Dado um espaço localmente compacto Hausdorff K e um espaço de Banach X, Co(K,X) representa o espaço de Banach das funções contínuas em K com valores em X que se anulam no infinito com a norma do supremo. No presente trabalho, unificaremos e melhoraremos várias generalizações do teorema clássico de Banach-Stone para os espaços Co(K,X) devidas a Cambern, Amir, Behrends e Jarosz. No caso em que X=lp com $ 2 p, nossos resultados são maximais. / Let K be a locally compact Hausdor space and X a Banach space. By Co(K,X) we denote the Banach space of all X-valued continuous functions dened on K which vanish at innity, provided with the supremum norm. In the present work, we unify and strengthen several generalizations obtained in recent years of the classical Banach-Stone theorem for Co(K,X) spaces. In the case where X = lp such that 2 p < 1, our results are optimal.

Este trabalho estuda o sistema de controle de um robô móvel, termo que designa veículos sem motorista humano e com capacidade de trafegar por rotas livremente escolhidas. As arquiteturas de controle utilizadas em robôs móveis são analisadas. A arquitetura adotada neste trabalho, de caráter funcional,é apresentada e discutida. O trabalho se concentra nos níveis mais simples de controle, relacionados ao controle de trajetória, cujo objetivo é garantir que o robô móvel percorra uma rota pré-definida. Um controlador de trajetória é proposto e projetado. O controlador resulta da aplicação da teoria de controle ótimo a um modelo de robô móvel em referencial fixo. Uma técnica recente de controle de robôs (técnica de atgs) é empregada para melhorar a robustez do controlador. O desempenho do controlador obtido com uso de atgs é comparado com o desempenho do controlador obtido inicialmente. Com o objetivo de validar esta proposta de controlador de trajetória, resultados fornecidos por simulações são discutidos. A análise do controlador foi respaldada com dados experimentais obtidos junto a um robô móvel, denominado Ariel, desenvolvido no laboratório de automação e sistemas (mecatrônica) da Escola Politécnica da Universidade de São Paulo. / This work studies the Control System of a Mobile Robot, term which refers to vehicles without human driver and with ability to follow arbitrary routes. This work analyses the Control Architectures frequently employed in Mobile Robots. The Architecture here adopted is a functional one, which is presented and described. This work focuses on the simplest levels of Control, those which are mainly related to the Trajectory Control, and whose objective is to guarantee that the Mobile Robot follows a specified route. A Trajectory Controller is proposed and designed. The Controller is based on Optimal Control Theory. A recently developed technique for robot control (called ATGS techinique) is used in order to improve the Controller robustness. Simulation results are discussed in order to validate the proposed Controller. The Controller analysis is tested in a real Mobile Robot (named Ariel) currently developed at Laboratório de Automação e Sistemas (Mecatrônica) , at Escola Politécnica of Universidade de São Paulo.

In this thesis we study a novel type of hydropower generation which uses a Venturi contraction to amplify the pressure drop across a turbine, allow- ing for cost-effective hydropower generation in situations where the head drop is small, such as in rivers and weirs. The efficiency is sensitive to how the secondary flow, which passes through the turbine, mixes with the accelerated primary flow, which is diverted around the turbine, within the confines of a closed geometry. In particular, it is important to understand the behaviour of the turbulent shear layers between the primary and sec- ondary flows, which grow downstream, mixing the flows together. The behaviour of the shear layers in the expanding part of the Venturi con- traction is strongly dependent on the shape of the channel. An important consequence of the channel shape, and hence the flow behaviour, is the degree of pressure amplification across the turbine, which determines the amount of generated power. We focus on mathematically modelling the mixing of the flows in turbu- lent shear layers, and we investigate two different ways to increase pres- sure amplification: optimising the shape of the channel, and using swirl to enhance mixing. The channel shape optimisation reveals an interest- ing balance between the effects of mixing and wall drag. Wide channel expansion tends to accentuate non-uniform flow, causing poor pressure amplification, whilst shallow expansion creates enhanced wall drag, which is also detrimental to pressure amplification. We show how the maximum power is generated with a channel shape that strikes a balance between these two effects. We find that swirl enhances mixing by increasing shear layer growth rates, but it produces large pressure losses in doing so, and for large amounts of swirl a slowly recirculating region can form along the channel centreline. Whilst swirl does not improve efficiency, there may be some inevitable swirl present in the flow, and we show how this affects the optimum channel shape. We also establish the criteria for the existence of such a recirculation region so that it may be avoided.

