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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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., ().   Google Scholar

[18]

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

[19]

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

[20]

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

[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.  Google Scholar

[22]

J. Shen, Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math, 68 (2007), 694-719. doi: 10.1137/060673254.  Google Scholar

[23]

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

[24]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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., ().   Google Scholar

[18]

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

[19]

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

[20]

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

[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.  Google Scholar

[22]

J. Shen, Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math, 68 (2007), 694-719. doi: 10.1137/060673254.  Google Scholar

[23]

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

[24]

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

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