July  2021, 26(7): 3693-3716. doi: 10.3934/dcdsb.2020253

Flocking and line-shaped spatial configuration to delayed Cucker-Smale models

Department of Mathematics, College of Liberal Arts and Sciences, National University of Defense Technology, Changsha, Hunan 410073, China

* Corresponding author: Yicheng Liu

Received  July 2019 Revised  February 2020 Published  July 2021 Early access  August 2020

Fund Project: This work was supported by the NSFC (11701267;11671011;11801562) and Hunan Natural Science Excellent Youth Fund (2020JJ3029)

As we known, it is popular for a designed system to achieve a prescribed performance, which have remarkable capability to regulate the flow of information from distinct and independent components. Also, it is not well understand, in both theories and applications, how self propelled agents use only limited environmental information and simple rules to organize into an ordered motion. In this paper, we focus on analysis the flocking behaviour and the line-shaped pattern for collective motion involving time delay effects. Firstly, we work on a delayed Cucker-Smale-type system involving a general communication weight and a constant delay $ \tau>0 $ for modelling collective motion. In a result, by constructing a new Lyapunov functional approach, combining with two delayed differential inequalities established by $ L^2 $-analysis, we show that the flocking occurs for the general communication weight when $ \tau $ is sufficiently small. Furthermore, to achieve the prescribed performance, we introduce the line-shaped inner force term into the delayed collective system, and analytically show that there is a flocking pattern with an asymptotic flocking velocity and line-shaped pattern. All results are novel and can be illustrated by numerical simulations using some concrete influence functions. Also, our results significantly extend some known theorems in the literature.

Citation: Zhisu Liu, Yicheng Liu, Xiang Li. Flocking and line-shaped spatial configuration to delayed Cucker-Smale models. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3693-3716. doi: 10.3934/dcdsb.2020253
References:
[1]

N. Bellomo, P. Degond and E. Tadmor, eds., Active Particles. Vol. 1, Advances in Theory, Models, and Applications, Birkhäuser/Springer, Cham, 2017.

[2]

N. Bellomo, P. Degond and E. Tadmor, eds., Active Particles. Vol. 2, Advances in Theory, Models, and Applications, Birkhäuser/Springer, Cham, 2019. doi: 10.1007/978-3-030-20297-2.

[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.

[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.

[5]

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.

[6]

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.

[7]

Y.-P. Choi, S. Ha and Z. Li, Emergent dynamics of the Cucker-Smale flocking model and its variants, In Active Particles, Vol. 1, Advances in Theory, Models, and Applications, Birkhäuser/Springer, Cham, (2017), 299–331.

[8]

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.

[9]

Y.-P. Choi and C. Pignotti, Emergent behavior of Cucker-Smale models with normalized weights and distributed time delays, Netw. Heterog. Media, 14 (2019), 789-804.  doi: 10.3934/nhm.2019032.

[10]

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

[11]

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

[12]

F. Cucker and J. Dong, A general collision-avoiding flocking framework, IEEE Trans. Automat. Control, 56 (2011), 1124-1129.  doi: 10.1109/TAC.2011.2107113.

[13]

F. Cucker and J.-G. Dong, On the critical exponent for flocks under hierarchical leadership, Math. Models Methods Appl. Sci., 19 (2009), 1391-1404.  doi: 10.1142/S0218202509003851.

[14]

J. DongS. -Y. HaD. Kim and J. Kim, Time-delay effect on the flocking in an ensemble of thermomechanical Cucker-Smale particles, J. Differential Equations, 266 (2019), 2373-2407.  doi: 10.1016/j.jde.2018.08.034.

[15]

J. DongS.-Y. HaD. Kim and J. 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.

[16]

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.

[17]

S.-Y. Ha and J. Liu, A simple proof of Cucker-Smale flocking dynamics and mean-field limit, Commun. Math. Sci., 7 (2009), 297-325.  doi: 10.4310/CMS.2009.v7.n2.a2.

[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.

[19]

S.-Y. Ha and M. Slemrod, Flocking dynamics of a singularly perturbed oscillator chain and the Cucker-Smale system, J. Dynam. Differential Equations, 22 (2010), 325-330.  doi: 10.1007/s10884-009-9142-9.

[20]

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.

