Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 9 (2013), 034, 31 pages      arXiv:1206.1101

Geometry of Optimal Control for Control-Affine Systems

Jeanne N. Clelland a, Christopher G. Moseley b and George R. Wilkens c
a) Department of Mathematics, 395 UCB, University of Colorado, Boulder, CO 80309-0395, USA
b) Department of Mathematics and Statistics, Calvin College, Grand Rapids, MI 49546, USA
c) Department of Mathematics, University of Hawaii at Manoa, 2565 McCarthy Mall, Honolulu, HI 96822-2273, USA

Received June 07, 2012, in final form April 03, 2013; Published online April 17, 2013

Motivated by the ubiquity of control-affine systems in optimal control theory, we investigate the geometry of point-affine control systems with metric structures in dimensions two and three. We compute local isometric invariants for point-affine distributions of constant type with metric structures for systems with 2 states and 1 control and systems with 3 states and 1 control, and use Pontryagin's maximum principle to find geodesic trajectories for homogeneous examples. Even in these low dimensions, the behavior of these systems is surprisingly rich and varied.

Key words: affine distributions; optimal control theory; Cartan's method of equivalence.

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