"A number of outstanding problems in robotic motion and manipulation involve tasks where degrees of freedom (DoF), be they part of the robot, an object being manipulated, or the surrounding environment, cannot be accurately controlled by the actuators of the robot alone. Rather, they are also controlled by physical properties or interactions - contact, robot dynamics, actuator behavior - that are influenced by the actuators of the robot. In particular, we focus on two important areas of poorly controlled robotic manipulation: motion planning for deformable objects and in deformable environments; and manipulation with uncertainty. Many everyday tasks we wish robots to perform, such as cooking and cleaning, require the robot to manipulate deformable objects. The limitations of real robotic actuators and sensors result in uncertainty that we must address to reliably perform fine manipulation. Notably, both areas share a common principle: contact, which is usually prohibited in motion planners, is not only sometimes unavoidable, but often necessary to accurately complete the task at hand. We make four contributions that enable robot manipulation in these poorly controlled tasks: First, an efficient discretized representation of elastic deformable objects and cost function that assess a ``cost of deformation' for a specific configuration of a deformable object that enables deformable object manipulation tasks to be performed without physical simulation. Second, a method using active learning and inverse-optimal control to build these discretized representations from expert demonstrations. Third, a motion planner and policy-based execution approach to manipulation with uncertainty which incorporates contact with the environment and compliance of the robot to generate motion policies which are then adapted during execution to reflect actual robot behavior. Fourth, work towards the development of an efficient path quality metric for paths executed with actuation uncertainty that can be used inside a motion planner or trajectory optimizer."

In this thesis, we present some contributions to the Markov jump linear systems theory in a context of partial information on the Markov chain. We consider that the state of the Markov chain cannot be measured, but instead there is only an observed variable that could model an asynchronous phenomenon between the application and the plant, or a simple fault detection and isolation device. In this formulation, we investigate the problem of designing controllers and filters depending only on the observed variable in the context of H2, H?, and mixed H2/H? control theory. Numerical examples and academic applications are presented for active-fault tolerant control systems and networked control systems. / Nesta tese, apresentamos algumas contribuições para a teoria de sistemas lineares com saltos markovianos em um contexto de observação parcial da cadeia de Markov. Consideramos que o estado da cadeia de Markov não pode ser medido, porém existe uma variável observada que pode modelar um fenômeno assíncrono entre a aplicação e a planta, ou ainda um dispositivo de detecção de falhas simples. Através desse modelo, investigamos o problema da síntese de controladores e filtros que dependem somente da variável observada no contexto das teorias de controle H2, H?, e misto H2/H?. Exemplos numéricos e aplicações acadêmicas são apresentadas no âmbito dos sistemas de controle tolerantes a falhas e dos sistemas de controle através da rede.

Esta dissertação visa a modelagem, simulação e controle de um veículo aéreo não tripulado (VANT) do tipo quadricóptero, utilizando-se as técnicas de controle ótimo e controle robusto no espaço de estados. O quadricóptero deve realizar as funções de decolagem, voo em cruzeiro e pouso de maneira autônoma. A dissertação inclui a síntese e análise comparativa entre um observador de estados de ordem plena de Luenberger e um filtro de Kalman. Além disso, um controlador linear quadrático gaussiano e um controlador robusto serão sintetizados e avaliados, procurando-se avaliar qual tem o melhor desempenho nas diversas tarefas do VANT. / This dissertation includes the modeling, simulation and control of a quadrotor unmanned aerial vehicle by using optimum control and robust control techniques on the space state. The quadrotor must perform the takeoff, cruise flight and landing in an autonomous way. This report also presents the synthesis and comparative analysis between a Luenberger full order state observer and a Kalman filter. A linear quadratic gaussian controller and a robust controller will be also synthetized and analyzed, to compare which one exhibits the best performance on the UAV tasks.

