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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
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 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.
Keywords: Control Theory, Newton’s Method, Deflation, Gröbner Bases, The Hermite Quadratic Form, Neural Networks, Algebraic Splines, Homological Algebra
College: Faculty of Science and Engineering
Funders: Swansea University Research Excellence Scholarships (SURES)

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