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Algebraic Methods in Feedback Control and Splines with Boundary Conditions / SAMUEL GUE

Swansea University Author: SAMUEL GUE

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DOI (Published version): 10.23889/SUThesis.71786

Abstract

This thesis applies methods from algebraic geometry and topology to two distinct problems: one in optimal control and one in the theory of spline functions.On the optimal control side, we use algebraic tools to develop a computational method for the synthesis of time-optimal feedback control laws fo...

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Published: Swansea 2026
Institution: Swansea University
Degree level: Doctoral
Degree name: Ph.D
Supervisor: Villamizar, N.
URI: https://cronfa.swan.ac.uk/Record/cronfa71786
first_indexed 2026-04-23T09:57:15Z
last_indexed 2026-04-24T07:12:30Z
id cronfa71786
recordtype RisThesis
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spelling 2026-04-23T11:21:27.3403384 v2 71786 2026-04-23 Algebraic Methods in Feedback Control and Splines with Boundary Conditions 49437a8bc11e39df06736f9b14b63e60 SAMUEL GUE SAMUEL GUE true false 2026-04-23 This thesis applies methods from algebraic geometry and topology to two distinct problems: one in optimal control and one in the theory of spline functions.On the optimal control side, we use algebraic tools to develop a computational method for the synthesis of time-optimal feedback control laws for nilpotent systems.In particular, we study the polynomial systems derived from nilpotent linear systems, and use Newton’s method and the Hermite quadratic form to solve them. We create a synthetic dataset with the solutions to these equations which we use to train a binary classifier neural network to solve nilpotent systems. To demonstrate the applicability of this tool, we solve chain of integrator systems of increasing dimension, focusing on the robustness of the method in the presence of perturbations.On the splines side, we derive a formula for the dimensions of vector spaces of splines with boundary conditions over simplicial complexes embedded in R2 for high enough polynomial degree. We use tools from algebraic topology to reframe some classic results from spline theory to account for the boundary conditions. We demonstrate the use of the formula by finding the dimensions of vector spaces of splines with boundary conditions over various example simplicial complexes. E-Thesis Swansea Control Theory, Newton’s Method, Deflation, Gröbner Bases, The Hermite Quadratic Form, Neural Networks, Algebraic Splines, Homological Algebra 6 1 2026 2026-01-06 10.23889/SUThesis.71786 COLLEGE NANME COLLEGE CODE Swansea University Villamizar, N. Doctoral Ph.D Swansea University Research Excellence Scholarships (SURES) Swansea University Research Excellence Scholarships (SURES) 2026-04-23T11:21:27.3403384 2026-04-23T10:50:30.5397658 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics SAMUEL GUE 1 71786__36566__e0ea984c6b504347964b9fb7bed80352.pdf 2026_Gue_S.final.71786.pdf 2026-04-23T10:56:06.7856528 Output 1831455 application/pdf E-Thesis – open access true Copyright: the author, Samuel Gue, 2026. Distributed under the terms of a Creative Commons Attribution 4.0 License (CC BY 4.0) true eng https://creativecommons.org/licenses/by/4.0/
title Algebraic Methods in Feedback Control and Splines with Boundary Conditions
spellingShingle Algebraic Methods in Feedback Control and Splines with Boundary Conditions
SAMUEL GUE
title_short Algebraic Methods in Feedback Control and Splines with Boundary Conditions
title_full Algebraic Methods in Feedback Control and Splines with Boundary Conditions
title_fullStr Algebraic Methods in Feedback Control and Splines with Boundary Conditions
title_full_unstemmed Algebraic Methods in Feedback Control and Splines with Boundary Conditions
title_sort Algebraic Methods in Feedback Control and Splines with Boundary Conditions
author_id_str_mv 49437a8bc11e39df06736f9b14b63e60
author_id_fullname_str_mv 49437a8bc11e39df06736f9b14b63e60_***_SAMUEL GUE
author SAMUEL GUE
author2 SAMUEL GUE
format E-Thesis
publishDate 2026
institution Swansea University
doi_str_mv 10.23889/SUThesis.71786
college_str Faculty of Science and Engineering
hierarchytype
hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Mathematics and Computer Science - Mathematics{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Mathematics
document_store_str 1
active_str 0
description This thesis applies methods from algebraic geometry and topology to two distinct problems: one in optimal control and one in the theory of spline functions.On the optimal control side, we use algebraic tools to develop a computational method for the synthesis of time-optimal feedback control laws for nilpotent systems.In particular, we study the polynomial systems derived from nilpotent linear systems, and use Newton’s method and the Hermite quadratic form to solve them. We create a synthetic dataset with the solutions to these equations which we use to train a binary classifier neural network to solve nilpotent systems. To demonstrate the applicability of this tool, we solve chain of integrator systems of increasing dimension, focusing on the robustness of the method in the presence of perturbations.On the splines side, we derive a formula for the dimensions of vector spaces of splines with boundary conditions over simplicial complexes embedded in R2 for high enough polynomial degree. We use tools from algebraic topology to reframe some classic results from spline theory to account for the boundary conditions. We demonstrate the use of the formula by finding the dimensions of vector spaces of splines with boundary conditions over various example simplicial complexes.
published_date 2026-01-06T06:39:00Z
_version_ 1867858984529035264
score 11.108426

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