doi: 10.3934/dcdss.2021169
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Collision-avoidance and flocking in the Cucker–Smale-type model with a discontinuous controller

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

* Corresponding author: Xiao Wang

Received  September 2021 Revised  November 2021 Early access December 2021

Fund Project: The third author is supported by National Natural Science Foundation of China (11671011)

The collision-avoidance and flocking of the Cucker–Smale-type model with a discontinuous controller are studied. The controller considered in this paper provides a force between agents that switches between the attractive force and the repulsive force according to the movement tendency between agents. The results of collision-avoidance are closely related to the weight function $ f(r) = (r-d_0)^{-\theta } $. For $ \theta \ge 1 $, collision will not appear in the system if agents' initial positions are different. For the case $ \theta \in [0,1) $ that not considered in previous work, the limits of initial configurations to guarantee collision-avoidance are given. Moreover, on the basis of collision-avoidance, we point out the impacts of $ \psi (r) = (1+r^2)^{-\beta } $ and $ f(r) $ on the flocking behaviour and give the decay rate of relative velocity. We also estimate the lower and upper bound of distance between agents. Finally, for the special case that agents moving on the 1-D space, we give sufficient conditions for the finite-time flocking.

Citation: Jianfei Cheng, Xiao Wang, Yicheng Liu. Collision-avoidance and flocking in the Cucker–Smale-type model with a discontinuous controller. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021169
References:
[1]

C. Anirban, Distributions of money in model markets of economy, International Journal of Modern Physics C, 13 (2002), 1315-1321.   Google Scholar

[2]

A. AttanasiA. CavagnaL. D. CastelloI. GiardinaT. S. GrigeraA. JelićS. Melillo1L. ParisiO. PohlE. Shen and M. Viale, Information transfer and behavioural inertia in starling flocks, Nature Physics, 10 (2014), 691-696.  doi: 10.1038/nphys3035.  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]

A. CavagnaA. CimarelliI. GiardinaG. ParisiR. SantagatiF. Stefanini and M. Viale, Scale-free correlations in starling flocks, Proceedings of the National Academy of Sciences, 107 (2010), 11865-11870.  doi: 10.1073/pnas.1005766107.  Google Scholar

[5]

J. ChengZ. 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

[6]

J. ChengL. RuX. Wang and Y. Liu, Collision-avoidance, aggregation and velocity-matching in a Cucker–Smale-type model, Appl. Math. Lett., 123 (2022), 107611.  doi: 10.1016/j.aml.2021.107611.  Google Scholar

[7]

F. Cucker and J. Dong, Avoiding collisions in flocks, IEEE Trans. Automat. Control, 55 (2010), 1238-1243.  doi: 10.1109/TAC.2010.2042355.  Google Scholar

[8]

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.  Google Scholar

[9]

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

[10]

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

[11]

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

[12]

S.-Y. HaK.-K. Kim and K. Lee, A mathematical model for multi-name credit based on community flocking, Quant. Finance, 15 (2015), 841-851.  doi: 10.1080/14697688.2012.744085.  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]

J. Haskovec, Flocking dynamics and mean-field limit in the Cucker–Smale-type model with topological interactions, Physica D, 261 (2013), 42-51.  doi: 10.1016/j.physd.2013.06.006.  Google Scholar

[15]

C. Jin, Flocking of the Motsch–Tadmor model with a cut-off interaction function, J. Stat. Phys., 171 (2018), 345-360.  doi: 10.1007/s10955-018-2006-0.  Google Scholar

[16]

J. KeJ. W. MinettC.-P. Au and W. S.-Y. Wang, Self-organization and selection in the emergence of vocabulary, Complexity, 7 (2002), 41-54.  doi: 10.1002/cplx.10030.  Google Scholar

[17]

X. LiD. Peng and J. Cao, Lyapunov stability for impulsive systems via event-triggered impulsive control, IEEE Trans. Automat. Control, 65 (2020), 4908-4913.  doi: 10.1109/TAC.2020.2964558.  Google Scholar

[18]

X. LiX. Yang and J. Cao, Event-triggered impulsive control for nonlinear delay systems, Automatic, 117 (2020), 108981.  doi: 10.1016/j.automatica.2020.108981.  Google Scholar

[19]

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

[20]

J. ParkH. 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

[21]