[21]

J. Haskovec and I. Markou, Delay Cucker-Smale model with and without noise revised, preprint, arXiv: 1810.01084v2.

[22]

L. LiL. Huang and J. Wu, Cascade flocking with free-will, Discrete Contin. Dyn. Syst. Ser.B, 21 (2016), 497-522.  doi: 10.3934/dcdsb.2016.21.497.

[23]

L. Li, W. Wang, L. Huang and J. Wu, Some weak flocking models and its application to target tracking, J. Math. Anal. Appl., 480 (2019), 123404. doi: 10.1016/j.jmaa.2019.123404.

[24]

X. LiY. Liu and J. Wu, Flocking and pattern motion in a modified Cucker-Smale model, Bull. Korean. Math. Soc., 53 (2016), 1327-1339.  doi: 10.4134/BKMS.b150629.

[25]

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.

[26]

Z. Li, Effectual leadership in flocks with hierarchy and individual preference, Discrete Contin. Dyn. Syst., 34 (2014), 3683-3702.  doi: 10.3934/dcds.2014.34.3683.

[27]

H. LiuX. WangY. Liu and X. Li, On non-collision flocking and line-shaped spatial configuration for a modified singular Cucker-Smale model, Commun. Nonlinear. Sci. Numer. Simul., 75 (2019), 280-301.  doi: 10.1016/j.cnsns.2019.04.006.

[28]

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.

[29]

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.

[30]

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

[31]

C. Pignotti and E. Trelat, 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.

[32]

C. Pignotti and I. 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.

[33]

C. Pignotti and I. 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. 32, Springer, Cham, 2019.

[34]

L. Ru and X. Xue, Multi-cluster flocking behavior of the hierarchical Cucker-Smale model, J. Franklin Inst., 354 (2017), 2371-2392.  doi: 10.1016/j.jfranklin.2016.12.018.

[35]

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

[36]

T. Vicsek and A. Zafeiris, Collective motion, Physics Reports, 517 (2012), 71-140.  doi: 10.1016/j.physrep.2012.03.004.

[37]

X. WangL. Wang and J. Wu, Impacts of time delay on flocking dynamics of a two-agent flock model, Commun. Nonlinear Sci. Numer. Simul., 70 (2019), 80-88.  doi: 10.1016/j.cnsns.2018.10.017.

show all references

References:
[1]

N. Bellomo, P. Degond and E. Tadmor, eds., Active Particles. Vol. 1, Advances in Theory, Models, and Applications, Birkhäuser/Springer, Cham, 2017.

[2]

N. Bellomo, P. Degond and E. Tadmor, eds., Active Particles. Vol. 2, Advances in Theory, Models, and Applications, Birkhäuser/Springer, Cham, 2019. doi: 10.1007/978-3-030-20297-2.

[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.

[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.

[5]

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.

[6]

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.

[7]

Y.-P. Choi, S. Ha and Z. Li, Emergent dynamics of the Cucker-Smale flocking model and its variants, In Active Particles, Vol. 1, Advances in Theory, Models, and Applications, Birkhäuser/Springer, Cham, (2017), 299–331.

[8]

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.

[9]

Y.-P. Choi and C. Pignotti, Emergent behavior of Cucker-Smale models with normalized weights and distributed time delays, Netw. Heterog. Media, 14 (2019), 789-804.  doi: 10.3934/nhm.2019032.

[10]

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

[11]

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

[12]

F. Cucker and J. Dong, A general collision-avoiding flocking framework, IEEE Trans. Automat. Control, 56 (2011), 1124-1129.  doi: 10.1109/TAC.2011.2107113.

[13]

F. Cucker and J.-G. Dong, On the critical exponent for flocks under hierarchical leadership, Math. Models Methods Appl. Sci., 19 (2009), 1391-1404.  doi: 10.1142/S0218202509003851.

[14]

J. DongS. -Y. HaD. Kim and J. Kim, Time-delay effect on the flocking in an ensemble of thermomechanical Cucker-Smale particles, J. Differential Equations, 266 (2019), 2373-2407.  doi: 10.1016/j.jde.2018.08.034.

[15]

J. DongS.-Y. HaD. Kim and J. 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.

[16]

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.

[17]

S.-Y. Ha and J. Liu, A simple proof of Cucker-Smale flocking dynamics and mean-field limit, Commun. Math. Sci., 7 (2009), 297-325.  doi: 10.4310/CMS.2009.v7.n2.a2.

[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.

