Geometry of Control-Affine Systems

Abstract

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.