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.

Nesta tese abordamos o problema do hedging de mínima variância de derivativos em mercados incompletos usando a teoria de controle ótimo estocástico com critério quadrático de otimização. Desenvolvemos um modelo geral de apreçamento e hedging de derivativos em mercados incompletos, a tempo discreto, que é capaz de acomodar qualquer tipo de payoff com característica européia que dependa de n ativos de risco. Nesse modelo, o mercado pode apresentar diferentes modos de operação, o que foi formalizado matematicamente por meio de uma cadeia de Markov. Também desenvolvemos um modelo geral de apreçamento e hedging de derivativos em mercados incompletos, a tempo discreto e espaço de estados contínuo, que é capaz de acomodar qualquer tipo de payoff com característica européia que dependa de um ativo de risco cujos retornos sejam representados por um processo de difusão com saltos. Desenvolvemos, ainda, expressões analíticas fechadas para o apreçamento e hedging de uma opção de compra européia vanilla em duas situações: (1) quando os retornos do ativo de risco são representados por um processo de difusão com saltos, e (2) quando os retornos do ativo de risco são representados por um processo de Wiener. Por fim, realizamos simulações numéricas para o controle (hedging) de uma opção de compra européia vanilla quando os retornos do ativo de risco são representados por um processo de Wiener, e comparamos os resultados obtidos com a estratégia de controle derivada do modelo de Black & Scholes. / In this thesis we approach the mean-variance hedging problem of derivatives in incomplete markets employing the theory of stochastic optimal control with quadratic optimization criteria. We developed a general derivatives pricing and hedging model in incomplete markets, in discrete time, capable of accommodating any type of European payoff contingent on n risky assets. In this model, the market may exhibit different operating modes, which were mathematically formalized by means of a Markov chain. We also developed a general derivatives pricing and hedging model in incomplete markets, in discrete time and continuous state space, capable of accommodating any type of European payoff contingent on one risky asset whose returns are described by a jump diffusion process. Even further, we developed closed-form analytical expressions for the pricing and hedging of a European vanilla call option in two situations: (1) when the risky asset returns are described by a jump diffusion process, and (2) when the risky asset returns are described by a Wiener process. Finally, we simulated the control (hedging) of a European vanilla call option when the risky asset returns are described by a Wiener process, and compared the results to those obtained with the control strategy derived from the Black & Scholes model.

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Variable Horizon Model Predictive Control (VH-MPC) is a form of predictive control that includes the horizon length as a decision variable in the constrained optimisation problem solved at each iteration. It has been recently applied to completion problems, where the system state is to be steered to a closed set in finite time. The behaviour of the system once completion has occurred is not considered part of the control problem. This thesis is concerned with three aspects of robustness and optimality in VH-MPC completion problems. In particular, the thesis investigates robustness to well defined but unpredictable changes in system and controller parameters, robustness to bounded disturbances in the presence of certain input parameterisations to reduce computational complexity, and optimal robustness to bounded disturbances using tightened constraints. In the context of linear time invariant systems, new theoretical contributions and algorithms are developed. Firstly, changing dynamics, constraints and control objectives are addressed by introducing the notion of feasible contingencies. A novel algorithm is proposed that introduces extra prediction variables to ensure that anticipated new control objectives are always feasible, under changed system parameters. In addition, a modified constraint tightening formulation is introduced to provide robust completion in the presence of bounded disturbances. Different contingency scenarios are presented and numerical simulations demonstrate the formulation’s efficacy. Next, complexity reduction is considered, using a form of input parameterisation known as move blocking. After introducing a new notation for move blocking, algorithms are presented for designing a move-blocked VH-MPC controller. Constraints are tightened in a novel way for robustness, whilst ensuring that guarantees of recursive feasibility and finite-time completion are preserved. Simulations are used to illustrate the effect of an example blocking scheme on computation time, closed-loop cost, control inputs and state trajectories. Attention is now turned towards mitigating the effect of constraint tightening policies on a VH-MPC controller’s region of attraction. An optimisation problem is formulated to maximise the volume of an inner approximation to the region of attraction, parameterised in terms of the tightening policy. Alternative heuristic approaches are also proposed to deal with high state dimensions. Numerical examples show that the new technique produces substantially improved regions of attraction in comparison to other proposed approaches, and greatly reduces the maximum required prediction horizon length for a given application. Finally, a case study is presented to illustrate the application of the new theory developed in this thesis to a non-trivial example system. A simplified nonlinear surface excavation machine and material model is developed for this purpose. The model is stabilised with an inner-loop controller, following which a VH-MPC controller for autonomous trajectory generation is designed using a discretised, linearised model of the stabilised system. Realistic simulated trajectories are obtained from applying the controller to the stabilised system and incorporating the ideas developed in this thesis. These ideas improve the applicability and computational tractability of VH-MPC, for both traditional applications as well as those that go beyond the realm of vehicle manœuvring.

