends upon the initial data in the short range communication case. Finally, with no restriction on the initial data we further establish other large time behavior of classical solutions.

Citation: Zili Chen, Xiuxia Yin. The delayed Cucker-Smale model with short range communication weights. Kinetic & Related Models, doi: 10.3934/krm.2021030
References:
[1]

M. CaponigroM. FornasierB. Piccoli and E. Trélat, Sparse stabilization and optimal control of the Cucker$-$Smale model, Math. Cont. Related Fields, 3 (2013), 447-466.  doi: 10.3934/mcrf.2013.3.447.  Google Scholar

[2]

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.  Google Scholar

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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.  Google Scholar

[5]

Z. Chen and X. Yin, The kinetic Cucker$-$Smale model: Well-posedness and asymptotic behavior, SIAM J. Math. Anal., 51 (2019), 3819-3853.  doi: 10.1137/18M1215001.  Google Scholar

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J. Cheng, Z. Li and J. Wu, Flocking in a two-agent Cucker$-$Smale model with large delay, Proc. Amer. Math. Soc., 149 (2021), 1711–1721. doi: 10.1090/proc/15295.  Google Scholar

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J. ChoS.-Y. HaF. HuangC. Jin and D. Ko, Emergence of bi-cluster flocking for the Cucker$-$Smale model, Math. Models Methods Appl. Sci., 26 (2016), 1191-1218.  doi: 10.1142/S0218202516500287.  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

[10]

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

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F. Cucker and C. Huepe, Flocking with informed agents, Math. Action, 1 (2008), 1-25.  doi: 10.5802/msia.1.  Google Scholar

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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

[13]

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

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[15]

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J. Haskovec, A simple proof of asymptotic consensus in the Hegselmann$-$Krause and Cucker$-$Smale models with normalization and delay, SIAM J. Appl. Dyn. Syst., 20 (2021), 130-148.  doi: 10.1137/20M1341350.  Google Scholar

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J. Haskovec and I. Markou, Asymptotic flocking in the Cucker$-$Smale model with reaction-type delays in the non-oscillatory regime, Kinet. Relat. Models, 13 (2020), 795-813.  doi: 10.3934/krm.2020027.  Google Scholar

[23]

J. Haskovec and I. Markou, Exponential asymptotic flocking in the Cucker$-$Smale model with distributed reaction delays, Math. Biosci. Eng., 17 (2020), 5651–5671. doi: 10.3934/mbe.2020304.  Google Scholar

[24]

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

[25]

S. Motsch and E. Tadmor, Heterophilious dynamics enhances consensus, SIAM Rev., 56 (2014), 577-621.  doi: 10.1137/120901866.  Google Scholar

[26]

P. B. Mucha and J. Peszek, The Cucker$-$Smale equation: Singular communication weight, measure-valued solutions and weak-atomic uniqueness, Arch. Ration. Mech. Anal., 227 (2018), 273-308.  doi: 10.1007/s00205-017-1160-x.  Google Scholar

[27]

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

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J. Peszek, Discrete Cucker$-$Smale flocking model with a weakly singular weight, SIAM J. Math. Anal., 47 (2015), 3671-3686.  doi: 10.1137/15M1009299.  Google Scholar

[29]

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

[30]

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.  Google Scholar

[31]

C. Pignotti and I. R. 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

[32]

C. Pignotti and I. R. 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., Vol. 32, Springer, Cham, 2019,233–253.  Google Scholar

[33]

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

[34]

X. Yin, Z. Gao, Z. Chen and Y. Fu, Non-existence of asymptotic flocking in the Cucker$-$Smale model with short range communication weights, IEEE Trans. Automat. Control, Available from: https://ieeexplore.ieee.org/document/9370113/. doi: 10.1109/TAC.2021.3063951.  Google Scholar

[35]

X. YinD. Yue and Z. Chen, Asymptotic behavior and collision avoidance in the Cucker$-$Smale model, IEEE Trans. Automat. Control, 65 (2020), 3112-3119.  doi: 10.1109/TAC.2019.2948473.  Google Scholar

[36]

X. YinD. Yue and S. Hu, Adaptive periodic event-triggered consensus for multi-agent systems subject to input saturation, Int. J. Control, 89 (2016), 653-667.  doi: 10.1080/00207179.2015.1088967.  Google Scholar

show all references

References:
[1]

M. CaponigroM. FornasierB. Piccoli and E. Trélat, Sparse stabilization and optimal control of the Cucker$-$Smale model, Math. Cont. Related Fields, 3 (2013), 447-466.  doi: 10.3934/mcrf.2013.3.447.  Google Scholar

[2]

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.  Google Scholar

[3]

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.  Google Scholar

[4]

