Figure 1.
$ \beta $ = 0.5, $ \tau $ = 0.01, the relative position of any two particles in System (2)-(3) is always bounded, and the velocity will asymptotically converge, i.e., the emergence of flocking
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February
2021, 26(2): 1223-1241.
doi: 10.3934/dcdsb.2020251
Collision-free flocking for a time-delay system
College of Liberal Arts and Sciences, National University of Defense Technology, Changsha 410073, China |
The co-existence of collision avoidance and time-asymptotic flocking of multi-particle systems with measurement delay is considered. Based on Lyapunov stability theory and some auxiliary differential inequalities, a delay-related sufficient condition is established for this system to admit a time-asymptotic flocking and collision avoidance. The estimated range of the delay is given, which may affect the flocking performance of the system. An analytical expression was proposed to quantitatively analyze the upper bound of this delay. Under the flocking conditions, the exponential decay of the relative velocity of any two particles in the system is characterized. Particularly, the collision-free flocking conditions are also given for the case without delay. This work verifies that both collision avoidance and flocking behaviors can be achieved simultaneously in a delay system.
Mathematics Subject Classification:
Primary: 34K20, 37N20; Secondary: 93D20.
Citation:
Maoli Chen, Xiao Wang, Yicheng Liu. Collision-free flocking for a time-delay system. Discrete and Continuous Dynamical Systems - B,
2021, 26
(2)
: 1223-1241.
doi: 10.3934/dcdsb.2020251
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References:
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J. A. Carrillo, Y.-P. Choi, P. B. Mucha and et al,
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. |
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M. L. Chen, X. Li, X. Wang and et al.,
Flocking and collision avoidance of a Cucker-Smale type system with singular weights, J. Appl. Anal. Comput., 10 (2019), 140-152.
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M. Chen and X. Wang,
Flocking dynamics for multi-agent system with measurement delay, Math. Comput. Simulat., 171 (2020), 187-200.
doi: 10.1016/j.matcom.2019.09.015. |
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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. |
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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. |
| [6] |
Y.-P. Choi and C. Pignotti, Emergent behavior of Cucker-Smale model with normalized weights and distributed time delays, Netw. Heterog. Media, 14 (2019), 789–804. arXiv: 1902.03819.
doi: 10.3934/nhm.2019032. |
| [7] |
F. Cucker and J.-G. Dong,
A general collision-avoiding flocking framework, IEEE Trans. Automat. Control, 56 (2011), 1124-1129.
doi: 10.1109/TAC.2011.2107113. |
| [8] |
F. Cucker and S. Smale,
Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.
doi: 10.1109/TAC.2007.895842. |
| [9] |
F. Cucker and S. Smale,
On the mathematics of emergence, Jpn. J. Math., 2 (2007), 197-227.
doi: 10.1007/s11537-007-0647-x. |
| [10] |
R. Erban, J. Haŝkovec and Y. Sun,
A Cucker–Smale model with noise and delay, SIAM J. Appl. Math., 76 (2016), 1535-1557.
doi: 10.1137/15M1030467. |
| [11] |
V. Gazi and K. M. Passino,
A class of attractions/repulsion functions for stable swarm aggregations, Internat. J. Control, 77 (2004), 1567-1579.
doi: 10.1080/00207170412331330021. |
| [12] |
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. |
| [13] |
H. K. Khalil and J. W. Grizzle, Nonlinear Systems, Upper Saddle River, NJ: Prentice hall, 2002. |
| [14] |
J. C. Kuang, Common Inequalities, Shandong Science and Technology Press, 2004.
|
| [15] |
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. |
| [16] |
J. Park, H. 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. |
| [17] |
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.
|
| [18] |
Z. T. Qiu,
General theory of delayed functional differential equations, Journal of National University of Defense Technology, 04 (1982), 132-143.
|
| [19] |
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] |
J. A. Carrillo, Y.-P. Choi, P. B. Mucha and et al,
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. |
| [2] |
M. L. Chen, X. Li, X. Wang and et al.,
Flocking and collision avoidance of a Cucker-Smale type system with singular weights, J. Appl. Anal. Comput., 10 (2019), 140-152.
|
| [3] |
M. Chen and X. Wang,
Flocking dynamics for multi-agent system with measurement delay, Math. Comput. Simulat., 171 (2020), 187-200.
doi: 10.1016/j.matcom.2019.09.015. |
| [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. |
| [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. |
| [6] |
Y.-P. Choi and C. Pignotti, Emergent behavior of Cucker-Smale model with normalized weights and distributed time delays, Netw. Heterog. Media, 14 (2019), 789–804. arXiv: 1902.03819.
doi: 10.3934/nhm.2019032. |
| [7] |
F. Cucker and J.-G. Dong,
A general collision-avoiding flocking framework, IEEE Trans. Automat. Control, 56 (2011), 1124-1129.
doi: 10.1109/TAC.2011.2107113. |
| [8] |
F. Cucker and S. Smale,
Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.
doi: 10.1109/TAC.2007.895842. |
| [9] |
F. Cucker and S. Smale,
On the mathematics of emergence, Jpn. J. Math., 2 (2007), 197-227.
doi: 10.1007/s11537-007-0647-x. |
| [10] |
R. Erban, J. Haŝkovec and Y. Sun,
A Cucker–Smale model with noise and delay, SIAM J. Appl. Math., 76 (2016), 1535-1557.
doi: 10.1137/15M1030467. |
| [11] |
V. Gazi and K. M. Passino,
A class of attractions/repulsion functions for stable swarm aggregations, Internat. J. Control, 77 (2004), 1567-1579.
doi: 10.1080/00207170412331330021. |
| [12] |
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. |
| [13] |
H. K. Khalil and J. W. Grizzle, Nonlinear Systems, Upper Saddle River, NJ: Prentice hall, 2002. |
| [14] |
J. C. Kuang, Common Inequalities, Shandong Science and Technology Press, 2004.
|
| [15] |
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. |
| [16] |
J. Park, H. 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. |
| [17] |
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.
|
| [18] |
Z. T. Qiu,
General theory of delayed functional differential equations, Journal of National University of Defense Technology, 04 (1982), 132-143.
|
| [19] |
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. |
Figure 2.
$ \beta $ = 0.6, $ \tau $ = 0.01, the relative position of any two particle in System (2)-(3) is always bounded, and the velocity will asymptotically converge, i.e., the emergence of flocking
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