Enlargement, geodesics, and collectives

28/10/2015
Publication GSI2015
OAI : oai:www.see.asso.fr:11784:14343

Résumé

We investigate optimal control of systems of particles on matrix Lie groups coupled through graphs of interaction, and characterize the limit of strong coupling. Following Brockett, we use an enlargement approach to obtain a convenient form of the optimal controls. In the setting of drift-free particle dynamics, the coupling terms in the cost functionals lead to a novel class of problems in subriemannian geometry of product Lie groups.

Enlargement, geodesics, and collectives

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application/pdf Enlargement, geodesics, and collectives Eric Justh, P. S. Krishnaprasad
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We investigate optimal control of systems of particles on matrix Lie groups coupled through graphs of interaction, and characterize the limit of strong coupling. Following Brockett, we use an enlargement approach to obtain a convenient form of the optimal controls. In the setting of drift-free particle dynamics, the coupling terms in the cost functionals lead to a novel class of problems in subriemannian geometry of product Lie groups.
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We investigate optimal control of systems of particles on matrix Lie groups coupled through graphs of interaction, and characterize the limit of strong coupling. Following Brockett, we use an enlargement approach to obtain a convenient form of the optimal controls. In the setting of drift-free particle dynamics, the coupling terms in the cost functionals lead to a novel class of problems in subriemannian geometry of product Lie groups.

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Enlargement, Geodesics and Collectives Eric W. Justh and P. S. Krishnaprasad Geometric Science of Information 2015 Ecole Polytechnique, Paris-Saclay October 28-20, 2015 Supported in part by ARL/ARO MURI Program Grant W911NF-13-1-0390, Air Force Office of Scientific Research under AFOSR Grant No. FA9550-10-1-0250, and by Office of Naval Research Flocking – mechanisms for behavior Collaboration with Andrea Cavagna Laboratory Waves in Flocks G. Beauchamp, Social Predation, Academic Press, Elsevier Inc., Amsterdam, 2014. From Biology to Control Allelomimetic behavior – activities in which performance of a behavior increases probability of that behavior being performed by other nearby animals (Wikipedia) Examples – switching activity, initiation of movement, vigilance bouts, schooling, flocking Function – there are benefits to social animals in behaving in a similar manner to others within their group; synchrony at a fine scale helps in achieving greater cohesion Allelomimesis is autocatalytic – positive feedback in the form of others copying me copying … Can we construct optimal control problems for collectives with solutions interpretable as allelomimesis? Can symmetry-reduction techniques be used in finding solutions? Are there useful hidden symmetries? From Biology to Self-steering Particles Deneubourg J-L, Goss S. 1989 Collective patterns and decision- making. Ethol. Ecol. Evol. 1, 295–311. Pays O et al. 2007 Prey synchronize their vigilant behaviour with other group members. Proc. R. Soc. B 274, 1287–1291. Proccacini et. al. (2011), Propagating waves in starling , Sturnus vulgaris, flock under predation, Animal Behaviour, 82, 759-765. EWJ and PSK (2011), Optimal Natural Frames, Comm. Info. and Syst., 11(1):17-34. EWJ and PSK (2014), Optimality, Reduction and Collective Motion, Proc. R. Soc. A, DOI: 10.1098/rspa.2014.0606, online 1 April 2015 Optimality on Products of Lie Groups Maximum Principle and Reduction Other Groups • Is there a path to deriving similar results in the limit of strong coupling in a more general setting? • Is there interesting geometry for intermediate coupling? • Are there explicitly solvable problems? The Extrinsic (enlargement) View Brockett RW. 1973 Lie theory and control systems defined on spheres. SIAM J. Appl. Math. 25, 213–225. (doi:10.1137/0125025) Extrinsic Collective Dynamics Minimize, Lagrangian, Pre-hamiltonian Pontryagin Pre-hamiltonian where, Then, Application of the Maximum Principle Define so that Defining Reduction and the Kirillov-Kostant-Souriau bracket Poisson Morphism Strong coupling limit Letting Subriemannian geodesics We have a family of subriemannian geometries parametrized by the coupling constant and the interconnection graph (under conditions of accessibility) Coupled Nonholonomic Integrators with fixed end point conditions, and cost functional Coupled Nonholonomic Integrators N = 1, subriemannian geodesic sphere for comparison purposes N = 2, a 2 dimensional slice of 5 dimensional subriemannian sphere EWJ and PSK (2015), “Subriemannian geodesics for coupled nonholonomic integrators”, submitted Coupled Nonholonomic Integrators Movies of Geodesic Dynamics THANK YOU FOR LISTENING