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Abstract: This thesis is devoted to the proof of several results on the existence and uniqueness of Monge solutions to the multi-marginal optimal transportation problem. These results are found in Chapters \\ref{Chapter3}, \\ref{Chapter4} and \\ref{Chapter5}, and represent joint work with Brendan Pass. The Chapters \\ref{Chapter1} and \\ref{Chapter2} are devoted to the introduction and preliminaries respectively. In Chapter \\ref{Chapter3} we study a multi-marginal optimal transportation problem with a cost function of the form $c(x_{1}, \\ldots,x_{m})=\\sum_{k=1}^{m-1}|x_{k}-x_{k+1}|^{2} + |x_{m}- F(x_{1})|^{2}$, where $F: \\mathbb{R}^n \\rightarrow \\mathbb{R}^n$ is a given map. %is a map from $X_{1}$ to $X_{1}$, $x_{k}\\in \\mathbb{R}^{n}$. When $m=4$, $F$ is a positive multiple of the identity mapping, and the first and last marginals are absolutely continuous with respect to Lebesgue measure, we establish that any solution of the Kantorovich problem is induced by a map; the solution is therefore unique. We go on to show that this result is sharp in a certain sense. Precisely, we exhibit examples showing that Kantorovich solutions may concentrate on higher dimensional sets if any of the following hold: 1) $F$ is any linear mapping other than a positive scalar multiple of the identity, 2) the last marginal is not absolutely continuous with respect to Lebesgue measure, or 3) the number of marginals $m \\geq 5$, even when $F$ is the identity mapping. In the fourth chapter we study a multi-marginal optimal transport problem with cost $c(x_{1}, \\ldots, x_{m})=\\sum_{\\{i,j\\}\\in P} |x_{i}- x_{j}|^{2}$, where $P\\subseteq Q:=\\{\\{i,j\\}: i, j \\in \\{1,2,...m\\}, i \\neq j\\}$. We reformulate this problem by associating each cost of this type with a graph with $m$ vertices whose set of edges is indexed by $P$. We then establish uniqueness and Monge solution results for two general classes of cost functions. Among many other examples, these classes encapsulate the Gangbo and \\'{S}wi\\c{e}ch cost \\cite{GangboSwicech1998} and the cost $c(x_{1}, \\ldots,x_{m})=\\sum_{k=1}^{m-1}|x_{k}-x_{k+1}|^{2} + |x_{m}- x_{1}|^{2}$ when $m\\leq 4$. In the final chapter we establish a general condition on the cost function to obtain uniqueness and Monge solutions in the multi-marginal optimal transport problem, under the assumption that a given collection of the marginals are absolutely continuous with respect to Lebesgue measure. When only the first marginal is assumed to be absolutely continuous, our condition is equivalent to the twist on splitting sets condition found in \\cite{KimPass2014}. In addition, it is satisfied by the special cost functions of Chapter \\ref{Chapter3} and \\ref{Chapter4} (found also in \\cite{PassVargas2021, PassVargas21}), when absolute continuity is imposed on certain other collections of marginals. We also present several new examples of cost functions which violate the twist on splitting sets condition but satisfy the new condition introduced here; we therefore obtain Monge solution and uniqueness results for these cost functions, under regularity conditions on an appropriate subset of the marginals.

