Geodesic
Geometry of Optimal Control for Control-Affine Systems
Symmetry, integrability and geometry: methods and applications, Tome 9 (2013) Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: affine distributions; optimal control theory; Cartan's method of equivalence. 
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     author = {Jeanne N. Clelland and Christopher G. Moseley and George R. Wilkens},
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     url = {http://geodesic.mathdoc.fr/item/SIGMA_2013_9_a33/}
}
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Jeanne N. Clelland; Christopher G. Moseley; George R. Wilkens. Geometry of Optimal Control for Control-Affine Systems. Symmetry, integrability and geometry: methods and applications, Tome 9 (2013). http://geodesic.mathdoc.fr/item/SIGMA_2013_9_a33/

[1] Clelland J. N., Moseley C. G., Wilkens G. R., “Geometry of control-affine systems”, SIGMA, 5 (2009), 095, 28 pp., arXiv: 0903.4932 | DOI | MR | Zbl

[2] Gardner R. B., The method of equivalence and its applications, CBMS-NSF Regional Conference Series in Applied Mathematics, 58, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989 | DOI | MR | Zbl

[3] Jurdjevic V., Geometric control theory, Cambridge Studies in Advanced Mathematics, 52, Cambridge University Press, Cambridge, 1997 | MR | Zbl

[4] Liu W., Sussman H. J., Shortest paths for sub-{R}iemannian metrics on rank-two distributions, Mem. Amer. Math. Soc., 118, no. 564, 1995, x+104 pp. | MR | Zbl

[5] Montgomery R., A tour of subriemannian geometries, their geodesics and applications, Mathematical Surveys and Monographs, 91, Amer. Math. Soc., Providence, RI, 2002 | MR | Zbl

[6] Moseley C. G., The geometry of sub-Riemannian Engel manifolds, Ph. D. thesis, University of North Carolina at Chapel Hill, 2001 | MR