The Schrödinger equations have had a profound impact on a wide range of fields of modern science, including quantum mechanics, superfluidity, geometrical optics, Bose-Einstein condensates, and the analysis of dispersive phenomena in the theory of PDE. The main purpose of this thesis is to explore two Schrödinger-type equations appearing in the so-called Bohmian formulation of quantum mechanics and in the study of exciton-polariton condensates. For the first topic, the linear Schrödinger equation is the starting point in the formulation of a phase-space model proposed in [1] for the Bohmian interpretation of quantum mechanics. We analyze this model, a nonlinear Vlasov-type equation, as a Hamiltonian system defined on an appropriate Poisson manifold built on Wasserstein spaces, the aim being to establish its existence theory. For this purpose, we employ results from the theory of PDE, optimal transportation, differential geometry and algebraic topology. The second topic of the thesis is the study of a nonlinear Schrödinger equation, called the complex Gross-Pitaevskii equation, appearing in the context of Bose-Einstein condensation of exciton-polaritons. This model can be roughly described as a driven-damped Gross-Pitaevskii equation which shares some similarities with the complex Ginzburg-Landau equation. The difficulties in the analysis of this equation stem from the fact that, unlike the complex Ginzburg-Landau equation, the complex Gross-Pitaevskii equation does not include a viscous dissipation term. Our approach to this equation will be in the framework of numerical computations, using two main tools: collocation methods and numerical continuation for the stationary solutions and a time-splitting spectral method for the dynamics. After performing a linear stability analysis on the computed stationary solutions, we are led to postulate the existence of radially symmetric stationary ground state solutions only for certain values of the parameters in the equation; these parameters represent the “strength” of the driving and damping terms. Moreover, numerical continuation allows us to show, for fixed parameters, the ground and some of the excited state solutions of this equation. Finally, for the values of the parameters that do not produce a stable radially symmetric solution, our dynamical computations show the emergence of rotating vortex lattices.

The main contribution of this thesis is to examine binary options within the British payoff mechanism introduced by Peskir and Samee. This includes British cash-or-nothing put, British asset-or-nothing put, British binary call and American barrier binary options. We assume the geometric Brownian motion model and reduce the optimal stopping problems to free-boundary problems under the Markovian nature of the underlying process. With the help of the local time-space formula on curves, we derive a closed form expression for the arbitrage-free price in terms of the rational exercise boundary and show that the rational exercise boundary itself can be characterised as the unique solution to a non-linear integral equation. We begin by investigating the binary options of American-type which are also called `one-touch' binary options. Then we move on to examine the British binary options. Chapter~2 reviews the existing work on all different types of the binary options and sets the background for the British binary options. We price and analyse the American-type (one-touch) binary options using the risk-neutral probability method. In Chapters~3 ~4 and ~5, we present the British binary options where the holder enjoys the early exercise feature of American binary options whereupon his payoff is the `best prediction' of the European binary options payoff under the hypothesis that the true drift equals a contract drift. Based on the observed price movements, if the option holder finds that the true drift of the stock price is unfavourable then he can substitute it with the contract drift and minimise his losses. The key to the British binary option is the protection feature as not only can the option holder exercise at unfavourable stock price to a substantial reimbursement of the original option price (covering the ability to sell in a liquid option market completely endogenously) but also when the stock price movements are favourable he will generally receive high returns. Chapters~3 and~4 focus on the British binary put options and Chapter~5 on call options. We also analyse the financial meaning of the British binary options and show that with the contract drift properly selected the British binary options become very attractive alternatives to the classic European/American options. Chapter~6 extends the binary options into barrier binary options and discusses the application of the optimal structure without a smooth-fit condition in the option pricing. We first review the existing work for the knock-in options and present the main results from the literature. Then we examine the method in \cite{dai2004knock} in the application to the knock-in binary options. For the American knock-out binary options, the smooth-fit property does not hold when we apply the local time-space formula on curves. We transfer the expectation of the local time term into a computational form under the basic properties of Brownian motion. Using standard arguments based on Markov processes, we analyse the properties of the value function.