Filtering and smoothing are classical estimation techniques that provide an estimate of the state of a classical system based on measurement information in the past (prior to the estimation time) and the past-future (prior and posterior to the estimation time), respectively. However, with the advent of quantum technologies, the need to estimate the state of an individual quantum system has also arisen. While the filtering technique is easily generalized and applied to quantum systems, it was not so simple for the smoothing technique. Applying the direct quantum analog of the smoothing theory often leads to an estimate that is unphysical, indicating that the classical theory is incompatible with quantum systems. The reason for this incompatibility is that the operators describing the future measurements on the system, called the retrofiltered effect, and the state conditioned on the past measurement, i.e., the filtered state, do not necessarily commute. One way to solve this issue is the quantum state smoothing theory. In order to deal with past and future information, the theory introduces the concept of a hidden measurement record, gathered by a secondary observer, say Bob, in order to define the true quantum state, a state containing maximal information about the system. With this concept of a true state, it is then possible to construct a valid smoothed quantum state, that is, a state conditioned on a past-future measurement record. In this thesis I delve into the quantum state smoothing theory. I begin by reformu-lating the quantum state smoothing theory as an optimal estimation problem, that is, minimizing a particular expected cost function. I show that the smoothed state is the optimal estimator for two cost functions, the trace-square deviation from and relative entropy with the true state. Additionally, I show, for a closely related cost function, the linear infidelity, that the smoothed state is suboptimal, while the pure state corre-sponding to the largest eigenvalue of the smoothed state is optimal. I then investigate under what conditions the smoothed quantum state reduces to a classically smoothed state, finding a sufficient condition. This sufficient condition requires the true state to be described probabilistically in a fixed basis. Subsequently, in an attempt to remove some of the restrictions on how Bob measures the system, I hypothesize a weaker suffi-cient condition of only requiring the filtered state and retrofiltered effect to be described probabilistically in a fixed basis. This hypothesis is disproven with a counter example. The remainder of the thesis is dedicated to a particular class of quantum system, the linear Gaussian quantum (LGQ) systems. I apply the quantum state smoothing theory to the LGQ systems, obtaining closed-form expressions for the smoothed quantum state.These closed-form expressions allow for numerous properties of the smoothed quantum state to be determined that would otherwise be arduous to even verify in the general setting. In particular, I investigate the behaviour of the smoothed quantum state in the low and high measurement efficiency limit. Furthermore, I derive a necessary and sufficient condition on the true state of the system that, in the event that Bob’s measurement is unknown, restricts the possible true states of the system based on the observer’s, say Alice’s, measurement choice. From the dynamical form of the LGQ state smoothing equations, I derive a necessary and sufficient condition for differentiable evolution of the smoothed quantum state. Lastly, I investigate the optimal measurement strategy for Alice and Bob in order to maximize the relative purity recovery. I pose three hypotheses and test the validity of each against two physical systems. One of these hypotheses provides an approximately optimal solution. To further verify this hypothesis, I generalize the hypothesis to qubit systems and test against an example system, again verifying the hypothesis.

Vacancy chain systems function as a method of resource distribution in domains such as housing and labor markets. Hermit crabs also employ vacancy chains as a method of shell exchange. Application of vacancy chain modelling in engineering has been attempted, but numerous flaws exist in the developed vacancy chain scheduling algorithm. This work addresses the lack of an appropriate vacancy chain cost function by developing a generalizable cost function based on hermit crab shell exchange behavior. The cost function’s purpose is enabling development of realistic engineering experiments and models based on real-world vacancy chain systems.

The finite control set model predictive control (FCS-MPC) for motor drives has been vigorously investigated during the past decade for control of current, torque, stator flux linkage, and other performances in a motor drive system. Model predictive torque control (MPTC) and flux control (MPFC) are two popular categories of FCS-MPC for motor drives. In FCS-MPTC, a finite number of possible voltage vectors are evaluated by a cost function in an iterative prediction loop. The cost function includes several control objectives, such as minimization of torque and flux errors, the neutral-point voltage of multi-level inverters and inverter switching frequency. The control algorithm determines the optimal voltage vector that minimizes the pre-defined cost function. Each variable included in the cost function has a weighting factor according to different magnitudes and units. Weighting factor tuning is a non-trivial and complicated task, particularly