This thesis introduces a selection of models for optimal execution of financial assets at the tactical level. As opposed to optimal scheduling, which defines a trading schedule for the trader, this thesis investigates how the trader should interact with the order book. If a trader is aggressive he will execute his order using market orders, which will negatively feedback on his execution price through market impact. Alternatively, the models we focus on consider a passive trader who places limit orders into the limit-order book and waits for these orders to be filled by market orders from other traders. We assume these models do not exhibit market impact. However, given we await market orders from other participants to fill our limit orders a new risk is borne: execution risk. We begin with an extension of Guéant et al. (2012b) who through the use of an exponential utility, standard Brownian motion, and an absolute decay parameter were able to cleverly build symmetry into their model which significantly reduced the complexity. Our model consists of geometric Brownian motion (and mean-reverting processes) for the asset price, a proportional control parameter (the additional amount we ask for the asset), and a proportional decay parameter, implying that the symmetry found in Guéant et al. (2012b) no longer exists. This novel combination results in asset-dependent trading strategies, which to our knowledge is a unique concept in this framework of literature. Detailed asymptotic analyses, coupled with advanced numerical techniques (informing the asymptotics) are exploited to extract the relevant dynamics, before looking at further extensions using similar methods. We examine our above mentioned framework, as well as that of Guéant et al. (2012), for a trader who has a basket of correlated assets to liquidate. This leads to a higher-dimensional model which increases the complexity of both numerically solving the problem and asymptotically examining it. The solutions we present are of interest, and comparable with Markowitz portfolio theory. We return to our framework of a single underlying and consider four extensions: a stochastic volatility model which results in an added dimension to the problem, a constrained optimisation problem in which the control has an explicit lower bound, changing the exponential intensity to a power intensity which results in a reformulation as a singular stochastic control problem, and allowing the trader to trade using both market orders and limit orders resulting in a free-boundary problem. We complete the study with an empirical analysis using limit-order book data which contains multiple levels of the book. This involves a novel calibration of the intensity functions which represent the limit-order book, before backtesting and analysing the performance of the strategies.

Localization and motion control (navigation) are two major tasks for a successful mobile robot navigation. The motion controller determines the appropriate action for the robot’s actuator based on its current state in an operating environment. A robot recognizes its environment through some sensors and executes physical actions through actuation mechanisms. However, sensory information is noisy and hence actions generated based on this information may be non-deterministic. Therefore, a mobile robot provides actions to its actuators with a certain degree of uncertainty. Moreover, when no prior knowledge of the environment is available, the problem becomes even more difficult, as the robot has to build a map of its surroundings as it moves to determine the position. Skilled navigation of a differential drive mobile robot (DDMR) requires solving these tasks in conjunction, since they are inter-dependent. Having resolved these tasks, mobile robots can be employed in many contexts in indoor and outdoor environments such as delivering payloads in a dynamic environment, building safety, security, building measurement, research, and driving on highways. This dissertation exploits the use of the emerging Radio Frequency IDentification (RFID) technology for the design and implementation of cost-effective and modular control techniques for navigating a mobile robot in an indoor environment. A successful realization of this process has been addressed with three separate navigation modules. The first module is devoted to the development of an indoor navigation system with a customized RFID reader. This navigation system is mainly pioneered by mounting a multiple antenna RFID reader on the robot and placing the RFID tags in three dimensional workspace, where the tags’ orthogonal position on the ground define the desired positions that the robot is supposed to reach. The robot generates control actions based on the information provided by the RFID reader for it to navigate those pre-defined points. On the contrary, the second and third navigation modules employ custom-made RFID tags (instead of the RFID reader) which are attached at different locations in the navigation environment (on the ceiling of an indoor office, or on posts, for instance). The robot’s controller generates appropriate control actions for it’s actuators based on the information provided by the RFID tags in order to reach target positions or to track pre-defined trajectory in the environment. All three navigation modules were shown to have the ability to guide a mobile robot in a highly reverberant environment with variant degrees of accuracy.

