Abstract
We study consensus and anti-consensus on the 3-sphere as the global optimization problems. The corresponding gradient descent algorithm is a dynamical systems on \(S^3\), that is known in Physics as non-Abelian Kuramoto model. This observation opens a slightly different insight into some previous results and also enables us to prove some novel results concerning consensus and balancing over the complete graph. In this way we fill some gaps in the existing theory. In particular, we prove that the anti-consensus algorithm over the complete graph on \(S^3\) converges towards a balanced configuration if a certain mild condition on initial positions of agents is satisfied. The form of this condition indicates an unexpected relation with some important constructions from Complex Analysis.
Similar content being viewed by others
References
Caponigro, M., Lai, A.C., Piccoli, B.: A nonlinear model of opinion formation on the sphere. Discrete Contin. Dyn. Syst. A 35(9), 4241–4268 (2015)
Chaturvedi, N.A., Sanyal, A.K., McClamroch, N.H.: Rigid-body attitude control. IEEE Control Syst. 31(3), 30–51 (2011)
Douady, A., Earle, C.J.: Conformally natural extension of homeomorphisms of the circle. Acta Math. 157(1), 23–48 (1986)
Hartley, R., Trumpf, J., Dai, Y., Li, H.: Rotation averaging. Int. J. Comput. Vis. 103(3), 267–305 (2013)
Jaćimović, V., Crnkić, A.: Low-dimensional dynamics in non-Abelian Kuramoto model on the 3-sphere. Chaos Interdiscip. J. Nonlinear Sci. 28(8), 083105 (2018)
Kato, S., McCullagh, P.: Conformal mapping for multivariate Cauchy families (2015). arXiv preprint arXiv:1510.07679
Kuramoto, Y.: Self-entrainment of a population of coupled nonlinear oscillators. In: Proceedings of International Symposium on Mathematical Problems in Theoretical Physics, pp. 420–422 (1975)
Lohe, M.A.: Non-Abelian Kuramoto models and synchronization. J. Phys. A Math. Theor. 42(39), 395101 (2009)
Lohe, M.A.: Quantum synchronization over quantum networks. J. Phys. A Math. Theor. 43(46), 465301 (2010)
Markdahl, J., Gonçalves, J.: Global converegence properties of a consensus protocol on the \(n\)-sphere. In: Proceedings of 55th IEEE Conference Decision and Control, pp. 3487–3492 (2016)
Markdahl, J., Thunberg, J., Gonçalves, J.: Almost global consensus on the \(n\)-sphere. IEEE Trans. Autom. Control 63(6), 1664–1675 (2018)
Marvel, S.A., Mirollo, R.E., Strogatz, S.H.: Identical phase oscillators with global sinusoidal coupling evolve by Möbius group action. Chaos Interdiscip. J. Nonlinear Sci. 19(4), 043104 (2009)
Nedic, A., Ozdaglar, A., Parrilo, P.A.: Constrained consensus and optimization in multi-agent networks. IEEE Trans. Autom. Control 55(4), 922–938 (2010)
Olfati-Saber, R., Fax, J.A., Murray, R.M.: Consensus and cooperation in networked multi-agent systems. Proc. IEEE 95(1), 215–233 (2007)
Olfati-Saber, R.: Swarms on sphere: a programmable swarm with synchronous behaviors like oscillator networks. In: Proceedings of 45th IEEE Conference Decision and Control, pp. 5060–5066 (2006)
Paley, D.A.: Stabilization of collective motion on a sphere. Automatica 45(1), 212–216 (2009)
Pikovsky, A., Rosenblum, M., Kurths, J.: Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge University Press, Cambridge (2001)
Sarlette, A.: Geometry and Symmetries in Coordination Control. Université de Liège, Liège (2009)
Sarlette, A., Sepulchre, R.: Consensus optimization on manifolds. SIAM J. Control Optim. 48(1), 56–76 (2009)
Sarlette, A., Sepulchre, R., Leonard, N.E.: Autonomous rigid body attitude synchronization. Automatica 45(2), 572–577 (2009)
Sarlette, A., Sepulchre, R.: Synchronization on the circle (2009). arXiv preprint arXiv:0901.2408
Sepulchre, R.: Consensus on nonlinear spaces. Ann. Rev. Control 35(1), 56–64 (2011)
Sepulchre, R., Paley, D.A., Leonard, N.E.: Stabilization of planar collective motion with limited communication. IEEE Trans. Autom. Control 53(3), 706–719 (2008)
Acknowledgements
The authors wish to thank anonymous referees for their valuable comments and suggestions. The second author acknowledges partial support of the Ministry of Science of Montenegro and the COST action CA16228 “European Network for Game Theory”.
Author information
Authors and Affiliations
Corresponding author
Electronic supplementary material
Below is the link to the electronic supplementary material.
Supplementary material 1 (mp4 22966 KB)
Supplementary material 2 (mp4 6143 KB)
Rights and permissions
About this article
Cite this article
Crnkić, A., Jaćimović, V. Consensus and balancing on the three-sphere. J Glob Optim 76, 575–586 (2020). https://doi.org/10.1007/s10898-018-0723-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-018-0723-1