With the increase of world population and industrial growth of developing countries, demand for energy, in particular electric power, has gone up at an unprecedented rate over the last decades. To meet the demand, electric power generation by use of fossil fuel has increased enormously thereby producing increased quantity of greenhouse gases. This contributes more and more to atmospheric pollution, which climate scientists believe can adversly affect the global climate, as well as health and the welfare of the world population. In view of these issues, there is global awareness to look for alternate sources of energy such as natural gas, hydropower, wind, solar, geothermal and biomass. It is recognized that this requires replacement of existing infrastructure with new systems, which cannot be achieved overnight. Optimal control theory has been widely used in diverse areas of physical sciences, medicine, engineering and economics. The main motivation of this thesis is to use this theory to find the optimum strategy for integration of all currently available renewable energy sources with the existing electric power generating systems. The ultimate goal is to eliminate fossil fuels. Eight main energy sources namely, Coal, Petroleum, Natural Gas, Conventional Hydro, Wind, Solar, Geothermal and Biomass are considered in a dynamic model. The state of the dynamic model represents the level of energy generation from each of the sources. Different objective functions are proposed in this thesis. These range from meeting the desired target level of power generation from each of the available sources at the end of a given plan period, to reducing the implementation and investment costs; from minimizing the production from polluted energy sources to meeting the electricity demand during a whole plan period. Official released data from the U.S. Energy Information Administration have been used as a case study. Based on real life data and the mathematics of optimal control theory, we present an optimal policy for integration of renewable energy sources to the national power grid.

With the increase of world population and industrial growth of developing countries, demand for energy, in particular electric power, has gone up at an unprecedented rate over the last decades. To meet the demand, electric power generation by use of fossil fuel has increased enormously thereby producing increased quantity of greenhouse gases. This contributes more and more to atmospheric pollution, which climate scientists believe can adversly affect the global climate, as well as health and the welfare of the world population. In view of these issues, there is global awareness to look for alternate sources of energy such as natural gas, hydropower, wind, solar, geothermal and biomass. It is recognized that this requires replacement of existing infrastructure with new systems, which cannot be achieved overnight. Optimal control theory has been widely used in diverse areas of physical sciences, medicine, engineering and economics. The main motivation of this thesis is to use this theory to find the optimum strategy for integration of all currently available renewable energy sources with the existing electric power generating systems. The ultimate goal is to eliminate fossil fuels. Eight main energy sources namely, Coal, Petroleum, Natural Gas, Conventional Hydro, Wind, Solar, Geothermal and Biomass are considered in a dynamic model. The state of the dynamic model represents the level of energy generation from each of the sources. Different objective functions are proposed in this thesis. These range from meeting the desired target level of power generation from each of the available sources at the end of a given plan period, to reducing the implementation and investment costs; from minimizing the production from polluted energy sources to meeting the electricity demand during a whole plan period. Official released data from the U.S. Energy Information Administration have been used as a case study. Based on real life data and the mathematics of optimal control theory, we present an optimal policy for integration of renewable energy sources to the national power grid.

As equações do modelo bidimensional de veículos autônomos subaquáticos fornecem um exemplo de sistema de controle não linear com o qual podemos ilustrar propriedades da teoria de controle ótimo. Apresentamos, sistematicamente, como os conceitos de formalismo hamiltoniano e teoria de Lie aparecem de forma natural neste contexto. Para tanto, estudamos brevemente o Princípio do Máximo de Pontryagin e discutimos características de sistemas afins. Tratamos com cuidado do Fenômeno Fuller, fornecendo critérios para decidir quando ele está ou não presente em junções, utilizando para isso uma linguagem algébrica. Apresentamos uma abordagem numérica para tratar problemas de controle ótimo e finalizamos com a aplicação dos resultados ao modelo bidimensional de veículo autônomo subaquático. / The equations of the two-dimensional model for autonomous underwater vehicles provide an example of a nonlinear control system which illustrates properties of optimal control theory. We present, systematically, how the concepts of the Hamiltonian formalism and the Lie theory naturally appear in this context. For this purpose, we briefly study the Pontryagin\'s Maximum Principle and discuss features of affine systems. We treat carefully the Fuller Phenomenon, providing criteria to detect its presence at junctions with an algebraic notation. We present a numerical approach to treat optimal control problems and we conclude with an application of the results in the bidimesional model of autonomous underwater vehicle.

