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 Lowdimensional Lie algebras and control theory
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Lowdimensional Lie algebras and control theory Mrani Zentar, Omar
Abstract
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 xyplane 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 fourdimensional Lie algebras with nontrivial 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.
Item Metadata
Title 
Lowdimensional Lie algebras and control theory

Creator  
Publisher 
University of British Columbia

Date Issued 
2019

Description 
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 xyplane 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 fourdimensional Lie algebras with nontrivial 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.

Genre  
Type  
Language 
eng

Date Available 
20190724

Provider 
Vancouver : University of British Columbia Library

Rights 
AttributionNonCommercialNoDerivatives 4.0 International

DOI 
10.14288/1.0380064

URI  
Degree  
Program  
Affiliation  
Degree Grantor 
University of British Columbia

Graduation Date 
201909

Campus  
Scholarly Level 
Graduate

Rights URI  
Aggregated Source Repository 
DSpace

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Item Citations and Data
Rights
AttributionNonCommercialNoDerivatives 4.0 International