Geometry of Control-Affine Systems

TL;DR

This paper introduces the concept of point-affine distributions on manifolds, computes local invariants for specific cases, and reveals that unlike linear distributions, these invariants can depend on arbitrary functions, especially in low dimensions.

Contribution

It defines point-affine distributions in the context of control-affine systems and derives local invariants, highlighting differences from linear distributions in low-dimensional cases.

Findings

Local invariants depend on arbitrary functions for certain low-dimensional cases.

Computed invariants for constant type distributions in specific dimensions and ranks.

Identified fundamental differences between linear and point-affine distributions in low dimensions.

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.