doi: 10.3934/dcds.2021195
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Cucker-Smale model with time delay

IMAS (UBA-CONICET) and Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, 1428 Buenos Aires, Argentina

Received  May 2021 Revised  October 2021 Early access December 2021

We study the flocking model for continuous time introduced by Cucker and Smale adding a positive time delay $ \tau $. The goal of this article is to prove that the same unconditional flocking result for the non-delayed case is valid in the delayed case. A novelty is that we do not need to impose any restriction on the size of $ \tau $. Furthermore, when the unconditional flocking occurs, velocities converge exponentially fast to a common one.

Citation: Mauro Rodriguez Cartabia. Cucker-Smale model with time delay. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021195
References:
[1]

S. AhnH.-O. BaeS.-Y. HaY. Kim and H. Lim, Application of flocking mechanism to the modeling of stochastic volatility, Math. Models Methods Appl. Sci., 23 (2013), 1603-1628.  doi: 10.1142/S0218202513500176.  Google Scholar

[2]

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

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Y.-P. ChoiS.-Y. Ha and Z. Li, Emergent dynamics of the cucker–smale flocking model and its variants, Active Particles, 1 (2017), 299-331.   Google Scholar

[4]

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

[6]

Y.-L. Chuang, Y. R. Huang, M. R. D'Orsogna and A. L. Bertozzi, Multi-vehicle flocking: Scalability of cooperative control algorithms using pairwise potentials, Proceedings 2007 IEEE International Conference on Robotics and Automation, (2007), 2292–2299. doi: 10.1109/ROBOT.2007.363661.  Google Scholar

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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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F. Cucker and S. Smale, On the mathematics of emergence, Jpn. J. Math., 2 (2007), 197-227.  doi: 10.1007/s11537-007-0647-x.  Google Scholar

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F. CuckerS. Smale and D.-X. Zhou, Modeling language evolution, Found. Comput. Math., 4 (2004), 315-343.  doi: 10.1007/s10208-003-0101-2.  Google Scholar

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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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R. ErbanJ. Haskovec 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

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S.-Y. Ha and D. Levy, Particle, kinetic and fluid models for phototaxis, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 77-108.  doi: 10.3934/dcdsb.2009.12.77.  Google Scholar

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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

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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

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J. Haskovec, Direct proof of unconditional asymptotic consensus in the Hegselmann–Krause model with transmission-type delay, Bull. Lond. Math. Soc., 53 (2021), 1312-1323.  doi: 10.1112/blms.12497.  Google Scholar

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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, Well posedness and asymptotic consensus in the Hegselmann-Krause model with finite speed of information propagation, Proc. Amer. Math. Soc., 149 (2021), 3425-3437.  doi: 10.1090/proc/15522.  Google Scholar

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R. Hegselmann, U. Krause and et al., Opinion dynamics and bounded confidence models, analysis, and simulation, Journal of Artificial Societies and Social Simulation, 5 (2002). Google Scholar

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S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys., 144 (2011), 923-947.  doi: 10.1007/s10955-011-0285-9.  Google Scholar

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S.-I. Niculescu, Delay Effects on Stability: A Robust Control Approach, Lecture Notes in Control and Information Sciences, 269. Springer-Verlag London, Ltd., London, 2001.  Google Scholar

[23]

R. Olfati-Saber, Flocking for multi-agent dynamic systems: Algorithms and theory, IEEE Trans. Automat. Control, 51 (2006), 401-420.  doi: 10.1109/TAC.2005.864190.  Google Scholar

[24]

J. ParkH. J. Kim and S.-Y. Ha, Cucker-Smale flocking with inter-particle bonding forces, IEEE Trans. Automat. Control, 55 (2010), 2617-2623.  doi: 10.1109/TAC.2010.2061070.  Google Scholar

[25]

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

[26]

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

[27]

J. P. PinascoM. Rodriguez Cartabia and N. Saintier, Interacting particles systems with delay and random delay differential equations, Nonlinear Anal., 214 (2022), 112524.  doi: 10.1016/j.na.2021.112524.  Google Scholar

[28]

C. W. Reynolds, Flocks, herds and schools: A distributed behavioral model, Seminal Graphics: Pioneering Efforts That Shaped the Field, (1998), 273–282. doi: 10.1145/280811.281008.  Google Scholar

[29]

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

[30] D. J. Sumpter, Collective Animal Behavior, Princeton University Press, 2010.   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. Zafeiris, Collective motion, Physics Reports, 517 (2012), 71-140.  doi: 10.1016/j.physrep.2012.03.004.  Google Scholar

show all references

References:
[1]

S. AhnH.-O. BaeS.-Y. HaY. Kim and H. Lim, Application of flocking mechanism to the modeling of stochastic volatility, Math. Models Methods Appl. Sci., 23 (2013), 1603-1628.  doi: 10.1142/S0218202513500176.  Google Scholar

[2]

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

[3]

