L'évolution importante des réseaux sans fil tend à fournir les supports nécessaires aux applications ubiquitaires émergentes dans les réseaux Mesh sans fil. Les réseaux mesh comprennent des nœuds stationnaires qui remplissent la fonction de routage et appelés routeurs Mesh sans fil (WMR) et qui consitutent le réseau backbone sans fil (WBMN) ainsi que des nœuds clients mesh sans fil (WMCs). Alors que les WMCs sont limités en termes de fonctions et de resources radio, les WMRS fournissent des fonctions de pont et de passerelle afin de connecter les réseaux WMNs aux autres réseaux comme les réseaux cellulaires, les réseaux IEEE 802.11, les réseaux IEEE 802.16, ou tout simplement à Internet. Par conséquent, les réseaux WMRs sont construits à partir sur la base d'équipement radio de communication rapide et/ou multi-radio et multi-canaux. Les routeurs WMRs sont supposes être auto-organisés, auto-configurés et constituant un réseau WNMN robuste ce qui nécessite de soutenir des volumes importants de trafic de données et sur de longues périodes. Cependant, répondre à attentes élevées en termes de services nécessite le développement d'approches décentralisés pour le control dynamique des puissances de transmission (DTPC). La présente thèse se focalise sur le problème DTPC pour les deux cas de réseaux; utilisant un canal de communication et multicanaux. Pour les réseaux exploitant un seul canal, le problème est formulé en termes de problème d'optimisation où l'objectif est de minimiser en même temps des critères convexes associés aux liens et aux réseaux. Afin de résoudre ce problème, des modèles et des algorithmes, appelés MATA (multiple access transmission aware), ont été proposés. Pour les réseaux WBMNs utilisant des systèmes multi-radio et multi-canaux (MRMC), le réseau est modélisé par un ensemble de graphes appelés UCGs (unified channel graphs), chacun consistant les utilisateurs connectés au point d'accès en utilisant le même canal fréquentiel. Pour chaque ensemble UCG, le problème à résoudre un problème quadratique et stochastique soumis aux contraintes des états des liens dynamiques LSI (Link State Information) de tous les UCGs. Le protocole PMMUP, mutli-radio et minimisant la consummation énegétique, est propose au niveau de la couche liaison. Algorithmes d'estimation prédictive base sur ce protocol sont proposes pour résoudre les problèmes d'optimisation associés aux UGCs. Les problèmes énergétiques, les instabilités des queues et les interférences, sont formulés en termes de problèmes de commande optimale couplée, appelés SPWC (singularly-perturbed weakly-coupled). Pour résoudre les problèmes SPWC caractérisant le problème de commande optimal des energies de transmission, l'algorithme HORA (aeneralized higher-order recursive algorithm) qui permet d'obtenir les solutions stables pour les équations de Riccati a été développé. Les performes des modèles et algorithmes proposés dans le cadre de la présente thèse ont été évalués tant sur le plan théorique qu'en simulation. Différentes simulations ont été effectuées sur un large ensemble de topologies réseaux générés aléatoirement. Les résultats de simulation et analytiques Simulation confirment l'efficacité des algorithmes proposés par rapport à la majorité des techniques existantes / The remarkable evolution of wireless networks into the next generation to provide ubiquitous and seamless broadband applications has recently triggered the emergence of Wireless Mesh Networks (WMNs). The WMNs comprise stationary Wireless Mesh Routers (WMRs) forming Wireless Backbone Mesh Networks (WBMNs) and mobile Wireless Mesh Clients (WMCs) forming the WMN access. While WMCs are limited in function and radio resources, the WMRs are expected to support heavy duty applications : that is, WMRs have gateway and bridge functions to integrate WMNs with other networks such as the Internet, cellular, IEEE 802.11, IEEE 802.15, IEEE 802.16, sensor networks, et cetera. Consequently, WMRs are constructed from fast switching radios or multiple radio devices operating on multiple frequency channels. WMRs are expected to be self-organized, self-configured and constitute a reliable and robust WBMN which needs to sustain high traffic volumes and long “online” time. However, meeting such stringent service expectations requires the development of decentralized dynamic transmission power control (DTPC) approaches. This thesis addresses the DTPC problem for both single and multiple channel WBMNs. For single channel networks, the problem is formulated as the minimization of both the link-centric and network-centric convex cost function. In order to solve this issue, multiple access transmission aware (MATA) models and algorithms are proposed. For multi-radio multi-channel (MRMC) WBMNs, the network is modelled as sets of unified channel graphs (UCGs), each consisting of interconnected active network