The development of a vaccine against some strains of the human papillomavirus (HPV) has led to many interesting public health questions [1]. We address some of these questions in the following work. We develop a compartmental mathematical model and examine the effect of waning immunity, vaccinating individuals prior to their becoming sexually active and the current government policy of vaccinating only females [2]. We calculate parameters based on data. We consider both time-dependent and age dependent ODE models and an age- and time-dependent PDE model and compare the results. We find the “effective” R0 value, Re0, for the time-dependent models. We introduce optimal control to both the time-dependent and age-dependent ODE models to assess the most cost-effective method for introducing the vaccine into a population. We find that the duration of protection offered by the vaccine can influence whether it is possible to eradicate infection from the population. We find the critical proportion to vaccinate to eradicate the disease. We see that introducing male vaccination would lead to a greater proportion of individuals to be vaccinated if the disease is to be eradicated. The PDE model shows that the proportion of females vaccinated has a large impact on the proportion of females infected. We show that it is cost-effective to vaccinate males and females. Our results support current government policy for age of vaccination [2]. We conclude that potential waning immunity will impact the success of the vaccine. We broadly support government policy for vaccination but recommend including male vaccination to most cost-effectively eradicate the disease.

We investigate algorithms for different steps in the decision making process, focusing on systems where we are uncertain about the outcomes but can quantify how probable they are using random variables. Any decision one makes in such a situation leads to a distribution of outcomes and requires a way to evaluate a decision. The standard approach is to marginalise the distribution of outcomes into a single number that tries in some way to summarise the value of each decision. After selecting a marginalisation approach, mathematicians and decision makers focus their analysis on the marginalised value but ignore the distribution. We argue that we should also be investigating the implications of the chosen mathematical approach for the whole distribution of outcomes. We illustrate the effect different mathematical formulations have on the distribution with one-stage and sequential decision problems. We show that different ways to marginalise the distributions can result in very similar decisions but each way has a different complexity and computational cost. It is often computationally intractable to approximate optimal decisions to high precision and much research goes into developing algorithms that are suboptimal in the marginalised sense, but work within the computational budget available. If the performance of these algorithms is evaluated they are mainly judged based on the marginalised values, however, comparing the performance using the full distribution provides interesting information: We provide numerical examples from dynamic pricing applications where the suboptimal algorithm results in higher profit than the optimal algorithm in more than half of the realisations, which is paid for with a more significant underperformance in the remaining realisations. All the problems discussed in this thesis lead to continuous optimisation problems. We develop a new algorithm that can be used on top of existing optimisation algorithms to reduce the cost of approximating solutions. The algorithm is tested on a range of optimisation problems and is shown to be competitive with existing methods.

Continuous-time systems are usually modelled by differential equations arising from physical laws. However, the use of these models in practice requires discretisation. In this thesis we consider sampled-data models for linear and nonlinear systems. We study some of the issues involved in the sampling process, such as the accuracy of the sampled-data models, the artifacts produced by the particular sampling scheme, and the relations to the underlying continuous-time system. We review, extend and present new results, making extensive use of the delta operator which allows a clearer connection between a sampled-data model and the underlying continuous-time system. In the first part of the thesis we consider sampled-data models for linear systems. In this case exact discrete-time representations can be obtained. These models depend, not only on the continuous-time system, but also on the artifacts involved in the sampling process, namely, the sample and hold devices. In particular, these devices play a key role in determining the sampling zeros of the discrete-time model. We consider robustness issues associated with the use of discrete-time models for continuous-time system identification from sampled data. We show that, by using restricted bandwidth frequency domain maximum likelihood estimation, the identification results are robust to (possible) under-modelling due to the sampling process. Sampled-data models provide a powerful tool also for continuous-time optimal control problems, where the presence of constraints can make the explicit solution impossible to find. We show how this solution can be arbitrarily approximated by an associated sampled-data problem using fast sampling rates. We also show that there is a natural convergence of the singular structure of the optimal control problem from discrete- to continuous-time, as the sampling period goes to zero. In Part II we consider sampled-data models for nonlinear systems. In this case we can only obtain approximate sampled-data models. These discrete-time models are simple and accurate in a well defined sense. For deterministic systems, an insightful observation is that the proposed model contains sampling zero dynamics. Moreover, these correspond to the same dynamics associated with the asymptotic sampling zeros in the linear case. The topics and results presented in the thesis are believed to give important insights into the use of sampled-data models to represent linear and nonlinear continuous-time systems. / PhD Doctorate

