Document Type

Dissertation

Date of Award

12-31-2021

Degree Name

Doctor of Philosophy in Mathematical Sciences - (Ph.D.)

Department

Mathematical Sciences

First Advisor

Roy Goodman

Second Advisor

David Shirokoff

Third Advisor

Cristina Frederick

Fourth Advisor

Richard O. Moore

Fifth Advisor

Alejandro Aceves

Abstract

This dissertation addresses some of the various issues which can arise when posing and solving optimization problems constrained by dispersive physics. Considered here are four technologically relevant experiments, each having their own unique challenges and physical settings including ultra-cold quantum fluids trapped by an external field, paraxial light propagation through a gradient index of refraction, light propagation in periodic photonic crystals, and surface gravity water waves over shallow and variable seabeds. In each of these settings, the physics can be modeled by dispersive wave equations, and the technological objective is to design the external trapping fields or propagation media such that a high fidelity or degree of coherence of the wave phenomena is achieved.

Optimal control theory is used as the analytical and computational framework in addressing these design problems. Optimal control problems are, generally speaking, challenging searches over infinite-dimensional spaces. Methods from Hamiltonian dynamical systems, asymptotic analysis, the integrability structure of the uncontrolled constraints, and simple physical intuition are employed to better guide these searches. By introducing the dimensional reductions afforded by these methods, our computational searches are significantly more efficient, over naively attempting to search the entire space of admissible controls, both in terms of the desired outcomes and in terms of expended computational resources.

The optimal control problems posed throughout this dissertation also have the additional challenge of being nonconvex optimization problems. In order to efficiently address the nonconvex nature of these problems, the program used is a global, nonconvex search which is then accelerated by fast local methods. This methodology is specifically tailored toward maintaining feasibility of implementing the computationally constructed control policies in technologically relevant settings.

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