[19]

S.-Y. Ha and M. Slemrod, Flocking dynamics of a singularly perturbed oscillator chain and the Cucker-Smale system, J. Dynam. Differential Equations, 22 (2010), 325-330.  doi: 10.1007/s10884-009-9142-9.

[20]

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.

[21]

J. Haskovec and I. Markou, Delay Cucker-Smale model with and without noise revised, preprint, arXiv: 1810.01084v2.

[22]

L. LiL. Huang and J. Wu, Cascade flocking with free-will, Discrete Contin. Dyn. Syst. Ser.B, 21 (2016), 497-522.  doi: 10.3934/dcdsb.2016.21.497.

[23]

L. Li, W. Wang, L. Huang and J. Wu, Some weak flocking models and its application to target tracking, J. Math. Anal. Appl., 480 (2019), 123404. doi: 10.1016/j.jmaa.2019.123404.

[24]

X. LiY. Liu and J. Wu, Flocking and pattern motion in a modified Cucker-Smale model, Bull. Korean. Math. Soc., 53 (2016), 1327-1339.  doi: 10.4134/BKMS.b150629.

[25]

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.

[26]

Z. Li, Effectual leadership in flocks with hierarchy and individual preference, Discrete Contin. Dyn. Syst., 34 (2014), 3683-3702.  doi: 10.3934/dcds.2014.34.3683.

[27]

H. LiuX. WangY. Liu and X. Li, On non-collision flocking and line-shaped spatial configuration for a modified singular Cucker-Smale model, Commun. Nonlinear. Sci. Numer. Simul., 75 (2019), 280-301.  doi: 10.1016/j.cnsns.2019.04.006.

[28]

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.

[29]

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.

[30]

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

[31]

C. Pignotti and E. Trelat, 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.

[32]

C. Pignotti and I. 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.

[33]

C. Pignotti and I. 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. 32, Springer, Cham, 2019.

[34]

L. Ru and X. Xue, Multi-cluster flocking behavior of the hierarchical Cucker-Smale model, J. Franklin Inst., 354 (2017), 2371-2392.  doi: 10.1016/j.jfranklin.2016.12.018.

[35]

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

[36]

T. Vicsek and A. Zafeiris, Collective motion, Physics Reports, 517 (2012), 71-140.  doi: 10.1016/j.physrep.2012.03.004.

[37]

X. WangL. Wang and J. Wu, Impacts of time delay on flocking dynamics of a two-agent flock model, Commun. Nonlinear Sci. Numer. Simul., 70 (2019), 80-88.  doi: 10.1016/j.cnsns.2018.10.017.

Figure 1.  Time-domain behaviors of the state variables $ x_{1}(t) $, $ x_{2}(t) $, $ v_{1}(t) $, $ v_{2}(t) $ in Example 4.1
Figure 2.  Time-domain behaviors of the state variables $ x_{1}(t) $, $ x_{2}(t) $, $ v_{1}(t) $, $ v_{2}(t) $ for the case $ \beta = \frac{1}{2} $ in Example 4.2
Figure 3.  Time-domain behaviors of the state variables $ x_{1}(t) $, $ x_{2}(t) $, $ v_{1}(t) $, $ v_{2}(t) $ for the case $ \beta = 1 $ in Example 4.2
Figure 4.  The first is the initial position of each particle in the system; the second is the population distribution after iteration $ 200 $ ($ 2s $). the third is the population distribution after iteration $ 2000 $ ($ 20s $). The value of each parameter is given as: $ N = 100, \lambda = 2, \tau = 0.2, \psi = \frac{1}{(1+r^2)^{1/3}} $, $ \gamma = 1 $
Figure 5.  The first is the initial position of each particle in the system; the second is the population distribution after iteration $ 2000 $ ($ 20s $); the third is the population distribution after iteration $ 5000 $ ($ 50s $). The value of each parameter is given as: $ N = 100, \lambda = 20, \tau = 0.2, \psi = \frac{1}{(1+r^2)^{1/2}} $, $ \gamma = 1 $
[1]

Mauro Rodriguez Cartabia. Cucker-Smale model with time delay. Discrete and Continuous Dynamical Systems, 2022, 42 (5) : 2409-2432. doi: 10.3934/dcds.2021195

[2]

Lining Ru, Xiaoping Xue. Flocking of Cucker-Smale model with intrinsic dynamics. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6817-6835. doi: 10.3934/dcdsb.2019168

[3]