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Enterococci bacteria that cannot be treated eectively with the antibiotic vancomycin are termed Vancomycin-Resistant Enterococci (VRE). In this thesis, we develop a mathematical framework for determining optimal strategies for prevention and treatment of VRE in an Intensive Care Unit (ICU). A system of ve ordinary dierential equations describes the movement of ICU patients in and out of dierent states related to VRE infection. Two control variables representing the prevention and treatment of VRE are incorporated into the system. An optimal control problem is formulated to minimize the VRE-related deaths and costs associated with controls over a nite time period. Pontryagin's Minimum Principle is used to characterize optimal controls by deriving a Hamiltonian expression and dierential equations for ve adjoint variables. Numerical solutions to the optimal control problem illustrate how hospital policy makers can use our mathematical framework to investigate optimal cost-eective prevention and treatment schedules during a VRE outbreak. / McAnulty College and Graduate School of Liberal Arts; / Computational Mathematics / MS; / Thesis;

Based on the Lyapunov stability theorem, an optimal model reference adaptive control (OMRAC) scheme with perturbation estimation is presented in this thesis to solve robust tracking problems. The plant considered belongs to a class of MIMO perturbed dynamic systems with input nonlinearity and time varying delay in the state. The proposed control scheme contains three types of controllers. The first one is a linear feedback controller, which is an optimal controller if there is no perturbation. The second one is an adaptive controller, it is used for adapting the unknown upper bound of perturbation estimation error. The last one is the perturbation estimation mechanism. The property of uniformly ultimately boundness is proved under the proposed control scheme, and the effects of each design parameter on the dynamic performance is analyzed. Two numerical examples are given for demonstrating the feasibility of the proposed methodology.

This dissertation begins with an introduction to finite-dimensional optimization and optimal control theory. It then proves lossless convexification for three problems: 1) a minimum time rendezvous using differential drag, 2) a maximum divert and landing, and 3) a general optimal control problem with linear state constraints and mixed convex and non-convex control constraints. Each is a unique contribution to the theory of lossless convexification. The first proves lossless convexification in the presence of singular controls and specifies a procedure for converting singular controls to the bang-bang type. The second is the first example of lossless convexification with state constraints. The third is the most general result to date. It says that lossless convexification holds when the state space is a strongly controllable subspace. This extends the controllability concepts used previously, and it recovers earlier results as a special case. Lastly, a few of the remaining research challenges are discussed. / text

It is often desired to have control over a process or a physical system, to cause it to behave optimally. Optimal control theory deals with analyzing and finding solutions for optimal control for a system that can be represented by a set of differential equations. This thesis examines such a system in the form of a set of matrix differential equations known as a continuous linear time-invariant system. Conditions on the system, such as linearity, allow one to find an explicit closed form finite solution that can be more efficiently computed compared to other known types of solutions. This is done by optimizing a quadratic cost function. The optimization leads to solving a Riccati equation. Conditions are discussed for which solutions are possible. In particular, we will obtain a solution for a stable and controllable system. Numerical examples are given for a simple system with 2x2 matrix coefficients.

The freight transport industry faces big challenges as the demand for transport and fuel prices are steadily increasing, whereas the environmental impact needs to be significantly reduced. Heavy-duty vehicle (HDV) platooning is a promising technology for a sustainable transportation system. By semi-autonomously governing each platooning vehicle at small inter-vehicle spacing, we can effectively reduce fuel consumption, emissions, and congestion, and relieve driver tension. Yet, it is not evident how to synthesise such a platoon control system and how constraints imposed by the road topography affect the safety or fuel-saving potential in practice. This thesis presents contributions to a framework for the design, implementation, and evaluation of HDV platooning. The focus lies mainly on establishing fuel-efficient platooning control and evaluating the fuel-saving potential in practice. A vehicle platoon model is developed together with a system architecture that divides the control problem into manageable subsystems. Presented results show that a significant fuel reduction potential exists for HDV platooning and it is favorable to operate the vehicles at a small inter-vehicle spacing. We address the problem of finding the minimum distance between HDVs in a platoon without compromising safety, by setting up the problem in a game theoretical framework. Thereby, we determine criteria for which collisions can be avoided in a worst-case scenario and establish the minimum safe distance to a vehicle ahead. A systematic design methodology for decentralized inter-vehicle distance control based on linear quadratic regulators is presented. It takes dynamic coupling and engine response delays into consideration, and the structure of the controller feedback matrix can be tailored to the locally available state information. The results show that a decentralized controller gives good tracking performance and attenuates disturbances downstream in the platoon for dynamic scenarios that commonly occur on highways. We also consider the problem of finding a fuel-efficient controller for HDV platooning based on road grade preview information under road and vehicle parameter uncertainties. We present two model predictive control policies and derive their fuel-saving potential. The thesis finally evaluates the fuel savings in practice. Experimental results show that a fuel reduction of 3.9–6.5 % can be obtained on average for a heterogenous platoon of HDVs on a Swedish highway. It is demonstrated how the savings depend on the vehicle position in the platoon, the behavior of the preceding vehicles, and the road topography. With the results obtained in this thesis, it is argued that a significant fuel reduction potential exists for HDV platooning. / <p>QC 20140527</p>

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