
SIGMA 9 (2013), 034, 31 pages arXiv:1206.1101
http://dx.doi.org/10.3842/SIGMA.2013.034
Geometry of Optimal Control for ControlAffine 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 803090395, 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 968222273, USA
Received June 07, 2012, in final form April 03, 2013; Published online April 17, 2013
Abstract
Motivated by the ubiquity of controlaffine systems in optimal control theory,
we investigate the geometry of
pointaffine control systems with metric structures in dimensions two and three.
We compute local isometric invariants for pointaffine 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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References
 Clelland J.N., Moseley C.G., Wilkens G.R., Geometry of controlaffine systems,
SIGMA 5 (2009), 095, 28 pages, arXiv:0903.4932.
 Gardner R.B., The method of equivalence and its applications, CBMSNSF
Regional Conference Series in Applied Mathematics, Vol. 58, Society for
Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989.
 Jurdjevic V., Geometric control theory, Cambridge Studies in Advanced
Mathematics, Vol. 52, Cambridge University Press, Cambridge, 1997.
 Liu W., Sussman H.J., Shortest paths for subRiemannian metrics on ranktwo
distributions, Mem. Amer. Math. Soc. 118 (1995), no. 564,
x+104 pages.
 Montgomery R., A tour of subriemannian geometries, their geodesics and
applications, Mathematical Surveys and Monographs, Vol. 91, Amer.
Math. Soc., Providence, RI, 2002.
 Moseley C.G., The geometry of subRiemannian Engel manifolds, Ph.D.
thesis, University of North Carolina at Chapel Hill, 2001.

