American Institute of Mathematical Sciences

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March  2014, 34(3): 1009-1020. doi: 10.3934/dcds.2014.34.1009

A conditional, collision-avoiding, model for swarming

Felipe Cucker 1, and  Jiu-Gang Dong 2,

1. 

Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon Tong, Hong Kong, China
2. 

Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, China

Received  October 2012 Revised  March 2013 Published  August 2013

We propose a model for swarming (i.e., cohesion preserving) that shares all the good properties of the CS-model for flocking. In particular, we show for this model that under strong interactions of the agents swarming unconditionally occurs and that, furthermore, it does so in a collision avoiding manner. We also show that under weak interactions the same holds true provided the initial state of the population (their positions and velocities) satisfies some explicit inequalities.
Keywords: swarming, collision avoidance., CS-model, Emergent behaviour.
Mathematics Subject Classification: Primary: 93C15; Secondary: 93C3.
Citation: Felipe Cucker, Jiu-Gang Dong. A conditional, collision-avoiding, model for swarming. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 1009-1020. doi: 10.3934/dcds.2014.34.1009
References:
[1]

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.

[2]

Y. Chuang, Y. Huang, M. D'Orsogna and A. Bertozzi, Multi-vehicle flocking: Scalability of cooperative control algorithms using pairwise potentials, IEEE International Conference on Robotics and Automation, (2007), 2292-2299. doi: 10.1109/ROBOT.2007.363661.

[3]

E. Cristiani, P. Frasca and B. Piccoli, Effects of anisotropic interactions on the structure of animal groups, J. Math. Biol., 62 (2011), 569-588. doi: 10.1007/s00285-010-0347-7.

[4]

F. Cucker and J.-G. Dong, A general collision-avoiding flocking framework, IEEE Trans. Autom. Control, 56 (2011), 1124-1129. doi: 10.1109/TAC.2011.2107113.

[5]

F. Cucker and C. Huepe, Flocking with informed agents, Mathematics in Action, 1 (2008), 1-25. doi: 10.5802/msia.1.

[6]

F. Cucker and E. Mordecki, Flocking in noisy environments, J. Math. Pures Appl., 89 (2008), 278-296. doi: 10.1016/j.matpur.2007.12.002.

[7]

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

[8]

F. Dalmao and E. Mordecki, Cucker-Smale flocking under hierarchical leadership and random interactions, SIAM. J. App. Math., 71 (2011), 1307-1316. doi: 10.1137/100785910.

[9]

V. Gazi and K. M. Passino, Stability analysis of swarms, IEEE Trans. Autom. Control, 48 (2003), 692-697. doi: 10.1109/TAC.2003.809765.

[10]

V. Gazi and K. M. Passino, Stability analysis of social foraging swarms, IEEE Trans. Syst., Man, Cybern. B, 34 (2004), 539-557. doi: 10.1109/TSMCB.2003.817077.

[11]

V. Gazi and K. M. Passino, A class of attractions/repulsion functions for stable swarm aggregations, Int. J. Control, 77 (2004), 1567-1579. doi: 10.1080/00207170412331330021.

[12]

S.-Y. Ha, T.-Y. Ha and J.-G. Kim, Emergent behavior of a Cucker-Smale type particle model with nonlinear velocity couplings, IEEE Trans. Autom. Control, 55 (2010), 1679-1683. doi: 10.1109/TAC.2010.2046113.

[13]

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.

[14]

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinetic and Related Models, 1 (2008), 415-435. doi: 10.3934/krm.2008.1.415.

[15]

M. Hirsch and S. Smale, "Differential Equations, Dynamical Systems, and Linear Algebra," 60 of Pure and Applied Mathematics. Academic Press, 1974.

[16]

A. Jadbabaie, J. Lin and A. S. Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Trans. on Autom. Control, 48 (2003), 988-1001. doi: 10.1109/TAC.2003.812781.

[17]

J. Kang, S.Y. Ha, E. Jeong and K. K. Kang, How do cultural classes emerge from assimilation and distinction? An extension of the Cucker-Smale flocking model, To Appear in J. Mathematical Sociology.

[18]

H. K. Khalil, "Nonlinear Systems," 3rd ed. Englewood Cliffs, NJ: Prentice-Hall, 2002.

[19]

N. Leonard and E. Fiorelli, Virtual leaders, artificial potentials and coordinated control of groups, Proc. 40th IEEE Conf. Decision Contr., (2001), 2968-2973.