L. PereaG. Gómez and P. Elosegui, Extension of the Cucker–Smale control law to space flight formations, Journal of Guidance, Control and Dynamics, 32 (2009), 526-536.  doi: 10.2514/1.36269.  Google Scholar

[22]

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

[23]

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

[24]

C. W. Reynolds, Flocks, herds and schools: A distributed behavioral model, Seminal Graphics: Pioneering Efforts That Shaped the Field, 21 (1998), 273-282.  doi: 10.1145/280811.281008.  Google Scholar

[25]

L. RuX. LiuY. Liu and X. Wang, Flocking of Cucker–Smale model with unit speed on general digraphs, Proc. Amer. Math. Soc., 149 (2021), 4397-4409.  doi: 10.1090/proc/15594.  Google Scholar

[26]

L. RuY. Liu and X. Wang, New conditions to avoid collision in the discrete Cucker–Smale model with singular interactions, Appl. Math. Lett., 114 (2021), 106906.  doi: 10.1016/j.aml.2020.106906.  Google Scholar

[27]

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.  Google Scholar

[28]

J. Wu and Y. Liu, Consensus and swarming behaviors for a proportional-derivative system with a cut-off interaction, At-Automatisierungstechnik, 69 (2021), 472-484.  doi: 10.1515/auto-2020-0068.  Google Scholar

[29]

X. Yin, Z. Gao, Z. Chen and Y. Fu, Non-existence of the asymptotic flocking in the Cucker–Smale model with short range communication weights, IEEE Transactions on Automatic Control, (2021), 1–1. doi: 10.1109/TAC.2021.3063951.  Google Scholar

[30]

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

show all references

References:
[1]

C. Anirban, Distributions of money in model markets of economy, International Journal of Modern Physics C, 13 (2002), 1315-1321.   Google Scholar

[2]

A. AttanasiA. CavagnaL. D. CastelloI. GiardinaT. S. GrigeraA. JelićS. Melillo1L. ParisiO. PohlE. Shen and M. Viale, Information transfer and behavioural inertia in starling flocks, Nature Physics, 10 (2014), 691-696.  doi: 10.1038/nphys3035.  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]

A. CavagnaA. CimarelliI. GiardinaG. ParisiR. SantagatiF. Stefanini and M. Viale, Scale-free correlations in starling flocks, Proceedings of the National Academy of Sciences, 107 (2010), 11865-11870.  doi: 10.1073/pnas.1005766107.  Google Scholar

[5]

J. ChengZ. 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

[6]

J. ChengL. RuX. Wang and Y. Liu, Collision-avoidance, aggregation and velocity-matching in a Cucker–Smale-type model, Appl. Math. Lett., 123 (2022), 107611.  doi: 10.1016/j.aml.2021.107611.  Google Scholar

[7]

F. Cucker and J. Dong, Avoiding collisions in flocks, IEEE Trans. Automat. Control, 55 (2010), 1238-1243.  doi: 10.1109/TAC.2010.2042355.  Google Scholar

[8]

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.  Google Scholar

[9]

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

[10]

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

[11]

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

[12]

S.-Y. HaK.-K. Kim and K. Lee, A mathematical model for multi-name credit based on community flocking, Quant. Finance, 15 (2015), 841-851.  doi: 10.1080/14697688.2012.744085.  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]

J. Haskovec, Flocking dynamics and mean-field limit in the Cucker–Smale-type model with topological interactions, Physica D, 261 (2013), 42-51.  doi: 10.1016/j.physd.2013.06.006.  Google Scholar

[15]

C. Jin, Flocking of the Motsch–Tadmor model with a cut-off interaction function, J. Stat. Phys., 171 (2018), 345-360.  doi: 10.1007/s10955-018-2006-0.  Google Scholar

[16]

J. KeJ. W. MinettC.-P. Au and W. S.-Y. Wang, Self-organization and selection in the emergence of vocabulary, Complexity, 7 (2002), 41-54.  doi: 10.1002/cplx.10030.  Google Scholar

[17]

X. LiD. Peng and J. Cao, Lyapunov stability for impulsive systems via event-triggered impulsive control, IEEE Trans. Automat. Control, 65 (2020), 4908-4913.  doi: 10.1109/TAC.2020.2964558.  Google Scholar

[18]

