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

SIGMA 5 (2009), 095, 28 pages      arXiv:0903.4932
Contribution to the Special Issue “Élie Cartan and Differential Geometry”

Geometry of 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 April 02, 2009, in final form September 28, 2009; Published online October 07, 2009

Motivated by control-affine systems in optimal control theory, we introduce the notion of a point-affine distribution on a manifold X – i.e., an affine distribution F together with a distinguished vector field contained in F. We compute local invariants for point-affine distributions of constant type when dim(X) = n, rank(F) = n–1, and when dim(X) = 3, rank(F) = 1. Unlike linear distributions, which are characterized by integer-valued invariants – namely, the rank and growth vector – when dim(X) ≤ 4, we find local invariants depending on arbitrary functions even for rank 1 point-affine distributions on manifolds of dimension 2.

Key words: affine distributions; control theory; exterior differential systems; Cartan's method of equivalence.

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