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.  Google Scholar

[5]

Z. Chen and X. Yin, The kinetic Cucker$-$Smale model: Well-posedness and asymptotic behavior, SIAM J. Math. Anal., 51 (2019), 3819-3853.  doi: 10.1137/18M1215001.  Google Scholar

[6]

J. Cheng, Z. Li and J. Wu, Flocking in a two-agent Cucker$-$Smale model with large delay, Proc. Amer. Math. Soc., 149 (2021), 1711–1721. doi: 10.1090/proc/15295.  Google Scholar

[7]

J. ChoS.-Y. HaF. HuangC. Jin and D. Ko, Emergence of bi-cluster flocking for the Cucker$-$Smale model, Math. Models Methods Appl. Sci., 26 (2016), 1191-1218.  doi: 10.1142/S0218202516500287.  Google Scholar

[8]

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

[9]

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

[10]

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

[11]

F. Cucker and C. Huepe, Flocking with informed agents, Math. Action, 1 (2008), 1-25.  doi: 10.5802/msia.1.  Google Scholar

[12]

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

[13]

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

[14]

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

[15]

M. FornasierJ. Haskovec and G. Toscani, Fluid dynamic description of flocking via the Povzner$-$Boltzmann equation, Phys. D, 240 (2011), 21-31.  doi: 10.1016/j.physd.2010.08.003.  Google Scholar

[16]

S.-Y. HaJ. KimJ. Park and X. Zhang, Complete cluster predictability of the Cucker$-$Smale flocking model on the real line, Arch. Ration. Mech. Anal., 231 (2019), 319-365.  doi: 10.1007/s00205-018-1281-x.  Google Scholar

[17]

S.-Y. HaD. Ko and Y. Zhang, Critical coupling strength of the Cucker$-$Smale model for flocking, Math. Models Methods Appl. Sci., 27 (2017), 1051-1087.  doi: 10.1142/S0218202517400097.  Google Scholar

[18]

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

[19]

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.  Google Scholar

[20]

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.  Google Scholar

[21]

J. Haskovec, A simple proof of asymptotic consensus in the Hegselmann$-$Krause and Cucker$-$Smale models with normalization and delay, SIAM J. Appl. Dyn. Syst., 20 (2021), 130-148.  doi: 10.1137/20M1341350.  Google Scholar

[22]

J. Haskovec and I. Markou, Asymptotic flocking in the Cucker$-$Smale model with reaction-type delays in the non-oscillatory regime, Kinet. Relat. Models, 13 (2020), 795-813.  doi: 10.3934/krm.2020027.  Google Scholar

[23]

J. Haskovec and I. Markou, Exponential asymptotic flocking in the Cucker$-$Smale model with distributed reaction delays, Math. Biosci. Eng., 17 (2020), 5651–5671. doi: 10.3934/mbe.2020304.  Google Scholar

[24]

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

[25]

S. Motsch and E. Tadmor, Heterophilious dynamics enhances consensus, SIAM Rev., 56 (2014), 577-621.  doi: 10.1137/120901866.  Google Scholar

[26]

P. B. Mucha and J. Peszek, The Cucker$-$Smale equation: Singular communication weight, measure-valued solutions and weak-atomic uniqueness, Arch. Ration. Mech. Anal., 227 (2018), 273-308.  doi: 10.1007/s00205-017-1160-x.  Google Scholar

[27]

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

[28]

J. Peszek, Discrete Cucker$-$Smale flocking model with a weakly singular weight, SIAM J. Math. Anal., 47 (2015), 3671-3686.  doi: 10.1137/15M1009299.  Google Scholar

[29]

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

[30]

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.  Google Scholar

[31]

C. Pignotti and I. R. 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

[32]

C. Pignotti and I. R. 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., Vol. 32, Springer, Cham, 2019,233–253.  Google Scholar

[33]

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

[34]

X. Yin, Z. Gao, Z. Chen and Y. Fu, Non-existence of asymptotic flocking in the Cucker$-$Smale model with short range communication weights, IEEE Trans. Automat. Control, Available from: https://ieeexplore.ieee.org/document/9370113/. doi: 10.1109/TAC.2021.3063951.  Google Scholar

[35]

X. YinD. Yue and Z. Chen, Asymptotic behavior and collision avoidance in the Cucker$-$Smale model, IEEE Trans. Automat. Control, 65 (2020), 3112-3119.  doi: 10.1109/TAC.2019.2948473.  Google Scholar

[36]

X. YinD. Yue and S. Hu, Adaptive periodic event-triggered consensus for multi-agent systems subject to input saturation, Int. J. Control, 89 (2016), 653-667.  doi: 10.1080/00207179.2015.1088967.  Google Scholar

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