Bracket-generating distributions and filtered structures

Abstract

Control theory studies motion systems in which the motion is restricted. Problems in control theory can be modelled by a distribution on a manifold, which is a subbundle of the tangent bundle.

In this thesis, we will start by discussing distributions and their properties. An important type of distribution are the bracket-generating distributions. For these distributions, the vector fields tangent to the distribution generate all vector fields on the manifold, by repeatedly applying the Lie bracket. One of the main results in the thesis is Chow's Theorem. Given a manifold endowed with a bracket-generating distribution, Chow's Theorem says that we can connect any two points on the manifold by a path tangent to the distribution.

In the later chapters of the thesis, we will focus on filtered structures. A filtered structure is a filtration of the tangent bundle by distributions. These will provide a weighted analysis for vector fields and smooth functions on the manifold. The main contribution of this thesis is a version of Mitchell's Theorem for filtered structures, saying that a manifold with a filtered structure looks infinitesimally like a Lie group with a left-invariant metric. Another original contribution of the thesis is a global metric induced by a filtered structure.