The geometry of optimal control solutions on some six dimensional lie groups
Biggs, J. and Holderbaum, W. (2005) The geometry of optimal control solutions on some six dimensional lie groups. In: 2005 44th IEEE Conference on Decision and Control & European Control Conference, Vols 1-8. IEEE Conference on Decision and Control - Proceedings. Ieee, New York, pp. 1427-1432. ISBN 0191-2216 0780395670
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This paper examines optimal solutions of control systems with drift defined on the orthonormal frame bundle of particular Riemannian manifolds of constant curvature. The manifolds considered here are the space forms Euclidean space E-3, the spheres S-3 and the hyperboloids H-3 with the corresponding frame bundles equal to the Euclidean group of motions SE(3), the rotation group SO(4) and the Lorentz group SO(1,3). The optimal controls of these systems are solved explicitly in terms of elliptic functions. In this paper, a geometric interpretation of the extremal solutions is given with particular emphasis to a singularity in the explicit solutions. Using a reduced form of the Casimir functions the geometry of these solutions are illustrated.