Pseudo-spectral optimal solvers are used to optimize numerically a performance index of a dynamical system with differential constraints. Although these solvers are commonly used for space vehicles and space launchers for trajectory optimization, few experimental papers exist on optimal control of small airships. The objective of this thesis is to evaluate the use of a pseudo-spectral optimal control solver for generating dynamically constrained, minimal time trajectories. A dynamical model of a small airship is presented, with its experimental virtual mass, drag and motor experimentally modeled. The problems are solved in PSOPT, a pseudo-spectral optimal control code. Experimental tests with a small scale model are performed to evaluate the generated paths. Although drift occurs, as a consequence of an open loop control, the vehicle is capable of following the path. This results of this thesis may find uses in verifying how close to optimal discreet path planners are, to plan complex trajectories on short distances, or to generate dynamic maneuverer such as take-off or landing. Ultimately, improving path planning of small airships will improve their safety, maneuverability and flight-time, which makes them fit for scientific monitoring, for search and rescue, or as mobile telecommunications platforms.

Une première partie de cette thèse est dédiée à l’étude de la régularité de la densité de transport sigma dans le problème de Monge entre deux mesures f^+ et f^- sur un domaine Omega. Tout d’abord, on étudie la question de la sommabilité L^p de cette densité de transport entre une mesure f^+ et sa projection sur le bord (P)# f^+, qui ne découle pas en fait des résultats connus (dus à De Pascale - Evans - Pratelli - Santambrogio) sur la densité de transport entre deux densités L^p, comme dans notre cas la mesure cible est singulière. Par une méthode de symétrisation, dès que Omega est convexe ou satisfait une condition de boule uniforme extérieure, nous prouvons les estimations L^p (si f^+ in L^p, alors sigma in L^p). En plus, nous analysons le cas où on paye des coûts supplémentaires g^± sur le bord, en prouvant que la densité de transport est dans L^p dès que f^± in L^p, Omega satisfait une condition de boule uniforme extérieure et, g^± sont lambda^± Lipschitiziens avec lambda^± < 1 et semi-concaves. Ensuite, on s’attaque à la régularité d’ordre supérieur (W^{1,p}, C^{0,alpha}, BV · · ·) de la densité de transport sigma entre deux densités régulières f^+ et f^-. Plus précisément, nous fournissons une famille de contre-exemples à la régularité supérieure: nous prouvons que la régularité W^{1,p} des mesures source et cible, f^+ et f^-, n’implique pas que la densité de transport est W^{1,p}, de même pour la régularité BV, et même f^± in C^infty n’implique pas que sigma est dans W^{1,p}, pour p grand. Ensuite, nous étudions la sommabilité L^p de la densité de transport entre deux mesures f^+ et f^- concentrées sur le bord. Plus précisément, nous prouvons que si f^+ et f^- sont dans L^p(partialOmega), alors la densité de transport sigma entre eux est dans L^p(Omega) dès que Omega est uniformément convexe et p leq 2; de plus, nous introduisons un contre-exemple montrant que ce résultat n’est plus vrai si p > 2. Cela fournit des résultats de régularité W^{1,p} sur la solution u du problème de gradient minimal avec donnée au bord g dans des domaines uniformément convexes (si g in W^{1,p}(partialOmega) alors u in W^{1,p}(Omega)).Dans une deuxième partie, nous étudions un problème de contrôle optimal motivé par un modèle de jeux à champ moyen. D’abord, nous montrons des résultats de différentiabilité et semi-concavité sur la fonction valeur associée au problème de contrôle (le résultat de semi-concavité est optimal en ce qui concerne les hypothèses sur la régularité en temps). Ensuite, nous démontrons que la densité des agents rho_t, dans le modèle MFG considéré, est dans L^p dès que la densité initiale rho_0 in L^p. En plus, nous arrivons à prouver l’existence d’un équilibre pour le problème MFG considéré dans un cas où la dynamique n’est pas régulière.Dernièrement, nous considérons le problème stationnaire associé au problème MFG. Nous montrons que la densité d’équilibre n’est rien d’autre que la densité de transport entre une densité source f et sa projection sur le bord en utilisant une métrique Riemannienne non-uniforme comme coût de transport. Cela nous permet de démontrer que la densité d’équilibre rho est dans L^p dès que la densité source f in L^p. Par conséquent, nous arrivons