when the control algorithm has two control-objectives which are prime variables i.e., torque and flux. In such a case, the control algorithm may choose a global optimal solution for torque which will be a sub-optimal solution for flux. In general, a conventional FCS-MPTC algorithm has a large computational burden due to a large number of available voltage vectors as in multi-level inverters or discrete SVM MPC technique and B) the presence of weighting factors in the cost function with several control objectives. This thesis developed a two-stage optimization-based FCS-MPC algorithm for an induction motor drive that uses reduced voltage control sets (RVCS) to evaluate the pre-defined cost function in the prediction loop. The proposed algorithm lowers the computational burden of the controller in two cascaded stages for a three-level neutral-point clamped voltage source inverter (3L-NPC VSI) fed IM drive. The voltage vector selection in the first stage of the proposed two-stage algorithm is executed using two different approaches: six long voltage vectors in the first stage and three long voltage vectors in the first stage. The first approach presented in Chapter 3 evaluates all six long voltage vectors in the first stage to obtain a long optimal voltage vector. In the second stage, the nearest 11 voltage vectors of the optimal long voltage vector are evaluated to reach the final optimal voltage vector. The cost function values from both stages are compared to each other to select the final optimal voltage vector. In chapter 4, the second approach evaluates three long voltage vectors instead of 6 in the first stage. It uses the sign of stator flux deviation and the position of stator flux to select the voltage vectors in the first stage. The second stage of this flux-error based approach is the same as Chapter 3. The proposed FCS-MPC two-stage optimization algorithms of Chapter 3-5 for an IM drive reduces the computational burden of the conventional algorithm which evaluates all available 27 voltage vectors. In Chapter 3, the proposed algorithm’s cost function combined torque and flux as prime control variables along with some other constraints i.e. neutral point voltage balance, switching frequency reduction, over current protection. This proposed algorithm reduced the computational burden of the conventional algorithm from 27 to 17 but suffered from sub-optimal decisions. The sub-optimality occurs when the reduced control set of the voltage vectors does not include the optimal vector and therefore the algorithm does not select the same voltage vector as the conventional all voltage vector-based MPC would have selected. The issue from Chapters 3 and 4 led to the development of a low-complexity finite control set model predictive flux control (FCS-MPFC) algorithm presented in Chapter 5 which eliminates the flux weighting factor in the cost function. A reference stator flux vector calculator (RSFVC) with an inner proportional-integral torque regulator is incorporated into the proposed two-stage optimization-based algorithm to convert the torque and flux amplitude references into an equivalent stator flux reference vector. The stator flux reference vector is then included in the cost function for flux deviation calculation. This allowed the elimination of flux weighting factor and elimination of flux-error based look-up table used in the first stage (the second approach) and solved the sub-optimality issue of the two-stage optimization-based FCS-MPTC algorithms. The developed FCS-MPFC algorithm experimentally demonstrated that this algorithm can be used for an induction motor drive to accurately control the electromagnetic torque, stator flux, and NPV with zero sub-optimal decision and reduced computation burden. Hence the original performance of the conventional FCS-MPFC is preserved with a lower computational burden. Operation with conventional FCS-MPC results in high torque and flux ripples in a 2-level inverter drive. Recently, MPC algorithms with integrated discrete space vector modulation (DSVM) for two-level inverter fed drives have been reported. DSVM technique creates a number of virtual voltage vectors using the real available voltage vectors. This allows for better optimization and minimization of the torque and flux errors/ripple due to the virtually created sub-divisions in the space vector region. However, all conventional DSVM integrated with FCS-MPTC yields a very high and complex calculation load. Therefore, these techniques are not practical for a cost-effective drive system. This thesis also proposes a simplified DSVM based FCS-MPFC algorithm for the two-level inverter (2L-VSI) fed IM drive system. This technique applies optimal voltage vectors that yield the desired zero sub-optimality and therefore the performance of the original DSVM-algorithm is retained. Lastly, a switching frequency reduction technique for the developed DSVM based FCS-MPFC is proposed which generates a minimum number of switching transitions by an effective synthesis of the virtual voltage vectors. This reduced switching is effective in any operating region for the 2L-VSI fed IM drive. This achieves low switching frequency of the inverter while preserving the same performances of the drive as the conventional algorithm. All the proposed methods which achieved zero sub-optimality have been experimentally tested in parallel with their conventional forms to detect the sub-optimality and compared with recently published algorithms to demonstrate their superior performances in a wide operating range.

