Geometry of Control-Affine Systems

Jeanne N. Clelland ; Christopher G. Moseley ; George R. Wilkens

Symmetry, integrability and geometry: methods and applications, Tome 5 (2009)

Résumé

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

Détail

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

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