Young-Pil Choi, Jan Haskovec. Cucker-Smale model with normalized communication weights and time delay. Kinetic and Related Models, 2017, 10 (4) : 1011-1033. doi: 10.3934/krm.2017040

[4]

Martial Agueh, Reinhard Illner, Ashlin Richardson. Analysis and simulations of a refined flocking and swarming model of Cucker-Smale type. Kinetic and Related Models, 2011, 4 (1) : 1-16. doi: 10.3934/krm.2011.4.1

[5]

Martin Friesen, Oleksandr Kutoviy. Stochastic Cucker-Smale flocking dynamics of jump-type. Kinetic and Related Models, 2020, 13 (2) : 211-247. doi: 10.3934/krm.2020008

[6]

Hyeong-Ohk Bae, Young-Pil Choi, Seung-Yeal Ha, Moon-Jin Kang. Asymptotic flocking dynamics of Cucker-Smale particles immersed in compressible fluids. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4419-4458. doi: 10.3934/dcds.2014.34.4419

[7]

Seung-Yeal Ha, Jinwook Jung, Peter Kuchling. Emergence of anomalous flocking in the fractional Cucker-Smale model. Discrete and Continuous Dynamical Systems, 2019, 39 (9) : 5465-5489. doi: 10.3934/dcds.2019223

[8]

Chun-Hsien Li, Suh-Yuh Yang. A new discrete Cucker-Smale flocking model under hierarchical leadership. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2587-2599. doi: 10.3934/dcdsb.2016062

[9]

Jiu-Gang Dong, Seung-Yeal Ha, Doheon Kim. Interplay of time-delay and velocity alignment in the Cucker-Smale model on a general digraph. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5569-5596. doi: 10.3934/dcdsb.2019072

[10]

Yu-Jhe Huang, Zhong-Fu Huang, Jonq Juang, Yu-Hao Liang. Flocking of non-identical Cucker-Smale models on general coupling network. Discrete and Continuous Dynamical Systems - B, 2021, 26 (2) : 1111-1127. doi: 10.3934/dcdsb.2020155

[11]

Young-Pil Choi, Samir Salem. Cucker-Smale flocking particles with multiplicative noises: Stochastic mean-field limit and phase transition. Kinetic and Related Models, 2019, 12 (3) : 573-592. doi: 10.3934/krm.2019023

[12]

Seung-Yeal Ha, Doheon Kim, Weiyuan Zou. Slow flocking dynamics of the Cucker-Smale ensemble with a chemotactic movement in a temperature field. Kinetic and Related Models, 2020, 13 (4) : 759-793. doi: 10.3934/krm.2020026

[13]

Jan Haskovec, Ioannis Markou. Asymptotic flocking in the Cucker-Smale model with reaction-type delays in the non-oscillatory regime. Kinetic and Related Models, 2020, 13 (4) : 795-813. doi: 10.3934/krm.2020027

[14]

Chiun-Chuan Chen, Seung-Yeal Ha, Xiongtao Zhang. The global well-posedness of the kinetic Cucker-Smale flocking model with chemotactic movements. Communications on Pure and Applied Analysis, 2018, 17 (2) : 505-538. doi: 10.3934/cpaa.2018028

[15]

Linglong Du, Xinyun Zhou. The stochastic delayed Cucker-Smale system in a harmonic potential field. Kinetic and Related Models, , () : -. doi: 10.3934/krm.2022022

[16]

Seung-Yeal Ha, Dongnam Ko, Yinglong Zhang, Xiongtao Zhang. Emergent dynamics in the interactions of Cucker-Smale ensembles. Kinetic and Related Models, 2017, 10 (3) : 689-723. doi: 10.3934/krm.2017028

[17]

Roberto Natalini, Thierry Paul. On the mean field limit for Cucker-Smale models. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2873-2889. doi: 10.3934/dcdsb.2021164

[18]

Agnieszka B. Malinowska, Tatiana Odzijewicz. Optimal control of the discrete-time fractional-order Cucker-Smale model. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 347-357. doi: 10.3934/dcdsb.2018023

[19]

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

[20]

Young-Pil Choi, Seung-Yeal Ha, Jeongho Kim. Propagation of regularity and finite-time collisions for the thermomechanical Cucker-Smale model with a singular communication. Networks and Heterogeneous Media, 2018, 13 (3) : 379-407. doi: 10.3934/nhm.2018017

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (253)
  • HTML views (295)
  • Cited by (0)

Other articles
by authors

[Back to Top]