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February  2021, 26(2): 1223-1241. doi: 10.3934/dcdsb.2020251

Collision-free flocking for a time-delay system

Maoli Chen , Xiao Wang , and  Yicheng Liu

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

* Corresponding author: Xiao Wang

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

Fund Project: This work was supported by the NSFC (Grant No.11401577 and No.11671011)

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.

Keywords: Measurement delay, time-asymptotic flocking, collision avoidance.
Mathematics Subject Classification: Primary: 34K20, 37N20; Secondary: 93D20.
Citation: 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
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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

H. K. Khalil and J. W. Grizzle, Nonlinear Systems, Upper Saddle River, NJ: Prentice hall, 2002. Google Scholar

[14] J. C. Kuang, Common Inequalities, Shandong Science and Technology Press, 2004.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[18]

Z. T. Qiu, General theory of delayed functional differential equations, Journal of National University of Defense Technology, 04 (1982), 132-143.   Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

H. K. Khalil and J. W. Grizzle, Nonlinear Systems, Upper Saddle River, NJ: Prentice hall, 2002. Google Scholar

[14] J. C. Kuang, Common Inequalities, Shandong Science and Technology Press, 2004.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[18]

Z. T. Qiu, General theory of delayed functional differential equations, Journal of National University of Defense Technology, 04 (1982), 132-143.   Google Scholar

[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.  Google Scholar

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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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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Figure 3.  $ \beta $ = 0.6, $ \tau $ = 0.1, System (2)-(3) fail to form a flock
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