Date of Award

May 2015

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Department

Mathematics

First Advisor

Craig Guilbault

Committee Members

Ric Ancel, Boris Okun, Tzu-Chu Lin, Peter Hinow

Abstract

State complexes are nonpositively curved cube complexes that model the state spaces of reconfigurable systems. The problem of determining a strategy for reconfiguring the system from a given initial state to a given goal state is equivalent to that of finding a path between two points in the state complex. The additional requirement that allowable paths must have a prescribed initial direction and minimal turning radius determines a Markov-Dubins problem with free terminal direction (MDPFTD).

Given a nonpositively curved, locally finite cube complex X, we consider the set of unit-speed paths which satisfy a certain smoothness condition in addition to the boundary conditions and curvature constraint that define a MDPFTD. We show that this set either contains a path of minimal length, or is empty.

We then focus on the case that X is a surface with a nonpositively curved cubical structure. We show that any solution to a MDPFTD in X must consist of finitely many geodesic segments and arcs of constant curvature, and we give an algorithm for determining those solutions to the MDPFTD in X which are CL paths, that is, made up of an arc of constant curvature followed by a geodesic segment. Finally, under the assumption that the 1-skeleton of X is d-regular, we give sufficient conditions for a topological ray in X of constant curvature to be a rose curve or a proper ray.

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