This thesis provides three general equilibrium overlapping generations models to analyze the macroeconomic effects of endogenous life expectancy. I find that endogenous life ex- pectancy has substantial effects on the effective discount rate, the demographic structure of the economy and productivity through the health channel, which subsquently affect human and physical capital accumulation, welfare and fiscal policy. In Chapter 1, I study the presence and magnitude of macroeconomic externalities associated with obesity. I argue that focusing solely on the economic costs on health care spending ig- nores the effects of obesity on net social security benefits caused by higher mortality among obese individuals. To estimate the size of this externality, I develop an overlapping gen- erations model with rational choice with respect to food consumption and weight as in Lakdawalla and Philipson (2009), endogeneizing life expectancy, labour productivity and health care costs. The life-time net contributions of the top 30% of the BMI distribution are negative but quantitatively small, despite the fact that the model generates substantial wealth and income inequality, consistent with the observed socioeconomic gradient of obe- sity (Baum and Ruhm, 2009), which results in lower lifetime contributions. Furthermore, I perform two policy experiments (i) eliminating childhood obesity and (ii) eliminating the VAT exemption of food consumption, both resulting in significant welfare gains, with the former eliminating the obesity externality. In Chapter 2, I study the effects of health on optimal taxation, where health affects the level of utility, the probability of survival and productivity. The results suggest that health affects optimal taxation in the Ramsey problem via three channels. First, since health is a stock that naturally deteriorates over time, the optimal level of taxation of medical spending is not constant over the life-cycle. Second, the productivity-enhancing aspect of health affects labour supply decisions over the life-cycle, where it is optimal for the government to use age-dependent labour income taxes to minimize distortions in the labour market. If the government cannot condition health care spending and labour income taxes on age, then a non-zero capital income tax can be implemented to achieve the optimal allocation. Finally, productivity growth in the medical sector which directly or indirectly affects longevity has a heterogeneous effect on each cohort, which in the absence of age-dependent taxation creates an evolutionary path of the optimal capital income taxation. In Chapter 3, I examine the macroeconomic effects of an increase in the retirement age as a response to an ageing population and deteriorating dependency ratios. An increase in retirement age induces agents to increase medical spending. Households invest in their level of health in order to be fit to work for longer, since older agents that are affected by the retirement age reform have a lower level of health and increased working hours lost due to illness. Furthermore, the higher level of health raises life expectancy, partially offsetting the effects of the retirement age reform with respect to dependency ratios.

Neste trabalho apresentamos uma nova classe de robos bipedes com pernas articuladas e um corpo central. O projeto de robo proposto faz uso de contrapesos em cada um de seus links, e apresenta propriedades que simplificam excepcionalmente as equações dinâmicas que regem seu movimento. Prova- mos que o sistema encontrado, sob certas hipóteses, é linearizavel por meio de uma realimentaçao não-linear de seus estados. Resolvemos o problema de otimização do tempo de percurso de uma trajetória predefinida para este robo, admitindo-se limitações em seus atuadores. Projetamos um sistema de controle inspirado no conceito de \"flatness\" a fim de rastrear esta política ótima de percurso da trajetória. Testamos a robustez deste sistema em simulações de alguns exemplos numéricos. / In this thesis we present a new class of biped robots with articulate legs and a torso. The proposed design is constructed by means of applying counter- balances to each link of the leg, and therefore it has some properties that simplifies dramatically the dynamics of the robot. We prove that the result- ing system, under certain assumptions, is exact linearizable by a nonlinear feedback. We describe the solution to the time-optimal tracking problem of a predefined trajectory for this robot, assuming that its actuators have torque limits. We designed a control system inspired on the concept of °flatness\"in order to track this reference optimal trajectory. We evaluated the robustness of such system during the simulations of some numerical examples.

There is an emerging need to develop new techniques for control system design that better address the challenges posed by modern large-scale cyber-physical systems. These systems are often massive networks of interconnected and interoperating subsystems that fuse physical processes, embedded computation, automation technologies, and communication. The resulting problems are dimensionally large, exhibit significant time-varying structural variations during operation, and feature complex dynamics, constraints and objectives across local and global-system scales. These properties are difficult to address using traditional control theoretic methods without substantial loss of performance and robustness. To overcome these limitations, this dissertation presents new concepts and methods for control of modern large-scale variably-structured systems through self-assembling and self-configuring control constructs that allow for fundamental restructuring of the control system’s topology in response to the current system structure. We present the System Component Graph (SCG) formulation as a mathematical framework that generalizes and extends directed graph methods from decentralized control. We present algorithms, methods, and metrics for real-time decentralization and control-structure optimization, utilizing the inclusion principle for addressing interconnected overlapping dynamics and optimal linear-quadratic (LQ) methods for local decentralized subsystem control. Global system control and performance is achieved through a centralized planner that provides continuous real-time optimized trajectories as guidance command inputs to each subsystem. We present the method of Random Subcomplement Trees (RST) for pseudo-optimal real-time trajectory planning of large-scale systems which formalizes and extends the method of rapidly-exploring random trees in a control optimization framework. The RST method defines transformations from the higher-dimension state space into an intermediate lower-dimensional search space, where optimal transitions between subspace states are defined. In the context of this approach, the resulting decentralized topology found within the SCG framework provides the RST subspace definition and requisite transformations, and optimal transitions in the search space are found through forward evaluation of the closed-loop decentralized subsystem dynamics. The methods developed in this thesis are applied to a set of real-world problems spanning various domains and demonstrate the application of these methods from first-principle modeling through control system analysis, design, implementation, and evaluation in experimental tests and simulation.

