Geometry of Control-Affine Systems

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.