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Title:  Local controllability of affine distributions 
Authors:  Aguilar, CESAR 

Keywords:  nonlinear control theory local controllability affine distribution jet bundles controlaffine systems Lie bracket highorder tangent vector 
Issue Date:  2010 
Series/Report no.:  Canadian theses 
Abstract:  In this thesis, we develop a feedbackinvariant theory of local controllability for affine distributions. We begin by developing an unexplored notion in control theory that we call proper smalltime local controllability (PSTLC). The notion of PSTLC is developed for an abstraction of the wellknown notion of a controlaffine 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 controlaffine systems.
The main object that we use to study PSTLC is a cone of highorder 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 CampbellBakerHausdorff 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. 
Description:  Thesis (Ph.D, Mathematics & Statistics)  Queen's University, 20100111 20:11:45.466 
URI:  http://hdl.handle.net/1974/5386 
Appears in Collections:  Queen's Graduate Theses and Dissertations Department of Mathematics and Statistics Graduate Theses

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