In this thesis a mathematical description and analysis of the Cumulative Prospect Theory is presented. Conditions that ensure well-posedness of the problem are provided, as well as existence results concerning optimal policies for discrete-time incomplete market models and for a family of diffusion market models. A brief outline of how this work is organised follows. In Chapter 2 important results on weak convergence and discrete time finance models are described, these facts form the main background to introduce in Chapter 3 the problem of optimal investment under the CPT theorem in a discrete time setting. We describe our model, present some assumptions and main results are derived. The second part of this work comprises the description of the martingale problem formulation of diffusion processes in Chapter 4. A key result on the limits and topological properties of the set of laws of a class of Itô processes is described in Chapter 5. Finally, we introduce a factor model that includes a class of stochastic volatility models, possibly with path-depending coefficients. Under this model, the problem of optimal investment with a behavioural investor is analysed and our main results on well-posedness and existence of optimal strategies are described under the framework of weak solutions. Further research and challenges when applying the techniques developed in this work are described.

The majority of the existing photovoltaic (PV) systems are dimensioned in such a way that no or only a small part of the production exceeds the buildings internal consumption. This is done because sold electricity to the grid has a lower economic value than if used internally in the building. Therefore commercial buildings, with high consumption during sunny hours, are to prefer when installing PV. Implementing a battery energy storage system in these facilities can lead to higher self consumption of the PV energy and reduced electricity bills. To take full advantage of this potential it requires optimal management of the battery. In this study an optimized battery algorithm was developed to show the full potential a perfectly managed battery can have to reduce cost of electricity within commercial buildings. There are three main charge and discharge patterns for a battery which can reduce the cost on the electricity bill: 1) Charge and discharge at different prices, 2) peak shaving and 3) overproduced PV is stored for later use. By utilizing a battery in an optimal way to reduce the costs as much as possible, batteries will reach break even at battery prices between 1350-2100 SEK/kWh, depending on which scenario evaluated. By implementing forecast based desicion-making, in which the battery operation is optimized with respect to PV and consumption forecasts, the system profitability declines rapidly, especially when using a consumption forecast. A real system would probably profit from basing the battery operation on both forecasts and real time measurements.

Le problème de filtrage consiste à estimer l'état d'un système dynamique, appelé signal qui est souvent un processus markovien, à partir d'observation bruitées des états passés du système. Dans ce mémoire, nous considérons un modèle de filtrage en temps continu pour le processus de diffusion. Le but est d'étudier la stabilité du filtre optimal par rapport à sa condition initiale au-delà de l'hypothèse de mélange (fort) pour le noyau de transition en ignorant l'ergodicité du signal / The filtering problem consists of estimating the state of a dynamic, called signal which is often a Markov process, from the noisy observation of the past states. In this thesis, we consider a filtering model in continuous time for the diffusion process. The aim is to study the stability of the optimal filter with respect to its initial condition beyond the mixing (or quasi – mixing) hypothesis for the transition kernel

The objective in this work is to build a rapid and automated numerical design method that makes optimal design of robots possible. In this work, two classes of optimal robot design problems were specifically addressed: (1) When the objective is to optimize a pre-designed robot, and (2) when the goal is to design an optimal robot from scratch. In the first case, to reach the optimum design some of the critical dimensions or specific measures to optimize (design parameters) are varied within an established range. Then the stress is calculated as a function of the design parameter(s), the design parameter(s) that optimizes a pre-determined performance index provides the optimum design. In the second case, this work focuses on the development of an automated procedure for the optimal design of robotic systems. For this purpose, Pro/Engineer© and MatLab© software packages are integrated to draw the robot parts, optimize them, and then re-draw the optimal system parts.

This thesis solves estimation and control problems in computational neuroscience, mathematically dealing with the first-passage times of diffusion stochastic processes. We first derive estimation algorithms for model parameters from first-passage time observations, and then we derive algorithms for the control of first-passage times. Finally, we solve an optimal design problem which combines elements of the first two: we ask how to elicit first-passage times such as to facilitate model estimation based on said first-passage observations. The main mathematical tools used are the Fokker-Planck partial differential equation for evolution of probability densities, the Hamilton-Jacobi-Bellman equation of optimal control and the adjoint optimization principle from optimal control theory. The focus is on developing computational schemes for the solution of the problems. The schemes are implemented and are tested for a wide range of parameters.