As equações do modelo bidimensional de veículos autônomos subaquáticos fornecem um exemplo de sistema de controle não linear com o qual podemos ilustrar propriedades da teoria de controle ótimo. Apresentamos, sistematicamente, como os conceitos de formalismo hamiltoniano e teoria de Lie aparecem de forma natural neste contexto. Para tanto, estudamos brevemente o Princípio do Máximo de Pontryagin e discutimos características de sistemas afins. Tratamos com cuidado do Fenômeno Fuller, fornecendo critérios para decidir quando ele está ou não presente em junções, utilizando para isso uma linguagem algébrica. Apresentamos uma abordagem numérica para tratar problemas de controle ótimo e finalizamos com a aplicação dos resultados ao modelo bidimensional de veículo autônomo subaquático. / The equations of the two-dimensional model for autonomous underwater vehicles provide an example of a nonlinear control system which illustrates properties of optimal control theory. We present, systematically, how the concepts of the Hamiltonian formalism and the Lie theory naturally appear in this context. For this purpose, we briefly study the Pontryagin\'s Maximum Principle and discuss features of affine systems. We treat carefully the Fuller Phenomenon, providing criteria to detect its presence at junctions with an algebraic notation. We present a numerical approach to treat optimal control problems and we conclude with an application of the results in the bidimesional model of autonomous underwater vehicle.

La gestion énergétique (EMS) pour véhicules hybrides a pour objectif de déterminer la répartition de puissance entre les différentes sources d'énergie de manière à minimiser la consommation de carburant et/ou les émissions polluantes. L'objectif de cette thèse est de développer un EMS en prenant en compte des températures internes (la température du moteur et/ou la température du système de post-traitement). Dans une première partie et en utilisant une connaissance préalable du cycle de conduite, le calcul d'un EMS est formulé comme un problème de commande optimale. Ensuite, le principe du minimum de Pontryagin (PMP) est utilisé pour résoudre ce problème d'optimisation.~En se basant sur les résultats numériques obtenus, un compromis entre les performances de la stratégie de commande et de la complexité du modèle utilisé pour la calculer est établi. Les différents problèmes étudiés dans cette thèse sont des exemples des simplifications successives de modèle qui peuvent être regroupées dans le concept des perturbations régulières en contrôle optimal sous contrainte de commande discuté ici. Dans une deuxième partie, la formulation de l'ECMS a été généralisée pour inclure les dynamiques thermiques. Ces extensions définissent des stratégies sous-optimales que nous avons testées numériquement et expérimentalement. / Energy management system (EMS) for hybrid vehicles consists on determining the power split between the different energy sources in order to minimize the overall fuel consumption and/or pollutant emissions of the vehicle. The objective of this thesis is to develop an EMS taking into account the internal temperatures (engine temperature and/or catalyst temperature). In a first part and using a prior knowledge of vehicle driving cycle, the EMS design is formulated as an optimal control problem. Then, the PMP is used to solve this optimization problem. Based on the obtained numerical results, some trade-off between performance of the control strategy and complexity of the model used to calculate this strategy is established. The various problems studied in this thesis are examples of successive model simplifications which can be recast in the concept of regular perturbations in optimal control under input constraints discussed here. In a second part, the feedback law of ECMS is generalized to include thermal dynamics. This defines sub-optimal feedback strategies which we have tested numerically and experimentally.

Ce mémoire de thèse présente une méthode permettant d'améliorer laconduite automobile. Le filtre de Kalman étendu est utilisé pour identifierun modèle de la voiture. Ce filtre est particulièrement étudié afin de prendreen compte la redondance des informations et leur mesure asynchrone.Un ensemble cohérent et bon marché de capteurs - incluant accéléromètres,GPS, temps d'ouverture des injecteurs et vitesse - a été développé et installédans plusieurs véhicules. Ces mesures sont utilisées afin de reconstituer lafonction d'efficacité du moteur. Cette méthodologie peut-être utilisée pourtoute étude requérant la connaissance de cette fonction.La fonction d'efficacité est approchée par une fonction polynomiale etle modèle obtenu est la base d'une optimisation utilisant le principe dumaximum de Pontryagin.Les résultats des tests en condition réelle sont donnés et montrent l'efficicacité de l'observateur et du contrôleur / The PhD presents a combined approach to improving individual car efficiency. An optimal observer, the Extended Kalman Filter, is used to create an efficiency model for the car. Particular attention was paid to handling the asynchronous and redundant nature of the measurement data. A low-cost sensor suite developed to measure data is described. This sensor suite was installed on multiple vehicles to good success. It employsan accelerometer, gps, fuel injector timer, and Vss input to measure all the data necessary to reconstruct the car's state. This observer and sensor suite can be used as the base for any study which requires car efficiency maps, allowing research to proceed without manufacturer supplied data. Once the efficiency map is found, it is then curve-fitted in order to reduce model complexity. The simplified model is then used as a basis for optimal control through Pontryagin's Maximum Principle. Real-world test results are given, both for efficiency mapping, and for optimal control. Detailed discussion of the observer and controller is presented, in order to ease understanding and save implementation time