X. LiX. Yang and J. Cao, Event-triggered impulsive control for nonlinear delay systems, Automatic, 117 (2020), 108981.  doi: 10.1016/j.automatica.2020.108981.  Google Scholar

[19]

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

[20]

J. ParkH. 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

[21]

L. PereaG. Gómez and P. Elosegui, Extension of the Cucker–Smale control law to space flight formations, Journal of Guidance, Control and Dynamics, 32 (2009), 526-536.  doi: 10.2514/1.36269.  Google Scholar

[22]

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

[23]

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

[24]

C. W. Reynolds, Flocks, herds and schools: A distributed behavioral model, Seminal Graphics: Pioneering Efforts That Shaped the Field, 21 (1998), 273-282.  doi: 10.1145/280811.281008.  Google Scholar

[25]

L. RuX. LiuY. Liu and X. Wang, Flocking of Cucker–Smale model with unit speed on general digraphs, Proc. Amer. Math. Soc., 149 (2021), 4397-4409.  doi: 10.1090/proc/15594.  Google Scholar

[26]

L. RuY. Liu and X. Wang, New conditions to avoid collision in the discrete Cucker–Smale model with singular interactions, Appl. Math. Lett., 114 (2021), 106906.  doi: 10.1016/j.aml.2020.106906.  Google Scholar

[27]

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.  Google Scholar

[28]

J. Wu and Y. Liu, Consensus and swarming behaviors for a proportional-derivative system with a cut-off interaction, At-Automatisierungstechnik, 69 (2021), 472-484.  doi: 10.1515/auto-2020-0068.  Google Scholar

[29]

X. Yin, Z. Gao, Z. Chen and Y. Fu, Non-existence of the asymptotic flocking in the Cucker–Smale model with short range communication weights, IEEE Transactions on Automatic Control, (2021), 1–1. doi: 10.1109/TAC.2021.3063951.  Google Scholar

[30]

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

Figure 1.  Flocking: the evolution of agents' position (left) and velocity (right)
Figure 2.  Flocking: the lower and upper bound of the distance between agents (left) and the logarithm of the maximum velocity difference (right)
Figure 3.  Swarming: the evolution of agents' position (left) and velocity (right)
Figure 4.  Swarming: the lower and upper bound of the distance between agents (left) and the maximum velocity difference (right)
Figure 5.  Finite-time flocking: the evolution of agents' position (left) and velocity (right)
Figure 6.  Finite-time flocking: the lower and upper bound of the distance between agents (left) and the logarithm of the maximum velocity difference (right)
Table 1.  Initial configurations of system (3)
$ i $ 1 2 3 4
$ (x_i(0)) $ (1, 1) (2, 2) (3, 3) (4, 4)
$ (v_i(0)) $ (-3.25, -1.25) (0.75, 3.75) (-1.25, 0.75) (3.75, -3.25)
$ i $ 1 2 3 4
$ (x_i(0)) $ (1, 1) (2, 2) (3, 3) (4, 4)
$ (v_i(0)) $ (-3.25, -1.25) (0.75, 3.75) (-1.25, 0.75) (3.75, -3.25)
Table 2.  Initial configurations of system (3)
$ i $ 1 2 3 4
$ (x_i(0)) $ (1, 1) (2, 2) (3, 3) (4, 4)
$ (v_i(0)) $ (-0.325, -0.125) (0.075, 0.375) (-0.125, 0.075) (0.375, -0.325)
$ i $ 1 2 3 4
$ (x_i(0)) $ (1, 1) (2, 2) (3, 3) (4, 4)
$ (v_i(0)) $ (-0.325, -0.125) (0.075, 0.375) (-0.125, 0.075) (0.375, -0.325)
Table 3.  Initial configurations of system (3) on the real line
$ i $ 1 2 3 4
$ (x_i(0),v_i(0)) $ (1, -3.25) (2, 0.75) (3, -1.25) (4, 3.75)
$ i $ 1 2 3 4
$ (x_i(0),v_i(0)) $ (1, -3.25) (2, 0.75) (3, -1.25) (4, 3.75)
[1]

Seung-Yeal Ha, Dohyun Kim, Jaeseung Lee, Se Eun Noh. Emergent dynamics of an orientation flocking model for multi-agent system. Discrete & Continuous Dynamical Systems, 2020, 40 (4) : 2037-2060. doi: 10.3934/dcds.2020105