Y.-P. ChoiS.-Y. Ha and Z. Li, Emergent dynamics of the cucker–smale flocking model and its variants, Active Particles, 1 (2017), 299-331.   Google Scholar

[4]

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

[5]

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

[6]

Y.-L. Chuang, Y. R. Huang, M. R. D'Orsogna and A. L. Bertozzi, Multi-vehicle flocking: Scalability of cooperative control algorithms using pairwise potentials, Proceedings 2007 IEEE International Conference on Robotics and Automation, (2007), 2292–2299. doi: 10.1109/ROBOT.2007.363661.  Google Scholar

[7]

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

[8]

F. Cucker and S. Smale, On the mathematics of emergence, Jpn. J. Math., 2 (2007), 197-227.  doi: 10.1007/s11537-007-0647-x.  Google Scholar

[9]

F. CuckerS. Smale and D.-X. Zhou, Modeling language evolution, Found. Comput. Math., 4 (2004), 315-343.  doi: 10.1007/s10208-003-0101-2.  Google Scholar

[10]

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

[11]

R. ErbanJ. Haskovec 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

[12]

S.-Y. Ha and D. Levy, Particle, kinetic and fluid models for phototaxis, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 77-108.  doi: 10.3934/dcdsb.2009.12.77.  Google Scholar

[13]

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

[14]

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

[15]

J. Haskovec, Direct proof of unconditional asymptotic consensus in the Hegselmann–Krause model with transmission-type delay, Bull. Lond. Math. Soc., 53 (2021), 1312-1323.  doi: 10.1112/blms.12497.  Google Scholar

[16]

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

[17]

J. Haskovec, Well posedness and asymptotic consensus in the Hegselmann-Krause model with finite speed of information propagation, Proc. Amer. Math. Soc., 149 (2021), 3425-3437.  doi: 10.1090/proc/15522.  Google Scholar

[18]

R. Hegselmann, U. Krause and et al., Opinion dynamics and bounded confidence models, analysis, and simulation, Journal of Artificial Societies and Social Simulation, 5 (2002). Google Scholar

[19]

V. Kolmanovskii and A. Myshkis, Introduction to the Theory and Applications of Functional Differential Equations, Mathematics and its Applications, 463. Kluwer Academic Publishers, Dordrecht, 1999. doi: 10.1007/978-94-017-1965-0.  Google Scholar

[20]

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

[21]

S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys., 144 (2011), 923-947.  doi: 10.1007/s10955-011-0285-9.  Google Scholar

[22]

S.-I. Niculescu, Delay Effects on Stability: A Robust Control Approach, Lecture Notes in Control and Information Sciences, 269. Springer-Verlag London, Ltd., London, 2001.  Google Scholar

[23]

R. Olfati-Saber, Flocking for multi-agent dynamic systems: Algorithms and theory, IEEE Trans. Automat. Control, 51 (2006), 401-420.  doi: 10.1109/TAC.2005.864190.  Google Scholar

[24]

J. ParkH. J. Kim and S.-Y. Ha, Cucker-Smale flocking with inter-particle bonding forces, IEEE Trans. Automat. Control, 55 (2010), 2617-2623.  doi: 10.1109/TAC.2010.2061070.  Google Scholar

[25]

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

[26]

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

[27]

J. P. PinascoM. Rodriguez Cartabia and N. Saintier, Interacting particles systems with delay and random delay differential equations, Nonlinear Anal., 214 (2022), 112524.  doi: 10.1016/j.na.2021.112524.  Google Scholar

[28]

C. W. Reynolds, Flocks, herds and schools: A distributed behavioral model, Seminal Graphics: Pioneering Efforts That Shaped the Field, (1998), 273–282. doi: 10.1145/280811.281008.  Google Scholar

[29]

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

[30] D. J. Sumpter, Collective Animal Behavior, Princeton University Press, 2010.   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. Zafeiris, Collective motion, Physics Reports, 517 (2012), 71-140.  doi: 10.1016/j.physrep.2012.03.004.  Google Scholar

Figure 3.  We plot two simulations using system 33. The imagines at the top represent the case of $ 4 $ particles with $ K = 5 $. The imagines at the bottom represent the case of $ 20 $ particles with $ K = 25 $. The black solid lines show the center of positions $ x_{\bf{c}} $ (left imagines) and the center of velocities $ v_{\bf{c}} $ (right imagines). In both cases we set $ \Delta t = 1/1000 $, $ \tau = 1 $ and the initial conditions are constant in the velocities, satisfying $ (d/dt) x^{(a)}_0 = v^{(a)}_0 $ for each $ a\in A $
Figure 1.  Numerical simulation of Example 4.1. We use system 33 and set $ \tau = 1 $, $ \Delta t = 1/1000 $, $ N = 2 $ and $ K = 1 $
Figure 2.  Numerical simulation of Example 4.2 with $ \tau = 1 $ and $ \varepsilon = 0.2 $. We use system 33 setting $ \Delta t = 1/1000 $, $ N = 2 $ and $ K = 1 $
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