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July  2022, 15(7): 1733-1748. doi: 10.3934/dcdss.2021169

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 Published  July 2022 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 and Continuous Dynamical Systems - S, 2022, 15 (7) : 1733-1748. 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. 

[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.

[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]

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.

[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.

[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.

[7]

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

[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.

[9]

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

[10]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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. 

[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.

[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]

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.

[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.

[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.

[7]

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

[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.

[9]

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

[10]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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)
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