users communicating on the same frequency channel. For each UCG set, the minimization of stochastic quadratic cost functions are developed subject to the dynamic Link State Information (LSI) equations from all UCGs. An energy-efficient multi-radio unification protocol (PMMUP) is then suggested at the Link-Layer (LL). Predictive estimation algorithms based on this protocol are proposed to solve such objective functions. To address transmission energy and packet instabilities, and interference across multiple channels, singularly-perturbed weakly-coupled (SPWC) control problems are formulated. In order to solve the SPWC transmission power control problem, a generalized higher-order recursive algorithm (HORA) that obtains the Riccati Stabilizing Solutions to the control problem is developed. The performance behaviours of the proposed models and algorithms are evaluated both analytically and through computer simulations. Several simulations are performed on a large number of randomly generated topologies. Simulation and analytical results confirm the efficacy of the proposed algorithms compared to the most recently studied techniques

Este trabalho tem por objetivo propor um método eficiente para avaliar gestão de estoque, aplicando conjuntamente a Teoria de Controle Ótimo (OCT), para obter funções de estocagem dinamicamente ótimas, e Análise Envoltória de Dados (DEA), para calcular as eficiências relativas. Tendo em vista esse objetivo foi desenvolvido um modelo integrado DEA-OCT para calcular a eficiência de custo otimizada ao longo do tempo, quando o sistema possui variáveis relacionadas entre si, como no caso de sistemas de controle de estoque, e para analisar produção e demanda (assim como a variável estoque, oriunda dessa relação), estendendo o modelo variacional. Este trabalho aplica o modelo proposto a 647 empresas das Américas do Sul e do Norte, depois faz uma comparação entre Brasil e Chile (países emergentes economicamente), posteriormente focando no setor de comércio, considerando seus sistemas produção-estoque com dados de variáveis contábeis. Os modelos minimizam os custos de produção e de estoque para calcular a eficiência de custo ao longo do tempo. O output (produto, ou variável de saída) é a demanda; o input (insumo, ou variável de entrada) é a produção, e o intermediate (variável intermediária) é o estoque. Seus custos são considerados na função objetivo. É acrescentada uma restrição variacional da OCT para descrever a relação entre demanda, produção e estoque. Em resumo, o modelo é relevante por calcular eficiência prevenindo a possibilidade de obter uma projeção que ignora a relação entre as variáveis, uma vez que essa relação sempre ocorre, na prática, em sistemas de controle de estoque. As principais contribuições são: possibilitar o uso de OCT como a ferramenta de benchmarking DEA no contexto de eficiência dinâmica, estender o modelo DEA variacional de Sengupta (1995), incluindo restrições de modelos mais recentes e possibilitar o cálculo de eficiência quando há relação entre as variáveis. / This work aims to propose an efficient method to evaluate inventory management, jointly applying optimal control theory (OCT), obtaining dynamically optimal production and inventory functions, and data envelopment analysis (DEA), calculating the relative efficiencies. With this objective in mind, it was developed a DEA-OCT integrated model to calculate allocative efficiency optimized over time, when systems have variable with relationship among themselves, like in the case of inventory control systems, and for analyzing production and demand (as the inventory variable obtained from this relationship), extending the variational model. This paper applies the proposed model to 647 companies from South and North America, after that it was made a comparison between Brazil and Chile (economically emerging countries), then focusing on the commercial sector, considering its production-inventory systems and data from accounting variables. The model minimizes the inventory and production costs to calculate the allocative efficiency over time. The output is demand; the input is production, and the intermediate variable is inventory. Their costs are considered in the objective function. A variational constraint OCT is added to describe the relationship among demand, production, and inventory. In summary, the model is relevant to calculate efficiency by preventing the possibility of finding a projection that ignores the relationship among variables, since this relationship always occur in practice in inventory control systems. The main contributions are: using OCT as the benchmarking tool DEA in the context of dynamic efficiency, extending the Sengupta (1995) variational DEA model, including constraints from recent model and making it possible to calculate efficiency when there is a relationship among variables.