à prouver aussi l’existence d’un équilibre stationnaire dans un cas où la dynamique n’est pas régulière. / A first part of this thesis is dedicated to the study of the regularity of the transport density sigma in the Monge problem between two measures f^+ and f^- on a domain Omega. First, we study the question of L^p summability of this transport density between a measure f^+ and its projection on the boundary (P)# f^+, which does not actually follow from the known results (due to De Pascale, Evans, Pratelli, Santambrogio) on the transport density between two L^p densities, as in our case the target measure is singular. By a symmetrization trick, if Omega is convex or satisfies a uniform exterior ball condition, we prove the L^p estimates (if f^+ in L^p, then sigma in L^p). In addition, we analyze the case where we pay additional costs g^± on the boundary, proving that the transport density sigma is in L^p as soon as f^± in L^p, Omega satisfies a uniform exterior ball condition and, g^± are lambda^± Lip with lambda^± < 1 and semi-concave. Then we attack the higher-order regularity (W^{1,p}, C^{0,alpha}, BV · · ·) of the transport density sigma between two regular densities f^+ and f^-. More precisely, we provide a family of counter-examples to the higher regularity: we prove that the W^{1,p} regularity of the source and target measures, f^+ and f^-, does not imply that the transport density is in W^{1,p}, the same for the BV regularity, and even f^± in C^infty does not imply that sigma is in W^{1,p}, for large p. Next, we study the L^p summability of the transport density between two measures, f^+ and f^-, concentrated on the boundary. More precisely, we prove that if f^+ and f^- are in L^p(partialOmega), then the transport density sigma between them is in L^p(Omega) as soon as Omega is uniformly convex and p leq 2; moreover, we introduce a counter-example showing that this result is no longer true if p > 2. This provides W^{1,p} regularity results on the solution u of the least gradient problem with boundary datum g in uniformly convex domains (if g in W^{1,p}(partialOmega) then u in W^{1,p}(Omega)).In a second part, we study an optimal control problem motivated by a model of mean field games. First, we show differentiability and semi-concavity results on the value function associated with the control problem (the semi-concavity result will be sharp in regards to the hypotheses on the regularity in time). Then we show that the density of agents rho_t, in the considered MFG model, is in L^p as soon as the initial density rho_0 in L^p. In addition, we prove existence of an equilibrium for the considered MFG problem in a case where the dynamic is non-regular.Lastly, we consider the stationary problem associated with the MFG model. We show that the equilibrium density is nothing but the transport density between a source density f and its projection on the boundary using a non-uniform Riemannian metric as a transport cost. This allows us to show that the equilibrium density rho is in L^p as soon as the source density f in L^p. Therefore, we also prove existence of a stationary equilibrium in a case where the dynamic is non-regular.

Managing large-scale dynamical systems (e.g., transportation systems, complex information systems, and power networks, etc.) in real-time is very challenging considering their complicated system dynamics, intricate network interactions, large scale, and especially the existence of various uncertainties. To address this issue, intelligent techniques which can quickly design decision-making strategies that are robust to uncertainties are needed. This dissertation aims to conquer these challenges by exploring a data-driven decision-making framework, which leverages big-data techniques and scalable uncertainty evaluation approaches to quickly solve optimal control problems. In particular, following techniques have been developed along this direction: 1) system modeling approaches to simplify the system analysis and design procedures for multiple applications; 2) effective simulation and analytical based approaches to efficiently evaluate system performance and design control strategies under uncertainty; and 3) big-data techniques that allow some computations of control strategies to be completed offline. These techniques and tools for analysis, design and control contribute to a wide range of applications including air traffic flow management, complex information systems, and airborne networks.

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