The main focus of this present work is to study how mechanical steady state traveling waves can be generated and propagated through one dimensional media by applying forces. By steady state traveling waves we refer to propagating mechanical waves in a finite medium that never exhibit reflections at the boundaries and continuously move from one end of the structure to the other. Mechanical waves can be classified as traveling, standing and hybrid waves that are the results of the interplay of excitation forces, applied force locations, and the boundary conditions of the structure. Traveling waves carry energy through a defined medium while standing waves keep energy at certain areas that are associated with the modes of excitation. To understand the interaction of systems that exhibit traveling waves with their surrounding media (i.e., swimming flagella, manta ray locomotion), it is crucial to first understand the wave propagation and what is desired in these structural systems. The parameters that affect the generation and propagation of waves should be welldefined to control and manipulate the desired system’s response. One-dimensional string and rod equations are studied with various boundary conditions to generate steady-state traveling waves in a string and longitudinal traveling waves in a rod. Two excitation forces are applied to a string and a rod near the boundaries to understand the generation and propagation of traveling and standing waves at various frequencies. The work examines the quality of the wave propagation in a string, and in a rod. A cost function approach is applied to identify the quality of such waves. Furthermore, steady-state square traveling waves are generated in a string and in-plane in a rod, both theoretically and experimentally. To the authors’ knowledge this is the first time this has been attempted in the literature. Determining the quality of traveling waves and understanding the parameters on the wave propagation of a string and rod can lead to further understand and leverage various engineering disciplines such as mechanical actuation mechanisms, propulsion of flagella, and the basilar membrane in the ear’s cochlea.

Robotic lower limb prostheses can improve the quality of life for amputees. Development of such devices, currently dominated by long prototyping periods, could be sped up by predictive simulations. In contrast to some amputee simulations, which track experimentally determined non-amputee walking kinematics, we can instead explicitly model the human-prosthesis interaction to produce a prediction of the user's walking kinematics. To accomplish this, we use large-scale trajectory optimization on a muscle-driven multi-body model of an amputee with a robotic prosthesis to obtain metabolic energy-minimizing walking gaits. While this computational framework can be applied to a wide range of passive or biomechatronic prosthetic, exoskeletal, and assistive devices, here, we focus on unilateral ankle-foot prostheses. We use this optimization to determine optimized prosthesis controllers by minimizing a weighted sum of human metabolic and prosthesis costs and develop Pareto optimal curves between human metabolic and prosthesis cost with various prostheses masses and at various speeds. We also use this optimization to obtain trends in the energetics and kinematics for various net prosthesis work rates produced by given prosthesis feedback controllers. We find that the net metabolic rate has a roughly quadratic relationship with the net prosthesis work rate. This simulation predicts that metabolic rate could be reduced below that of a non-amputee, although such gaits are highly asymmetric and not seen in experiments with amputees. Walking simulations with bilateral symmetry in kinematics or ground reaction forces have higher metabolic rates than asymmetric gaits, suggesting a potential reason for asymmetries in amputee walking. Our findings suggest that a computational framework such as one presented here could augment the experimental approaches to prosthesis design iterations, although quantitatively accurate predictions of experiments from simulation remains an open problem. We run a series of optimizations to examine additional objective functions, which may improve the prediction. These objective functions include mechanical muscle costs and socket interaction costs. Finally, we consider a simple point-mass model of a unilateral amputee, finding that the point-mass models make broad qualitative predictions similar to those of the complex model: as the prosthesis produces more net work, the metabolic cost to the person is reduced and the bilateral asymmetry of the gait increases; favoring the affected side.