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Les métamodèles statistiques sont régulièrement confrontés au manque de données qui engendre des difficultés à estimer les paramètres. Le paradigme bayésien fournit un moyen élégant de contourner le problème en décrivant la connaissance que nous avons des paramètres par une loi de probabilité a posteriori au lieu de la résumer par une estimation ponctuelle. Cependant, ce paradigme nécessite de définir une loi a priori adéquate, ce qui est un exercice difficile en l'absence de jugement d'expert. L'école bayésienne objective propose des priors par défaut dans ce genre de situation telle que le prior de référence de Berger-Bernardo. Un tel prior a été calculé par Berger, De Oliveira and Sansó [2001] pour le modèle de krigeage avec noyau de covariance isotrope. Une extension directe au cas des noyaux anisotropes poserait des problèmes théoriques aussi bien que pratiques car la théorie de Berger-Bernardo ne peut s'appliquer qu'à un jeu de paramètres ordonnés. Or dans ce cas de figure, tout ordre serait nécessairement arbitraire. Nous y substituons une solution bayésienne objective fondée sur les posteriors de référence conditionnels. Cette solution est rendue possible par une théorie du compromis entre lois conditionnelles incompatibles. Nous montrons en outre qu'elle est compatible avec le krigeage trans-gaussien. Elle est appliquée à un cas industriel avec des données non-stationnaires afin de calculer des Probabilités de Détection de défauts (POD de l'anglais Probability Of Detection) par tests non-destructifs dans les tubes de générateur de vapeur de centrales nucléaires. / A recurring problem in surrogate modelling is the scarcity of available data which hinders efforts to estimate model parameters. The Bayesian paradigm offers an elegant way to circumvent the problem by describing knowledge of the parameters by a posterior probability distribution instead of a pointwise estimate. However, it involves defining a prior distribution on the parameter. In the absence of expert opinion, finding an adequate prior can be a trying exercise. The Objective Bayesian school proposes default priors for such can be a trying exercise. The Objective Bayesian school proposes default priors for such situations, like the Berger-Bernardo reference prior. Such a prior was derived by Berger, De Oliveira and Sansó [2001] for the Kriging surrogate model with isotropic covariance kernel. Directly extending it to anisotropic kernels poses theoretical as well as practical problems because the reference prior framework requires ordering the parameters. Any ordering would in this case be arbitrary. Instead, we propose an Objective Bayesian solution for Kriging models with anisotropic covariance kernels based on conditional reference posterior distributions. This solution is made possible by a theory of compromise between incompatible conditional distributions. The work is then shown to be compatible with Trans-Gaussian Kriging. It is applied to an industrial case with nonstationary data in order to derive Probability Of defect Detection (POD) by non-destructive tests in steam generator tubes of nuclear power plants.