Este trabalho estuda o sistema de controle de um robô móvel, termo que designa veículos sem motorista humano e com capacidade de trafegar por rotas livremente escolhidas. As arquiteturas de controle utilizadas em robôs móveis são analisadas. A arquitetura adotada neste trabalho, de caráter funcional,é apresentada e discutida. O trabalho se concentra nos níveis mais simples de controle, relacionados ao controle de trajetória, cujo objetivo é garantir que o robô móvel percorra uma rota pré-definida. Um controlador de trajetória é proposto e projetado. O controlador resulta da aplicação da teoria de controle ótimo a um modelo de robô móvel em referencial fixo. Uma técnica recente de controle de robôs (técnica de atgs) é empregada para melhorar a robustez do controlador. O desempenho do controlador obtido com uso de atgs é comparado com o desempenho do controlador obtido inicialmente. Com o objetivo de validar esta proposta de controlador de trajetória, resultados fornecidos por simulações são discutidos. A análise do controlador foi respaldada com dados experimentais obtidos junto a um robô móvel, denominado Ariel, desenvolvido no laboratório de automação e sistemas (mecatrônica) da Escola Politécnica da Universidade de São Paulo. / This work studies the Control System of a Mobile Robot, term which refers to vehicles without human driver and with ability to follow arbitrary routes. This work analyses the Control Architectures frequently employed in Mobile Robots. The Architecture here adopted is a functional one, which is presented and described. This work focuses on the simplest levels of Control, those which are mainly related to the Trajectory Control, and whose objective is to guarantee that the Mobile Robot follows a specified route. A Trajectory Controller is proposed and designed. The Controller is based on Optimal Control Theory. A recently developed technique for robot control (called ATGS techinique) is used in order to improve the Controller robustness. Simulation results are discussed in order to validate the proposed Controller. The Controller analysis is tested in a real Mobile Robot (named Ariel) currently developed at Laboratório de Automação e Sistemas (Mecatrônica) , at Escola Politécnica of Universidade de São Paulo.

Este trabalho apresenta aspectos teóricos e práticos relevantes do desenvolvimento do Sistema de Navegação e Controle (SNC) a ser implementado em um Veículo Submarino Semi-Autônomo (VSSA), tipo não carenado e auto propelido, que está em desenvolvimento e construção na Escola Politécnica da USP, para a Petrobrás. Os três graus de liberdade horizontais são controlados para seguirem trajetórias pré-definidas, enviadas como sinais de referência para navegação por uma estação de apoio localizada na superfície, responsável pela guiagem do veículo. Os sinais de referência enviados são acústicos propagados pela água. A implementação física do SNC e o controle dos três graus de liberdade verticais não fazem parte do escopo deste trabalho. O SNC consiste em um controlador determinístico, um seguidor de trajetórias linear quadrático alimentado por um vetor de estados estimado assintoticamente. Por segurança, em caso de falha de algum sensor, e para filtrar ruídos nos sinais medidos, um estimador de estados de ordem plena é utilizado conjuntamente. Pela simplicidade de síntese e implementação, esta arquitetura de controle é considerada a melhor alternativa para capacitar o VSSA a executar os movimentos semi-autônomos desejados. As técnicas de controle utilizadas requerem a linearização do modelo matemático não-linear que descreve o comportamento dinâmico do veículo. O modelo é obtido de maneira simplificada. Os resultados são gerados por simulações com o modelo não-linear. / This work presents theoretical and practical aspects of the development of the Navigation and Control System (NCS) to be implemented into a Petrobras\' Semi-Autonomous Underwater Vehicle (SAUV), an open-frame and self-propelled type, which is being developed and built at Escola Politécnica of the University of São Paulo (EPUSP). The three horizontal Degrees-of-Freedom (DoF) are controlled so that they can follow a pre-defined trajectory sent as navigation reference signals to the NCS by a support ship, responsible for the guidance of the vehicle and placed on the ocean surface. Reference signals are sent as acoustic signals through the ocean water. The implementation and the control of the three vertical DoF are not in the scope of the present work. The NCS is based upon a deterministic controller, a Linear Quadratic (LQ) trajectory follower fed by an asymptotically estimated state vector, even though all the state variables are available by direct measurents. For safety, if some sensor fails, and for filtering noise on measured signals, a full-order state estimator is also designed. Since the LQ controller architecture is rather simple to design and implement, it was elected to control the SAUV manoeuvers. The control techniques require a linear model of the dynamics of the vehicle. Hence, a linearization procedure is applied to the system of nonlinear differential equations that describe the dynamic behavior of the SAUV. The results presented are provided by computer-aided simulations with the nonlinear model of the plant.