Submitted by Maria Aparecida (cidazen@gmail.com) on 2017-09-04T13:42:58Z No. of bitstreams: 1 Watson Robert.pdf: 2699368 bytes, checksum: cf204eec3df50b251f4adbbbd380ffd0 (MD5) / Made available in DSpace on 2017-09-04T13:42:58Z (GMT). No. of bitstreams: 1 Watson Robert.pdf: 2699368 bytes, checksum: cf204eec3df50b251f4adbbbd380ffd0 (MD5) Previous issue date: 2014-08-21 / Due to the demand for high-performance equipments and the rising cost of energy, the industrial sector is developing equipments to attend minimization of the theirs operational costs. The implementation of these requirements generate a demand for projects and implementations of high-performance control systems. The optimal control theory is an alternative to solve this problem, because in its design considers the normative specifications of the system design, as well as those that are related to the operational costs. Motivated by these perspectives, it is presented the study of methods and the development of algorithms to the approximated solution of the Equation Hamilton-Jacobi-Bellman, in the form of discrete Riccati equation, model free and dependent of the dynamic system. The proposed solutions are developed in the context of adaptive dynamic programming that are based on the methods for online design of optimal control systems, Discrete Linear Quadratic Regulator type. The proposed approach is evaluated in multivariable models of the dynamic systems to evaluate the perspectives of the optimal control law for online implementations. / Devido a demanda por equipamentos de alto desempenho e o custo crescente da energia, o setor industrial desenvolve equipamentos que atendem a minimização dos seus custos operacionais. A implantação destas exigências geram uma demanda por projetos e implementações de sistemas de controle de alto desempenho. A teoria de controle ótimo é uma alternativa para solucionar este problema, porque considera no seu projeto as especificações normativas de projeto do sistema, como também as relativas aos seus custos operacionais. Motivado por estas perspectivas, apresenta-se o estudo de métodos e o desenvolvimento de algoritmos para solução aproximada da Equação Hamilton-Jacobi-Bellman, do tipo Equação Discreta de Riccati, livre e dependente de modelo do sistema dinâmico. As soluções propostas são desenvolvidas no contexto de programação dinâmica adaptativa (ADP) que baseiam-se nos métodos para o projeto on-line de Controladores Ótimos, do tipo Regulador Linear Quadrático Discreto. A abordagem proposta é avaliada em modelos de sistemas dinâmicos multivariáveis, tendo em vista a implementação on-line de leis de controle ótimo.

Este trabalho tem por objetivo propor um método eficiente para avaliar gestão de estoque, aplicando conjuntamente a Teoria de Controle Ótimo (OCT), para obter funções de estocagem dinamicamente ótimas, e Análise Envoltória de Dados (DEA), para calcular as eficiências relativas. Tendo em vista esse objetivo foi desenvolvido um modelo integrado DEA-OCT para calcular a eficiência de custo otimizada ao longo do tempo, quando o sistema possui variáveis relacionadas entre si, como no caso de sistemas de controle de estoque, e para analisar produção e demanda (assim como a variável estoque, oriunda dessa relação), estendendo o modelo variacional. Este trabalho aplica o modelo proposto a 647 empresas das Américas do Sul e do Norte, depois faz uma comparação entre Brasil e Chile (países emergentes economicamente), posteriormente focando no setor de comércio, considerando seus sistemas produção-estoque com dados de variáveis contábeis. Os modelos minimizam os custos de produção e de estoque para calcular a eficiência de custo ao longo do tempo. O output (produto, ou variável de saída) é a demanda; o input (insumo, ou variável de entrada) é a produção, e o intermediate (variável intermediária) é o estoque. Seus custos são considerados na função objetivo. É acrescentada uma restrição variacional da OCT para descrever a relação entre demanda, produção e estoque. Em resumo, o modelo é relevante por calcular eficiência prevenindo a possibilidade de obter uma projeção que ignora a relação entre as variáveis, uma vez que essa relação sempre ocorre, na prática, em sistemas de controle de estoque. As principais contribuições são: possibilitar o uso de OCT como a ferramenta de benchmarking DEA no contexto de eficiência dinâmica, estender o modelo DEA variacional de Sengupta (1995), incluindo restrições de modelos mais recentes e possibilitar o cálculo de eficiência quando há relação entre as variáveis. / This work aims to propose an efficient method to evaluate inventory management, jointly applying optimal control theory (OCT), obtaining dynamically optimal production and inventory functions, and data envelopment analysis (DEA), calculating the relative efficiencies. With this objective in mind, it was developed a DEA-OCT integrated model to calculate allocative efficiency optimized over time, when systems have variable with relationship among themselves, like in the case of inventory control systems, and for analyzing production and demand (as the inventory variable obtained from this relationship), extending the variational model. This paper applies the proposed model to 647 companies from South and North America, after that it was made a comparison between Brazil and Chile (economically emerging countries), then focusing on the commercial sector, considering its production-inventory systems and data from accounting variables. The model minimizes the inventory and production costs to calculate the allocative efficiency over time. The output is demand; the input is production, and the intermediate variable is inventory. Their costs are considered in the objective function. A variational constraint OCT is added to describe the relationship among demand, production, and inventory. In summary, the model is relevant to calculate efficiency by preventing the possibility of finding a projection that ignores the relationship among variables, since this relationship always occur in practice in inventory control systems. The main contributions are: using OCT as the benchmarking tool DEA in the context of dynamic efficiency, extending the Sengupta (1995) variational DEA model, including constraints from recent model and making it possible to calculate efficiency when there is a relationship among variables.