En astronautique, une question importante est de contrôler le mouvement d’un satellite soumis à la gravitation des corps célestes de telle sorte que certains indices de performance soient minimisés (ou maximisés). Dans cette thèse, nous nous intéressons à la minimisation de la norme L¹ du contrôle pour le problème circulaire restreint des trois corps. Les conditions nécessaires à l’optimalité sont obtenues en utilisant le principe du maximum de Pontryagin, révélant l’existence de contrôles bang-bang et singuliers. En s’appuyant sur les résultats de Marchal [1] et Zelikin et al. [2], la présence du phénomène de Fuller est mise en évidence par l’analyse des es extrêmales singulières. La contrôlabilité pour le problème à deux corps (un cas dégénéré du problème circulaire restreint des trois corps) avec un contrôle prenant des valeurs dans une boule euclidienne est caractérisée dans le chapitre 2. Le résultat de contrôlabilité est facilement étendu au problème des trois corps puisque le champ de vecteurs correspondant à la dérive est récurrent. En conséquence, si les trajectoires contrôlées admissibles restent dans un compact fixé, l’existence des solutions du problème de minimisation L¹ peut être obtenu par une combinaison du théorème de Filippov (voir [4, chapitre 10]) et une procédure appropriée de convexification (voir [5]). En dimension finie, le problème de minimisation L¹ est bien connu pour générer des solutions où le contrôle s’annule sur certains intervalles de temps. Bien que le principe du maximum de Pontryagin soit un outil puissant pour identifier les solutions candidates pour le problème de minimisation L¹, il ne peut pas garantir que ces candidats sont au moins localement optimaux sauf si certaines conditions d’optimalité suffisantes sont satisfaites. En effet, il est une condition préalable pour établir (et pour être capable de vérifier) les conditions d’optimalité nécessaires et suffisantes pour résoudre le problème de minimisation L¹. Dans cette thèse, l’idée cruciale pour obtenir de telles conditions est de construire une famille paramétrée d’extrémales telle que l’extrémale de référence peut être intégrée dans un champ d’extrémales. Deux conditions de non-pliage pour la projection canonique de la famille paramétrée d’extrémales sont proposées. En ce qui concerne le cas de points terminaux fixés, ces conditions de non-pliage sont suffisantes pour garantir que l’extrémale de référence est localement minimisante tant que chaque point de commutation est régulier (cf. chapitre 3). Si le point terminal n’est pas fixe mais varie sur une sous-variété lisse, une condition suffisante supplémentaire impliquant la géométrie de variété de cible est établie (cf. chapitre 4). Bien que diverses méthodes numériques, y compris celles considérées comme directes [6, 7], indirectes [5, 8], et hybrides [11], dans la littérature sont en mesure de calculer des solutions optimales, nous ne pouvons pas attendre d’un satellite piloté par le contrôle optimal précalculé (ou le contrôle nominal) de se déplacer sur la trajectoire optimale précalculée (ou trajectoire nominale) en raison de perturbations et des erreurs inévitables. Afin d’éviter de recalculer une nouvelle trajectoire optimale une fois que la déviation de la trajectoire nominale s’est produite, le contrôle de rétroaction optimale voisin, qui est probablement l’application pratique la plus importante de la théorie du contrôle optimal [12, Chapitre 5], est obtenu en paramétrant les extrémales voisines autour de la nominale (cf. chapitre 5). Étant donné que la fonction de contrôle optimal est bang-bang, le contrôle optimal voisin comprend non seulement la rétroaction sur la direction de poussée, mais aussi celle sur les instants de commutation. En outre, une analyse géométrique montre qu’il est impossible de construire un contrôle optimal voisin une fois que le point conjugué apparaisse ou bien entre ou bien à des instants de commutation. / In astronautics, an important issue is to control the motion of a satellite subject to the gravitation of celestial bodies in such a way that certain performance indices are minimized (or maximized). In the thesis, we are interested in minimizing the L¹-norm of control for the circular restricted three-body problem. The necessary conditions for optimality are derived by using the Pontryagin maximum principle, revealing the existence of bang-bang and singular controls. Singular extremals are analyzed, and the Fuller phenomenon shows up according to the theories developed by Marchal [1] and Zelikin et al. [2, 3]. The controllability for the controlled two-body problem (a degenerate case of the circular restricted three-body problem) with control taking values in a Euclidean ball is addressed first (cf. Chapter 2). The controllability result is readily extended to the three-body problem since the drift vector field of the three-body problem is recurrent. As a result, if the admissible controlled trajectories remain in a fixed compact set, the existence of the solutions of the L¹-minimizaion problem can be obtained by a combination of Filippov theorem (see [4, Chapter 10], e.g.) and a suitable convexification procedure (see, e.g., [5]). In finite dimensions, the L¹-minimization problem is well-known to generate solutions where the control vanishes on some time intervals. While the Pontryagin maximum principle is a powerful tool to identify candidate solutions for L1-minimization problem, it cannot guarantee that the these candidates are at least locally optimal unless sufficient optimality conditions are satisfied. Indeed, it is a prerequisite to establish (as well as to be able to verify) the necessary and sufficient optimality conditions in order to solve the L¹-minimization problem. In this thesis, the crucial idea for establishing such conditions is to construct a parameterized family of extremals such that the reference extremal can be embedded into a field of extremals. Two no-fold conditions for the canonical projection of the parameterized family of extremals are devised. For the scenario of fixed endpoints, these no-fold conditions are sufficient to guarantee that the reference extremal is locally minimizing provided that each switching point is regular (cf. Chapter 3). If the terminal point is not fixed but varies on a smooth submanifold, an extra sufficient condition involving the geometry of the target manifold is established (cf. Chapter 4). Although various numerical methods, including the ones categorized as direct [6, 7], in- direct [5, 8, 9], and hybrid [10], in the literature are able to compute optimal solutions, one cannot expect a satellite steered by the precomputed optimal control (or nominal control) to move on the precomputed optimal trajectory (or nominal trajectory) due to unavoidable perturbations and errors. In order to avoid recomputing a new optimal trajectory once a deviation from the nominal trajectory occurs, the neighboring optimal feedback control, which is probably the most important practical application of optimal control theory [11, Chapter 5], is derived by parameterizing the neighboring extremals around the nominal one (cf. Chapter 5). Since the optimal control function is bang-bang, the neighboring optimal control consists of not only the feedback on thrust direction but also that on switching times. Moreover, a geometric analysis shows that it is impossible to construct the neighboring optimal control once a conjugate point occurs either between or at switching times.