L'objectif de cette thèse est principalement d'appliquer la théorie du contrôle optimal à des problématiques que soulèvent la maladie du diabète et celle de la propagation de la rumeur sur les réseaux sociaux.Pour la première application, à savoir la maladie du diabète, nous développons deux études. Dans une première étude, à un modèle qui examine les diabétiques avec et sans complications, nous associons un problème de contrôle optimal. Nous montrons qu'il n'existe pas de comportement cyclique entre le groupe des diabétiques avec complications et celui des diabétiques sans complications, et que le point d'équilibre associé au problème existe et est un point selle. Dans une seconde étude, nous modifions un modèle de glucose-insuline à temps différé par l'ajout d'actions extérieures avec retard. Puis, pour minimiser la glycémie d'un diabétique, nous les contrôlons séparément puis simultanément, afin d'en donner une caractérisation à l'aide du principe du maximum de Pontryagin.Pour la deuxième application, la problématique de la propagation de la rumeur sur les réseaux sociaux, nous proposons aussi deux approches. Premièrement, nous mettons en place des stratégies optimales, par l'ajout d'actions extérieures sur un modèle d'e-rumeur de type SIR que nous contrôlons séparément puis simultanément, pour minimiser la propagation d'une fausse information. Et, dans une deuxième approche, nous construisons un nouveau modèle d'e-rumeur pour lequel nous étudions les points d'équilibres admissibles en mettant en évidence leurs conditions de stabilité, ainsi que les critères de persistance du modèle. / The aim of this thesis is mainly to apply optimal control theory to problems raised by diabetes disease and the spread of rumors on social networks.For the first application, namely diabetes disease, we develop two studies. In a first one, from a model that examines diabetics with and without complications, we associate an optimal control problem. We show that there is no cyclical behavior between the group of diabetics with complications and the one without complications, and that the associated equilibrium point exists and is a saddle point. In a second study, we modify a model of delayed glucose-insulin by adding external actions with delay. Then, in order to minimize the glycemia of a diabetic, we control them separately and simultaneously in order to give a characterization of the optimal actions with the Pontryagin maximum principle.For the second application, the issue of spreading rumors on social networks, we also give two approaches. First, we introduce some optimal strategies, by adding external actions on a e-rumor model of SIR type that we control separately and simultaneously to minimize the spread of fake news. Then, in a second approach, we build a new e-rumor model for which we study the admissible equilibrium points by highlighting their stability conditions, as well as the criteria of persistence of the model.

Orientador: Valeriano Antunes de Oliveira / Resumo: Neste trabalho, primeiramente, serão apresentados problemas de otimização nos quais a função objetivo é de múltiplas variáveis e de valor intervalar e as restrições de desigualdade são dadas por funcionais clássicos, isto é, de valor real. Serão dadas as condições de otimalidade usando a E−diferenciabilidade e, depois, a gH−diferenciabilidade total das funções com valor intervalar de várias variáveis. As condições necessárias de otimalidade usando a gH−diferenciabilidade total são do tipo KKT e as suficientes são do tipo de convexidade generalizada. Em seguida, serão estabelecidos problemas de controle ótimo nos quais a funçãao objetivo também é com valor intervalar de múltiplas variáveis e as restrições estão na forma de desigualdades e igualdades clássicas. Serão fornecidas as condições de otimalidade usando o conceito de Lipschitz para funções intervalares de várias variáveis e, logo, a gH−diferenciabilidade total das funções com valor intervalar de várias variáveis. As condições necessárias de otimalidade, usando a gH−diferenciabilidade total, estão na forma do célebre Princípio do Máximo de Pontryagin, mas desta vez na versão intervalar. / Abstract: In this work, firstly, it will be presented optimization problems in which the objective function is interval−valued of multiple variables and the inequality constraints are given by classical functionals, that is, real−valued ones. It will be given the optimality conditions using the E−differentiability and then the total gH−differentiability of interval−valued functions of several variables. The necessary optimality conditions using the total gH−differentiability are of KKT−type and the sufficient ones are of generalized convexity type. Next, it will be established optimal control problems in which the objective function is also interval−valued of multiple variables and the constraints are in the form of classical inequalities and equalities. It will be furnished the optimality conditions using the Lipschitz concept for interval−valued functions of several variables and then the total gH−differentiability of interval−valued functions of several variables. The necessary optimality conditions using the total gH−differentiability is in the form of the celebrated local Pontryagin Maximum Principle, but this time in the intervalar version. / Doutor