In dieser Arbeit wird eine Methode zur Leistungsoptimierung der Kolbenpfade von Stirling-Motoren entwickelt, die auf der Theorie der optimalen Steuerung beruht. Für die effiziente praktische Umsetzbarkeit ist dabei ein geringer numerischer Aufwand des eingesetzten thermodynamischen Modells entscheidend. In detaillierten Modellen von Stirling-Motoren resultiert ein Großteil des numerischen Aufwandes aus der Beschreibung des Regenerators, einem gasdurchströmten Kurzzeit-Wärmespeicher. Im ersten Teil der Arbeit wird der Fokus deshalb auf die Entwicklung eines effizienten Regeneratormodells gelegt. Hierbei wird ein neuartiger Ansatz gewählt, der sich aus der Perspektive der Endoreversiblen Thermodynamik ergibt: Der Regenerator wird als endoreversibles Teilsystem betrachtet, welches an zwei Kontaktpunkten durch irreversible Interaktionen mit den benachbarten Teilsystemen Gasteilchen, Entropie und Energie austauscht. Innere Irreversibilitäten des Regenerators werden als Entropiequellterme in die Modellierung einbezogen. Im zweiten Teil der Arbeit wird dann ein iterativer Optimierungsalgorithmus erarbeitet, der die Leistung von Stirling-Motoren unter periodischen Randbedingungen für eine vorgegebene Periodendauer maximieren kann. Der Algorithmus startet mit vorgegeben initialen Kolbenpfaden, die im Laufe der Iterationen graduell verschoben und so den optimalen Pfaden angenähert werden. Um diese graduelle Verschiebung zu bestimmen, muss in jedem Iterationsschritt neben dem Differentialgleichungssystem, das die Thermodynamik des Stirling-Motors beschreibt, ein konjugiertes Differentialgleichungssystem gelöst werden. Der erarbeitete Algorithmus nutzt dabei die Existenz eines Grenzzyklus des konjugierten Differentialgleichungssystems unter Zeitumkehr zu dessen Lösung für periodische Randbedingungen aus. Unter Verwendung des endoreversiblen Regeneratormodells wird mit diesem iterativen Optimierungsalgorithmus die Theorie der optimalen Steuerung erstmals für die Kolbenpfadoptimierung eines beispielhaften Stirling-Motors in α-Konfiguration eingesetzt. / In this thesis a method for power optimization of the piston paths of Stirling engines is developed, which is based on Optimal Control Theory. For the efficient practical feasibility of this task, low numerical effort of the utilized thermodynamic model is crucial. In detailed models of Stirling engines, a large part of the numerical effort results from the description of the regenerator, which is a short-time heat storage. Therefore, in the first part of this thesis the focus is on the development of an efficient regenerator model. Here, a novel ansatz is chosen which arises from the perspective of Endoreversible Thermodynamics: The regenerator is described as an endoreversible subsystem that has two contact points, at which it exchanges particles, entropy, and energy with the adjacent subsystems through irreversible interactions. Internal irreversibilities of the regenerator are included in the model as entropy source terms. In the second part of the thesis an iterative optimization algorithm is worked out, which can maximize the power output of Stirling engines under periodic boundary conditions for given cycle time. The algorithm starts with predefined initial piston paths, which are gradually shifted over the course of the iterations and thus approach the optimal paths. To determine this gradual shift, in every iteration not only the system of differential equations describing the thermodynamics of the Stirling engine needs to be solved, but also a conjugate system of differential equations. The algorithm here exploits the existence of a limit cycle of the conjugate system under time reversal to solve it for periodic boundary conditions. By means of the endoreversible regenerator model, with this iterative optimization algorithm Optimal Control Theory is applied for the first time to optimize the piston paths of an exemplary Stirling engine in α-configuration.