On February 8, 1996, the U.S. Congress enacted the Telecommunications Act of 1996 to promote competition and reduce regulation in local telephone service, in order to secure lower prices and higher quality of service. The promise of competition in local service, however, has largely remained unfulfilled, and there is still much unknown about the cost structure of the telephone industry at the local level. The question of whether natural monopoly characteristics have ever existed has never been fully answered, and is still a valid one. The purpose of this research is to expand earlier empirical research on telephone costs by accounting for the site-specific (hedonic) characteristics of the service territory. In past research, these characteristics have been approximated by population density, customer density, or service territory area. In order to achieve a better understanding of the cost structure of the telephone industry, both total cost and disaggregate capital investment cost functions have been developed and econometrically estimated while including site-specific physical and geographical characteristics, such as (1) soil, (2) slope, (3) environmental quality, (4) service territory size, (5) street pattern, (6) population density, (7) land uses, and (8) exchange proximity.A translog total cost function is estimated at the company level, using 1980 data for 41 telephone companies operating within the state of New York. Disaggregate capital investment cost functions are estimated at the local exchange level for five different plant components: central office equipment, buildings, cables, pole lines, and underground conduits. A separate equation for the share of underground investment in total cable (underground and overhead) investment is also estimated. The size of the samples used in these estimations varies between 65 and 615, depending upon the data availability for the selected variables. Additive, log-log, and Box-Cox functional forms are considered, and the optimal functional form is selected using log-likelihood ratio tests.The empirical results show that site-specific factors (1) are significant determinants of costs in the telephone industry, and (2) provide substantial advantages (or disadvantages) to telephone companies regarding the output levels where economies of scale or density are exhausted. The results also show that significant cost savings and scale economies are achieved by expansion through densification (fixed service territory size). At the company sample mean, economies of scale are exhausted for a market of 61,543 telephone units. Economies of density, however, are exhausted at 610,836 units, a much higher level. These thresholds vary with the site-specific cost factors. Among the plant components, the strongest economies of density are achieved in pole (ε=0.60) and cable (ε=0.74) capital investments, followed by central office equipment (ε=0.92) and underground conduit (ε=0.90) investments. The only component where the exchanges experience diseconomies of density is buildings (ε=1.12). The total economies of density are 0.83 at the sample mean, implying that monopoly is the optimal form of production.It is concluded that monopoly is the optimal form of production for predominantly small to mid-size markets, when the expansion is achieved through densification, which implies a need for mergers among existing small and mid-size companies and exchanges. However, for larger markets and for newly emerging service areas, competitive entry to the market should be encouraged.

The New Zealand Department of Conservation has so far classified 2,373 species and subspecies of those so far assessed as being threatened with extinction. Annual expenditure for management services for protected species and island habitats was NZ$35.8 million in 2001/02. Some threatened species programmes, however, require far more funding than other programmes for species that are also at risk of extinction. Until now, the contribution of economics to threatened species conservation has focused on areas such as the value of threatened species and the opportunity costs of threatened species conservation in terms of economic development, and not the costs of management for threatened species. The aim of this research is to improve the formal understanding of the management costs by investigating the specific form of the cost function for threatened species programmes. The cost function is based on Swanson (1994) and describes the Present Value (PV) of the cost of a threatened species programme as a factor, inter alia, in a cost-benefit ranking criterion, which conceptualises threatened species conservation as a dynamic optimisation problem. It is proposed that the cost of a programme in a single time period is determined by the costs of the base natural resources and the management services needed to maximise the conservation of a threatened species; and that the cost of a programme over time is determined by the costs in each time period and a species' extant population and recovery rate, which together act as a controlling mechanism on these costs. To investigate the specific form of the cost function, this research conducted a cross-case analysis of the costs of New Zealand threatened species programmes. Cost data was collected and analysed from a survey of the Department of Conservation's Recovery Group Leaders for eleven programmes from 2003 until 2012 and used to test hypotheses developed from the theorised characteristics of the cost function. Although the results of the cross-case analysis are subject to uncertainty, habitat area and a species' taxon are identified as two factors that determine the specific costs of New Zealand threatened species programmes. The results also indicate that many threatened species programmes receive minimal or partial funding and, as a consequence, the conservation of species may be delayed, which could increase the risk of further decline, or even extinction, of species and the total cost of the programme. It is recommended that estimates of costs are included in recovery plans, cost-effectiveness analysis of threatened species programmes is conducted, cost and a species' possible recovery rate are included as factors in priority ranking systems, and the costs of threatened species programmes are used in funding applications for threatened species conservation.