[20]

W. Li, Stability analysis of swarms with general topology, IEEE Trans. Syst., Man, Cybern. B, 38 (2008), 1084-1097.

[21]

L. Perea, P. Elosegui and G. Gómez, 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. Shen, Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math, 68 (2007), 694-719. doi: 10.1137/060673254.

[23]

J. J. E. Slotine and W. Li, "Applied Nonlinear Control," Englewood Cliffs, NJ: Prentice-Hall, 1991.

[24]

T. Vicsek and A. Zafeiris, Collective motion, Physics Reports, 517 (2012), 71-140. doi: 10.1016/j.physrep.2012.03.004.

show all references

References:
[1]

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.

[2]

Y. Chuang, Y. Huang, M. D'Orsogna and A. Bertozzi, Multi-vehicle flocking: Scalability of cooperative control algorithms using pairwise potentials, IEEE International Conference on Robotics and Automation, (2007), 2292-2299. doi: 10.1109/ROBOT.2007.363661.

[3]

E. Cristiani, P. Frasca and B. Piccoli, Effects of anisotropic interactions on the structure of animal groups, J. Math. Biol., 62 (2011), 569-588. doi: 10.1007/s00285-010-0347-7.

[4]

F. Cucker and J.-G. Dong, A general collision-avoiding flocking framework, IEEE Trans. Autom. Control, 56 (2011), 1124-1129. doi: 10.1109/TAC.2011.2107113.

[5]

F. Cucker and C. Huepe, Flocking with informed agents, Mathematics in Action, 1 (2008), 1-25. doi: 10.5802/msia.1.

[6]

F. Cucker and E. Mordecki, Flocking in noisy environments, J. Math. Pures Appl., 89 (2008), 278-296. doi: 10.1016/j.matpur.2007.12.002.

[7]

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

[8]

F. Dalmao and E. Mordecki, Cucker-Smale flocking under hierarchical leadership and random interactions, SIAM. J. App. Math., 71 (2011), 1307-1316. doi: 10.1137/100785910.

[9]

V. Gazi and K. M. Passino, Stability analysis of swarms, IEEE Trans. Autom. Control, 48 (2003), 692-697. doi: 10.1109/TAC.2003.809765.

[10]

V. Gazi and K. M. Passino, Stability analysis of social foraging swarms, IEEE Trans. Syst., Man, Cybern. B, 34 (2004), 539-557. doi: 10.1109/TSMCB.2003.817077.

[11]

V. Gazi and K. M. Passino, A class of attractions/repulsion functions for stable swarm aggregations, Int. J. Control, 77 (2004), 1567-1579. doi: 10.1080/00207170412331330021.

[12]

S.-Y. Ha, T.-Y. Ha and J.-G. Kim, Emergent behavior of a Cucker-Smale type particle model with nonlinear velocity couplings, IEEE Trans. Autom. Control, 55 (2010), 1679-1683. doi: 10.1109/TAC.2010.2046113.

[13]

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.

[14]

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinetic and Related Models, 1 (2008), 415-435. doi: 10.3934/krm.2008.1.415.

[15]

M. Hirsch and S. Smale, "Differential Equations, Dynamical Systems, and Linear Algebra," 60 of Pure and Applied Mathematics. Academic Press, 1974.

[16]

A. Jadbabaie, J. Lin and A. S. Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Trans. on Autom. Control, 48 (2003), 988-1001. doi: 10.1109/TAC.2003.812781.

[17]

J. Kang, S.Y. Ha, E. Jeong and K. K. Kang, How do cultural classes emerge from assimilation and distinction? An extension of the Cucker-Smale flocking model, To Appear in J. Mathematical Sociology.

[18]

H. K. Khalil, "Nonlinear Systems," 3rd ed. Englewood Cliffs, NJ: Prentice-Hall, 2002.

[19]

N. Leonard and E. Fiorelli, Virtual leaders, artificial potentials and coordinated control of groups, Proc. 40th IEEE Conf. Decision Contr., (2001), 2968-2973.

[20]

W. Li, Stability analysis of swarms with general topology, IEEE Trans. Syst., Man, Cybern. B, 38 (2008), 1084-1097.

[21]

L. Perea, P. Elosegui and G. Gómez, 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. Shen, Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math, 68 (2007), 694-719. doi: 10.1137/060673254.

[23]

J. J. E. Slotine and W. Li, "Applied Nonlinear Control," Englewood Cliffs, NJ: Prentice-Hall, 1991.

[24]

T. Vicsek and A. Zafeiris, Collective motion, Physics Reports, 517 (2012), 71-140. doi: 10.1016/j.physrep.2012.03.004.

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