La physique moderne repose sur deux théories fondamentales distinctes, la relativité générale et la mécanique quantique. Toutes les deux décrivent d’une part les phénomènes macroscopiques et cosmologiques tels que les ondes gravitationnelles et les trous noirs et d’autre part les phénomènes microscopiques comme la superfluidité ou le spin des particules. L’unification de ces deux théories reste, jusqu’à présent, un problème non résolu. Il est intéressant de noter que les différentes théories de gravité quantique prédisent une violation des principes de la relativité générale à différents niveaux.Il est donc hautement intéressant de détecter les violations de ces principes et de déterminer à quel niveau elles se produisent.De récentes propositions pour effectuer des tests du principe d’ équivalence d’Einstein suggèrent une amélioration spectaculaire des performances en utilisant des capteurs atomiques `a ondes de matière.Dans ce contexte, il est nécessaire de concevoir des états d’entrée de l’interferomètre avec des conditions initiales bien définies. Un test de pointe de l’universalité de la chute libre (Universality of FreeFall en anglais (UFF) ) nécessiterait, par exemple,un contrôle des positions et des vitesses avec une précision de l’ordre de 1 μm et 1 μm.s⁻¹ , respectivement.De plus, les systématiques liées à la taille du paquet d’ondes limitent le taux d’expansion maximum possible à 100 μm.s⁻¹. La création initiale des états d’entrée de l’interféromètre doit être assez rapide,de l’ordre de quelques centaines de ms au maximum,pour que le temps de cycle de l’expérience soit pertinent d’un point de vue métrologique. Dans cette thèse j’ai développé des séquences optimisées s’appuyant sur l’excitation du centre de masse et de la taille d’un ou plusieurs ensembles d’atomes refroidis ainsi que dégénérés. Certaines séquences proposé dans cette thèse ont déjà été implémenté dans des expériences augmentant de manière significative le contrôle des ensembles atomiques. / Modern physics relies on two distinct fundamental theories, General Relativity and Quantum Mechanics. Both describe on one hand macroscopic and cosmological phenomena such as gravitational waves and black holes and on the other hand microscopic phenomena as superfluidity or the spin of particles. The unification of these two theories remains, so far, an unsolved problem. Interestingly, candidate Quantum Gravity theories predict a violation of the principles of General Relativity at different levels. It is, therefore, of a timely interest to detect violations of these principles and determine at which level they occur. Recent proposals to perform Einstein Equivalence Principle tests suggest a dramatic performance improvement using matter-wave atomic sensors. In this context, the design of the input states with well defined initial conditions is required. A state-of-the-art test of the universality of free fall (UFF) would, for example, require a control of positions and velocities at the level of 1 µm and 1 µm.s⁻¹, respectively. Moreover, sizerelated systematics constrain the maximum expansion rate possible to the 100 µm.s⁻¹level. This initial engineering of the input states has to be quite fast, of the order of few hundred ms at maximum, for the experiment’s duty cycle to be metrologically-relevant. In this thesis I developed optimized sequences based on the excitation of the center of mass and the size excitation of one or two cooled atomic sample as well as degenerated gases. Some sequences proposed in this thesis have already been implemented in experiments and significantly increase the control of atomic ensembles.

