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Upper and weak-lower semicontinuity of pullback attractors to impulsive evolution processes
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 |
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.
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. Carrillo, Y.-P. Choi, P. 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. Carrillo, M. Fornasier, J. 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. Cho, S.-Y. Ha, F. Huang, C. 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. Dong, S. -Y. Ha, D. 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. Dong, S.-Y. Ha, D. 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. 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. |
[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. 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. |
[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. Ha, J. Kim, J. 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. Google Scholar |
[22] |
L. Li, L. 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. Li, Y. 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. Liu, X. Wang, Y. 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. Wang, L. 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. Carrillo, Y.-P. Choi, P. 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. Carrillo, M. Fornasier, J. 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. Cho, S.-Y. Ha, F. Huang, C. 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. Dong, S. -Y. Ha, D. 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. Dong, S.-Y. Ha, D. 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. 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. |
[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. 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. |
[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. Ha, J. Kim, J. 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. Google Scholar |
[22] |
L. Li, L. 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. Li, Y. 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. Liu, X. Wang, Y. 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. Wang, L. 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. |





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