Linear Programming Based Approach To Infinite Horizon Optimal Control Problems with Time Discounting Criteria

Author: Neil Thatcher

Thatcher, Neil, 2017 Linear Programming Based Approach To Infinite Horizon Optimal Control Problems with Time Discounting Criteria, Flinders University, School of Computer Science, Engineering and Mathematics

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Abstract

The aim of this thesis is to develop mathematical tools for the analysis and solution of infinite horizon optimal control problems with a time discounting criteria based on the fact that the latter are equivalent to certain infinite dimensional linear programming problems. We establish that near-optimal solutions of these infinite dimensional linear programming problems and their duals can be obtained via approximation with semi-infinite linear programming problems and subsequently with finite-dimensional (``standard'') linear programming problems and their respective duals. We show that near-optimal controls of the underlying optimal control problems can be constructed on the basis of solutions of these standard linear programming problems. The thesis consists of two parts. In Part I, theoretical results are presented. These include results about semi-infinite and finite dimensional approximations of the infinite dimensional linear programming problems, results about the construction of near-optimal controls and results establishing the possibility of using solutions of optimal control problems with time discounting criteria for the construction of stabilising controls. In Part II, results of numerical experiments are presented. These results include finding near-optimal controls for the optimisation of a damped mass-spring system, the optimisation of a Neck and Dockner model and the problem of finding stabilising controls for a Lotka-Volterra system.

Keywords: optimisation, optimal control, linear programming problem, time discounting criteria, infinite horizon, infinite dimensional, semi-infinite dimensional, finite dimensional, stablising controls, numerical approximation.
Subject: Mathematics thesis

Thesis type: Doctor of Philosophy
Completed: 2017
School: School of Computer Science, Engineering and Mathematics
Supervisor: Professor Vladimir Gaitsgory