Driven by the need for automation and autonomy as well as the need to reduce resources and emissions, the automotive industry is currently undergoing a major transformation. Technologically, this transformation is addressing a wide range of challenges and opportunities. The optimal control of all components is significant for the sustainable development and eco-friendly operation of vehicles. Additionally, robust control of the actuators forms the basis for the development of driver assistance systems and functions for autonomous driving. The actuators of the steering system are particularly important, as they enable safe and comfortable lateral vehicle control. Therefore, the model-based development and virtual simulation of an innovative highly robust control approach for modern Steer-by-Wire systems were conducted in this thesis. The approaches and algorithms described in this thesis allow the design of robust Steer-by-Wire systems and offer the opportunity to conduct many investigations in a computer-aided virtual environment at an early stage in the development process. This reduces time- and cost-intensive testing on prototypes, avoids unnecessary iterations in the design and significantly increases the efficiency and quality of the development. The desired high degree of robustness of the steering control also ensures that the parameterization of the steering feel generator can be freely selected for the individual application. This enables safe and comfortable vehicle lateral control.In summary, the research results described in this thesis accelerate the development of new, modern Steer-by-Wire systems whose robust design forms the basis for the realization of functions for highly automated and autonomous driving.

Optimal control is a paradigm for solving optimization problems involving dynamical systems, which are to be controlled. It is able to solve fish harvesting problems, in which we want to optimize harvesting out-take by considering fishing as a control function that acts on the state of the dynamical system, which represents the growth of fish species in the environment. Other modelling aspects of optimal control are defining terminal costs and running costs, e.g. maximizing profit. We keep the terminal condition comparable for a different number of species. It is based on the initial population. By using the optimal control Hamiltonian and Pontryagin’s Maximum Principle we can calculate the optimal state trajectories corresponding to suitable optimal controls. The Hamiltonian is dependent on the state equation and the running costs. We present two approaches of modelling the running costs. An approach that is not directly translatable to the fish harvesting problem, but it leads to a smooth Hamiltonian, which greatly simplifies derivation and computation. The other, which is equivalent to maximizing profit, leads to a non-smooth Hamiltonian. This leads to jump-discontinuous derivatives needed for computation. We propose to regularize the derivatives of the Hamiltonian using suitable smooth functions, such that it is equivalent to regularizing the Hamiltonian directly. We give details for implementing both approaches up to systems of n competing species. After which we go into detail on algorithms and programming structure implemented. Finally, in modest numerical experiments, for one and two species, we show the relation between the optimal control and the terminal costs. But more interestingly, that the smooth Hamiltonian models are inadequate and regularized Hamiltonian models are the preferred choice. Intriguingly, the latter approach results in steady state solution, wherethe control acts as a stabilizer. / Optimal kontroll är ett paradigm för att lösa optimeringsproblem som omfattar dynamiska system som ska kontrolleras. Den kan lösa problem med skörd av fisk där vi vill optimera skörd av fisk genom att betrakta fisket som en kontrollfunktion som verkar på tillståndet i det dynamiska systemet, som representerar tillväxten av fiskarter i miljön. Andra modelleringsaspekter av optimal styrning är att definiera slutkostnader och löpande kostnader, t.ex. maximering av vinsten. Vi håller terminalvillkoret jämförbart för ett antal olika arter. Det baseras på den ursprungliga populationen.Genom att använda Hamiltonianen för optimal styrning och Pontryagins maximiprincip kan vi beräkna de optimala tillståndsbanorna som motsvarar lämpliga optimala styrningar. Hamiltonianen är beroende av tillståndsekvationen och driftskostnaderna. Vi presenterar två metoder för att modellera driftskostnaderna. Ett tillvägagångssätt som inte är direkt överförbart till problemet med skörd av fisk, men som leder till en slät Hamiltonian, vilket förenklar härledning och beräkning avsevärt. Den andra metoden, som är likvärdig med vinstmaximering, leder till en icke slät Hamiltonian. Detta leder till hopp-diskontinuerliga derivator som behövs för beräkningen. Vi föreslår att man reglerar Hamiltonianens derivator med hjälp av lämpliga släta funktioner, så att det är likvärdigt med att reglera Hamiltonianen direkt. Vi ger detaljer för genomförandet av bå-da tillvägagångssätten upp till system med n konkurrerande arter. Därefter går vi in i detalj på algoritmer och den implementerade programmeringsstrukturen. Slutligen visar vi genom numeriska experiment, för en och två arter, sambandet mellan den optimala kontrollen och slutkostnaderna. Men mer intressant är att de släta hamiltoniska modellerna är otillräckliga, vilket ger upphov till att reglerade hamiltoniska modeller är att föredra. Intressant nog resulterar det senare tillvägagångssättet i en stabil lösning, där kontrollen fungerar som en stabilisator.