[2]

Rui Li, Yingjing Shi. Finite-time optimal consensus control for second-order multi-agent systems. Journal of Industrial & Management Optimization, 2014, 10 (3) : 929-943. doi: 10.3934/jimo.2014.10.929

[3]

Maoli Chen, Xiao Wang, Yicheng Liu. Collision-free flocking for a time-delay system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1223-1241. doi: 10.3934/dcdsb.2020251

[4]

Rong Yang, Li Chen. Mean-field limit for a collision-avoiding flocking system and the time-asymptotic flocking dynamics for the kinetic equation. Kinetic & Related Models, 2014, 7 (2) : 381-400. doi: 10.3934/krm.2014.7.381

[5]

Giulia Cavagnari, Antonio Marigonda, Benedetto Piccoli. Optimal synchronization problem for a multi-agent system. Networks & Heterogeneous Media, 2017, 12 (2) : 277-295. doi: 10.3934/nhm.2017012

[6]

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

[7]

Richard Carney, Monique Chyba, Chris Gray, George Wilkens, Corey Shanbrom. Multi-agent systems for quadcopters. Journal of Geometric Mechanics, 2021  doi: 10.3934/jgm.2021005

[8]

Xi Zhu, Meixia Li, Chunfa Li. Consensus in discrete-time multi-agent systems with uncertain topologies and random delays governed by a Markov chain. Discrete & Continuous Dynamical Systems - B, 2020, 25 (12) : 4535-4551. doi: 10.3934/dcdsb.2020111

[9]

Zhongkui Li, Zhisheng Duan, Guanrong Chen. Consensus of discrete-time linear multi-agent systems with observer-type protocols. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 489-505. doi: 10.3934/dcdsb.2011.16.489

[10]

Hong Man, Yibin Yu, Yuebang He, Hui Huang. Design of one type of linear network prediction controller for multi-agent system. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 727-734. doi: 10.3934/dcdss.2019047

[11]

Zhongqiang Wu, Zongkui Xie. A multi-objective lion swarm optimization based on multi-agent. Journal of Industrial & Management Optimization, 2022  doi: 10.3934/jimo.2022001

[12]

Arno Berger. On finite-time hyperbolicity. Communications on Pure & Applied Analysis, 2011, 10 (3) : 963-981. doi: 10.3934/cpaa.2011.10.963

[13]

Nadia Loy, Andrea Tosin. Boltzmann-type equations for multi-agent systems with label switching. Kinetic & Related Models, 2021, 14 (5) : 867-894. doi: 10.3934/krm.2021027

[14]

Arno Berger, Doan Thai Son, Stefan Siegmund. Nonautonomous finite-time dynamics. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 463-492. doi: 10.3934/dcdsb.2008.9.463

[15]

Le Li, Lihong Huang, Jianhong Wu. Cascade flocking with free-will. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 497-522. doi: 10.3934/dcdsb.2016.21.497

[16]

Le Li, Lihong Huang, Jianhong Wu. Flocking and invariance of velocity angles. Mathematical Biosciences & Engineering, 2016, 13 (2) : 369-380. doi: 10.3934/mbe.2015007

[17]

Brendan Pass. Multi-marginal optimal transport and multi-agent matching problems: Uniqueness and structure of solutions. Discrete & Continuous Dynamical Systems, 2014, 34 (4) : 1623-1639. doi: 10.3934/dcds.2014.34.1623

[18]

Matthias Gerdts, René Henrion, Dietmar Hömberg, Chantal Landry. Path planning and collision avoidance for robots. Numerical Algebra, Control & Optimization, 2012, 2 (3) : 437-463. doi: 10.3934/naco.2012.2.437

[19]

Claudia Totzeck. An anisotropic interaction model with collision avoidance. Kinetic & Related Models, 2020, 13 (6) : 1219-1242. doi: 10.3934/krm.2020044

[20]

Juanjuan Huang, Yan Zhou, Xuerong Shi, Zuolei Wang. A single finite-time synchronization scheme of time-delay chaotic system with external periodic disturbance. Mathematical Foundations of Computing, 2019, 2 (4) : 333-346. doi: 10.3934/mfc.2019021

2020 Impact Factor: 2.425

Article outline

Figures and Tables

[Back to Top]