The Diploma thesis deals with the problem of energy-efficient train control. It presents the basic survey of mathematical models used in the problem of energy-efficient train control, analysis of optimal driving regimes, determining optimal switching times between optimal driving regimes and timetabling of the train. The mathematical formulation of the problem is done using Newton's second law of motion and other known physical laws. To analyse optimal driving regimes and determine the switching times between optimal driving regimes, we apply tools of optimal control theory, particularly Pontryagin's Maximum Principle. The timetabling of the train is discussed from the numerical solution of the settled non-linear programming problem.

Cette thèse comprend deux parties indépendantes.Dans la première partie nous contribuons une nouvelle méthode qui, étant donnée un volume de tolérance, génère un maillage triangulaire surfacique garanti d’être dans le volume de tolérance, sans auto-intersection et topologiquement correct. Un algorithme flexible est conçu pour capturer la topologie et découvrir l’anisotropie dans le volume de tolérance dans le but de générer un maillage de faible complexité.Dans la seconde partie nous contribuons une nouvelle approche pour calculer une fonction de correspondance entre deux surfaces. Tandis que la plupart des approches précédentes procède par composition de correspondance avec un domaine simple planaire, nous calculons une fonction de correspondance en optimisant directement une fonction de sorte à minimiser la variance d’un plan de transport entre les surfaces / This thesis is divided into two independent parts.In the first part, we introduce a method that, given an input tolerance volume, generates a surface triangle mesh guaranteed to be within the tolerance, intersection free and topologically correct. A pliant meshing algorithm is used to capture the topology and discover the anisotropy in the input tolerance volume in order to generate a concise output. We first refine a 3D Delaunay triangulation over the tolerance volume while maintaining a piecewise-linear function on this triangulation, until an isosurface of this function matches the topology sought after. We then embed the isosurface into the 3D triangulation via mutual tessellation, and simplify it while preserving the topology. Our approach extends toDépôt de thèseDonnées complémentairessurfaces with boundaries and to non-manifold surfaces. We demonstrate the versatility and efficacy of our approach on a variety of data sets and tolerance volumes.In the second part we introduce a new approach for creating a homeomorphic map between two discrete surfaces. While most previous approaches compose maps over intermediate domains which result in suboptimal inter-surface mapping, we directly optimize a map by computing a variance-minimizing mass transport plan between two surfaces. This non-linear problem, which amounts to minimizing the Dirichlet energy of both the map and its inverse, is solved using two alternating convex optimization problems in a coarse-to-fine fashion. Computational efficiency is further improved through the use of Sinkhorn iterations (modified to handle minimal regularization and unbalanced transport plans) and diffusion distances. The resulting inter-surface mapping algorithm applies to arbitrary shapes robustly and efficiently, with little to no user interaction.