En Imagerie par Résonance Magnétique (IRM), l’augmentation du champ magnétique statique permet en théorie de fournir un rapport signal sur bruit accru, améliorant la qualité des images. L’objectif de l’IRM à ultra haut champ est d’atteindre une résolution spatiale suffisamment haute pour pouvoir distinguer des structures si fines qu’elles sont actuellement impossibles à visualiser de façon non-invasive. Cependant, à de telles valeurs de champs magnétiques, la longueur d’onde du rayonnement électromagnétique envoyé pour basculer les spins des protons de l’eau est du même ordre de grandeur que l’objet dont on souhaite faire l’image. Des phénomènes d’interférences sont observés, ce qui se traduit par l’inhomogénéité de ce champ radiofréquence (RF) au sein de l’objet. Ces interférences engendrent des artefacts de signal et/ou de contraste dans les images IRM, et rendent ainsi leur exploitation délicate. Il est donc crucial de fournir des solutions pour atténuer la non-uniformité de l’excitation des spins, à défaut de quoi de tels systèmes ne pourront atteindre leurs pleins potentiels. Pour obtenir des diagnostics pertinents à très haut champ, il est donc nécessaire de créer des impulsions RF homogénéisant l'excitation de l'ensemble des spins (ici du cerveau humain), optimisées pour chaque individu. Pour cela, un système de transmission parallèle (pTX) à 8 canaux a été installé au sein de notre imageur à 7 Tesla. Alors que la plupart des systèmes IRM cliniques n’utilisent qu’un seul canal d’émission, l’extension pTX permet de jouer différentes formes d’impulsions RF de concert. La somme résultante de ces interférences doit alors être optimisée pour atténuer la non-uniformité observée classiquement. L’objectif de cette thèse est donc de synthétiser ce type d’impulsions, en utilisant la pTX. Ces impulsions auront pour contrainte supplémentaire le respect des limitations internationales concernant l'exposition à des champs radiofréquence, qui induit une hausse de température dans les tissus. En ce sens, de nombreuses simulations électromagnétiques et de températures ont été réalisées en introduction de cette thèse, afin d’évaluer la relation entre les seuils recommandés d’exposition RF et l’élévation de température prédite dans les tissus. Cette thèse porte plus spécifiquement sur la conception de l’ensemble des impulsions RF refocalisantes utilisées dans des séquences IRM non-sélectives, basées sur l’écho de spin. Dans un premier temps, seule une impulsion RF a été générée, pour une application simple : l’inversion du déphasage des spins dans le plan transverse. Dans un deuxième temps, sont considérées les séquences à long train d’échos de refocalisation appliquées à l’in vivo. Ici, l’opérateur mathématique agissant sur la magnétisation, et non pas son état final comme il est fait classiquement, est optimisé. Le gain en imagerie à très haut champ est clairement visible puisque les opérations mathématiques (la rotation des spins) voulues sont réalisées avec plus de fidélité que dans le cadre des méthodes de l’état de l’art. Pour cela, la génération de ces impulsions RF combine une méthode d’excitation des spins avec navigation dans l’espace de Fourier, les kT-points, et un algorithme d’optimisation, appelé Gradient Ascent Pulse Engineering (GRAPE), utilisant le contrôle optimal. Cette conception est rapide grâce à des calculs analytiques plus directs que des méthodes de différences finies. La prise en compte d’un grand nombre de paramètres nécessite l’usage de GPUs (Graphics Processing Units) pour atteindre des temps de calcul compatibles avec un examen clinique. Cette méthode de conception d’impulsions RF a été validée expérimentalement sur l’imageur 7 Tesla de NeuroSpin, sur une cohorte de volontaires sains. / In Magnetic Resonance Imaging (MRI), the increase of the static magnetic field strength is used to provide in theory a higher signal-to-noise ratio, thereby improving the overall image quality. The purpose of ultra-high-field MRI is to achieve a spatial image resolution sufficiently high to be able to distinguish structures so fine that they are currently impossible to view in a non-invasive manner. However, at such static magnetic fields strengths, the wavelength of the electromagnetic waves sent to flip the water proton spins is of the same order of magnitude than the scanned object. Interference wave phenomena are then observed, which are caused by the radiofrequency (RF) field inhomogeneity within the object. These generate signal and/or contrast artifacts in MR images, making their exploitation difficult, if not impossible, in certain areas of the body. It is therefore crucial to provide solutions to mitigate the non-uniformity of the spins excitation. Failing this, these imaging systems with very high fields will not reach their full potential.For relevant high field clinical diagnosis, it is therefore necessary to create