**Original manuscript:** 2010/01/21

In this thesis, we develop a feedback-invariant theory of local
controllability for affine distributions. We begin by developing an
unexplored notion in control theory that we call *proper small-time local
controllability* (PSTLC). The notion of PSTLC is developed for an
abstraction of the well-known notion of a control-affine system, which we
call an *affine system*. Associated to every affine system is an
*affine distribution*, an adaptation of the notion of a distribution.
Roughly speaking, an affine distribution is PSTLC if the local behaviour of
every affine system that locally approximates the affine distribution is
locally controllable in the standard sense. We prove that, under a
regularity condition, the PSTLC property can be characterized by studying
control-affine systems.

The main object that we use to study PSTLC is a cone of high-order tangent
vectors, or *variations*, and these are defined using the vector fields
of the affine system. To better understand these variations, we study how
they depend on the jets of the vector fields by studying the Taylor expansion
of a composition of flows. Some connections are made between labeled rooted
trees and the coefficients appearing in the Taylor expansion of a composition
of flows. Also, a relation between variations and the formal
Campbell-Baker-Hausdorff formula is established.

After deriving some algebraic properties of variations, we define a variational cone for an affine system and relate it to the local controllability problem. We then study the notion of neutralizable variations and give a method for constructing subspaces of variations.

Finally, using the tools developed to study variations, we consider two important classes of systems: driftless and homogeneous systems. For both classes, we are able to characterize the PSTLC property.

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