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Low-dimensional Lie algebras and control theory

University of British Columbia

Lie groups and Lie algebras are important mathematical constructs first developed by Sophus Lie in the late nineteenth century to unify and extend known methods used to solve differential equations. The problem considered in this thesis emphasizes one way Lie groups and Lie algebras can be used in control theory. Suppose an apparatus has mechanisms for moving in a limited number of ways with other movements generated by compositions of allowed motions. The question is then how to get a targeted motion using a minimal number of the allowed motions. Motions can often be represented by Lie groups which have associated Lie algebras as their building blocks. This research shows explicitly how one can obtain elements of Lie groups as compositions of products of other elements based on the structure of the associated Lie algebras. Here, the structure of a Lie algebra refers to its commutators which are the results that one gets by applying an operation known as the ”commutator” to each pair of elements of a Lie algebra. Two concrete examples of this problem, in control theory, are: (1) the restricted parallel parking problem where the commutator of the Lie algebra element representing translations in y and that representing rotations in the xy-plane yields translations in x. Here the control problem involves a vehicle that can only perform a series involving translations in y and rotations with the aim of efficiently obtaining a pure translation in x; (2) involves an apparatus that can only perform rotations about two axes and the aim is to perform a pure rotation about a third axis. Both examples involve threedimensional Lie algebras. In this thesis, the composition problem is solved for the nine three and four-dimensional Lie algebras with non-trivial solutions. Three different solution methods are presented. Two of these methods depend on operator and matrix representations of a Lie algebra. The other method is a differential equation method that depends solely on the commutator properties of a Lie algebra. Remarkably, for these distinguished Lie algebras the solutions involve arbitrary functions and can be expressed in terms of elementary functions.

Text

2019-07-24

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/71090

Mathematics

University of British Columbia

UBCV

http://creativecommons.org/licenses/by-nc-nd/4.0/