RF pulses homogenizing the excitation of all spins (here of the human brain), and optimized for each individual to be imaged. For this, an 8-channel parallel transmission system (pTX) was installed in our 7 Tesla scanner. While most clinical MRI systems only use a single transmission channel, the pTX extension allows to simultaneously playing various forms of RF pulses on all channels. The resulting sum of the interference must be optimized in order to reduce the non-uniformity typically seen.The objective of this thesis is to synthesize this type of tailored RF pulses, using parallel transmission. These pulses will have as an additional constraint the compliance with the international exposure limits for radiofrequency exposure, which induces a temperature rise in the tissue. In this sense, many electromagnetic and temperature simulations were carried out as an introduction of this thesis, in order to assess the relationship between the recommended RF exposure limits and the temperature rise actually predicted in tissues.This thesis focuses specifically on the design of all RF refocusing pulses used in non-selective MRI sequences based on the spin-echo. Initially, only one RF pulse was generated for a simple application: the reversal of spin dephasing in the transverse plane, as part of a classic spin echo sequence. In a second time, sequences with very long refocusing echo train applied to in vivo imaging are considered. In all cases, the mathematical operator acting on the magnetization, and not its final state as is done conventionally, is optimized. The gain in high field imaging is clearly visible, as the necessary mathematical operations (that is to say, the rotation of the spins) are performed with a much greater fidelity than with the methods of the state of the art. For this, the generation of RF pulses is combining a k-space-based spin excitation method, the kT-points, and an optimization algorithm, called Gradient Ascent Pulse Engineering (GRAPE), using optimal control.This design is relatively fast thanks to analytical calculations rather than finite difference methods. The inclusion of a large number of parameters requires the use of GPUs (Graphics Processing Units) to achieve computation times compatible with clinical examinations. This method of designing RF pulses has been experimentally validated successfully on the NeuroSpin 7 Tesla scanner, with a cohort of healthy volunteers. An imaging protocol was developed to assess the image quality improvement using these RF pulses compared to typically used non-optimized RF pulses. All methodological developments made during this thesis have contributed to improve the performance of ultra-high-field MRI in NeuroSpin, while increasing the number of MRI sequences compatible with parallel transmission.

<div>Optimal control theory (OCT) exists since the 1950s. However, with the advent of modern computers, the design community delegated the task of solving the optimal control problems (OCPs) largely to computationally intensive direct methods instead of methods that use OCT. Some recent work showed that solvers using OCT could leverage parallel computing resources for faster execution. The need for near real-time, high quality solutions for OCPs has therefore renewed interest in OCT in the design community. However, certain challenges still exist that prohibits its use for solving complex practical aerospace problems, such as landing human-class payloads safely on Mars.</div><div><br></div><div>In order to advance OCT, this thesis introduces Epsilon-Trig regularization method to simply and efficiently solve bang-bang and singular control problems. The Epsilon-Trig method resolves the issues pertaining to the traditional smoothing regularization method. Some benchmark problems from the literature including the Van Der Pol oscillator, the boat problem, and the Goddard rocket problem verified and validated the Epsilon-Trig regularization method using GPOPS-II.</div><div><br></div><div>This study also presents and develops the usage of trigonometry for incorporating control bounds and mixed state-control constraints into OCPs and terms it as Trigonometrization. Results from literature and GPOPS-II verified and validated the Trigonometrization technique using certain benchmark OCPs. Unlike traditional OCT, Trigonometrization converts the constrained OCP into a two-point boundary value problem rather than a multi-point boundary value problem, significantly reducing the computational effort required to formulate and solve it. This work uses Trigonometrization to solve some complex aerospace problems including prompt global strike, noise-minimization for general aviation, shuttle re-entry problem, and the g-load constraint problem for an impactor. Future work for this thesis includes the development of the Trigonometrization technique for OCPs with pure state constraints.</div>