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

SIGMA 9 (2013), 034, 31 pages      arXiv:1206.1101      https://doi.org/10.3842/SIGMA.2013.034

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

Abstract
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.

pdf (494 kb)   tex (107 kb)

References

  1. Clelland J.N., Moseley C.G., Wilkens G.R., Geometry of control-affine systems, SIGMA 5 (2009), 095, 28 pages, arXiv:0903.4932.
  2. Gardner R.B., The method of equivalence and its applications, CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 58, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989.
  3. Jurdjevic V., Geometric control theory, Cambridge Studies in Advanced Mathematics, Vol. 52, Cambridge University Press, Cambridge, 1997.
  4. Liu W., Sussman H.J., Shortest paths for sub-Riemannian metrics on rank-two distributions, Mem. Amer. Math. Soc. 118 (1995), no. 564, x+104 pages.
  5. Montgomery R., A tour of subriemannian geometries, their geodesics and applications, Mathematical Surveys and Monographs, Vol. 91, Amer. Math. Soc., Providence, RI, 2002.
  6. Moseley C.G., The geometry of sub-Riemannian Engel manifolds, Ph.D. thesis, University of North Carolina at Chapel Hill, 2001.

Previous article  Next article   Contents of Volume 9 (2013)