title: Optimality, reduction and collective motion.
abstract: The planar self-steering particle model of agents in a collective gives rise to dynamics on theN-fold direct product ofSE(2), the rigid motion group in the plane. Assuming a connected, undirected graph of interaction between agents, we pose a family of symmetric optimal control problems with a coupling parameter capturing the strength of interactions. The Hamiltonian system associated with the necessary conditions for optimality is reducible to a Lie-Poisson dynamical system possessing interesting structure. In particular, the strong coupling limit reveals additional (hidden) symmetry, beyond the manifest one used in reduction: this enables explicit integration of the dynamics, and demonstrates the presence of a 'master clock' that governs all agents to steer identically. For finite coupling strength, we show that special solutions exist with steering controls proportional across the collective. These results suggest that optimality principles may provide a framework for understanding imitative behaviours observed in certain animal aggregations.
authors.affiliation: Naval Research Laboratory , Washington DC 20375, USA.
authors.forename: Eric W
authors.initials: EW
authors.lastname: Justh
authors.affiliation: Institute for Systems Research, University of Maryland, College Park, MD 20742, USA; Department of Electrical and Computer Engineering, University of Maryland, College Park, MD 20742, USA.
authors.forename: P S
authors.initials: PS
authors.lastname: Krishnaprasad
keywords: Poisson structure; collective behaviour; explicit integrability; maximum principle; optimal control; symmetry reduction