O propósito desta tese é estudar um método numérico indireto para a solução de problemas de otimização de trajetórias espaciais em campo gravitacional central Newtoniano. Neste método, a resolução do problema de valor de contorno em dois pontos é baseada no conceito de variação segunda, motivo pelo qual ele é denominado de método da variação segunda.Inicialmente, este trabalho apresenta alguns fundamentos e resultados matemáticos da Teoria de Controle Ótimo. Estes conceitos são então utilizados no desenvolvimento matemático e formulação do método da variação segunda, incluindo sua extensão para problemas com vínculos nas variáveis de controle e a descrição do algoritmo final. Este algoritmo é então aplicado a problemas de transferências espaciais ótimas entre órbitas circulares coplanares próximas, considerando dois modelos clássicos de sistemas propulsivos: sistemas a velocidade de ejeção constante (VEC) e empuxo limitado; e sistemas com velocidade de ejeção modulável e potência limitada (PL). De modo geral, os resultados obtidos para os problemas são muito bons. Para o sistema PL, há grande concordância deles com resultados analíticos, provenientes de uma teoria linear, e numéricos, obtidos através de outros métodos. Para o sistema VEC, os resultados estão em acordo com a teoria de transferências orbitais ótimas e verifica-se que as duas formulações utilizadas, linearizada e não-linearizada, fornecem resultados bastante próximos, para problemas de transferência entre órbitas próximas.

Este trabalho apresenta estudos e proposições sobre guiamento de veículos espaciais, especialmente o VLS-1, com ênfase em tratamentos para dispersões em valores de parâmetros pré-estabelecidos para vôo, no âmbito de possibilidade de atuação do guiamento. Introdutoriamente, é feita uma sinopse sobre a missão de vôo do VLS-1 e a contextualização do guiamento nessa missão. Para fundamentar teoricamente o desenvolvimento neste trabalho, e com enfoque no estritamente necessário para isso, são expostos alguns conceitos de cálculo variacional; o que envolve tópicos em controle ótimo, princípio de máximo de Pontryagin e método de extremais vizinhas. O desenvolvimento aplicado ao guiamento começa com o estabelecimento de condições requeridas, inclusive com relação a dispersões. Então, com base nos mesmos métodos matemáticos, são desenvolvidos dois modelos de guiamento, denotados Guiwa e GuiAa. São analisadas diversas características estabelecidas na concepção e na realização dos modelos, especialmente em relação ao contexto do VLS-1. Segue-se a avaliação de resultados de vários ensaios de simulação, efetuados com protótipos implementados para cada modelo de guiamento. Dois casos de missão, denotados Hsp615 e Hsp750, foram preparados para essas simulações com ambos os modelos. Incluíram-se nas simulações e análises decorrentes, critérios de controle específicos para trajetória após o instante previsto de atingimento de objetivo, bem como antecipação desse instante. Constata-se a eficácia de condição, introduzida no guiamento, contra efeitos de dispersões. Constata-se também a grande conveniência do critério denotado ProHs, para trajetória após o atingimento de objetivo, para ampliação dessa eficácia. Essa condição contra efeitos de dispersões e esse critério ProHs podem ser incorporados a diferentes modelos de guiamento. Quanto aos dois modelos de guiamento experimentados, o Guiwa tem desempenho superior. Contudo, outras alternativas também utilizando cálculo variacional são sugeridas, inclusive com agregação de outras atribuições funcionais relevantes, e merecem ser desenvolvidas e experimentadas.