La qualité des prévisions météorologiques repose principalement sur la qualité du modèle utilisé et de son état initial. Cet état initial est reconstitué en combinant les informations provenant du modèle et des observations disponibles en utilisant des techniques d'assimilation de données. Historiquement, les prévisions et l'assimilation sont réalisées dans l'atmosphère et l'océan de manière découplée. Cependant, les centres opérationnels développent et utilisent de plus en plus des modèles couplés océan-atmosphère. Or, assimiler des données de manière découplée n'est pas satisfaisant pour des systèmes couplés. En effet, l'état initial ainsi obtenu présente des inconsistances de flux à l'interface entre les milieux, engendrant des erreurs de prévision. Il y a donc besoin d'adapter les méthodes d'assimilation aux systèmes couplés. Ces travaux de thèse s'inscrivent dans ce contexte et ont été effectués dans le cadre du projet FP7 ERA-Clim2, visant à produire une réanalyse globale du système terrestre.Dans une première partie, nous introduisons les notions d'assimilation de données, de couplage et les différentes méthodologies existantes appliquées au problème de l'assimilation couplée. Ces méthodologies n’étant pas satisfaisantes en terme de qualité de couplage ou de coût de calcul, nous proposons, dans une seconde partie, des méthodes alternatives. Nous faisons le choix de méthodes d'assimilation basées sur la théorie du contrôle optimal. Ces alternatives se distinguent alors par le choix de la fonction coût à minimiser, des variables contrôlées et de l’algorithme de couplage utilisé. Une étude théorique de ces algorithmes a permis de déterminer un critère nécessaire et suffisant de convergence dans un cadre linéaire. Pour conclure cette seconde partie, les performances des différentes méthodes introduites sont évaluées en terme de qualité de l’analyse produite et de coût de calcul à l’aide d’un modèle couplé linéaire 1D. Dans une troisième et dernière partie, un modèle couplé non-linéaire 1D incluant des paramétrisations physique a été développé et implémenté dans OOPS (textit{Object-Oriented Prediction System}) qui est une surcouche logicielle permettant la mise en œuvre d’un ensemble d’algorithmes d’assimilation de données. Nous avons alors pu évaluer la robustesse de nos algorithmes dans un cadre plus réaliste, et conclure sur leurs performances vis à vis de méthodes existantes. Le fait d’avoir développé nos méthodes dans le cadre de OOPS devrait permettre à l’avenir de les appliquer aisément à des modèles réalistes de prévision. Nous exposons enfin quelques perspectives d'amélioration de ces algorithmes. / In the context of operational meteorology and oceanography, forecast skills heavily rely on the model used and its initial state. This initial state is produced by a proper combination of model dynamics and available observations via data assimilation techniques. Historically, numerical weather prediction is made separately for the ocean and the atmosphere in an uncoupled way. However, in recent years, fully coupled ocean-atmosphere models are increasingly used in operational centres. Yet the use of separated data assimilation schemes in each medium is not satisfactory for coupled problems. Indeed, the result of such assimilation process is generally inconsistent across the interface, thus leading to unacceptable artefacts. Hence, there is a strong need for adapting existing data assimilation techniques to the coupled framework. This PhD thesis is related to this context and is part of the FP7 ERA-Clim2 project, which aim to produce an earth system global reanalysis.We first introduce data assimilation and model coupling concepts, followed by some existing algorithms of coupled data assimilation. Since these methods are not satisfactory in terms of coupling strengh or numerical cost, we suggest, in a second part, some alternatives. These are based on optimal control theory and differ by the choice of the cost function to minimize, controled variable and coupling algorithm used. A theoretical study of these algorithms exhibits a necessary and sufficient convergence criterion in a linear case. To conclude about this second part, the different methods are compared in terms of analysis quality and numerical cost using a 1D linear model. In a third part, a 1D non-linear model with subgrid parametrizations was developed and implemented in OOPS (Object-Oriented Prediction System), a software overlay allowing the implementation of a set of data assimilation algorithms. We then assess the robustness of the different algorithms in a more realistic case, and concluded about their performances against existing methods. By implementing our methods in OOPS, we hope it should be easier to use them with operational forecast models. Finally, we expose some propects for improving these algorithms.

Inicialmente, os conceitos fundamentais e a problemática básica subjacentes ao Controle de Sistemas Quânticos são apresentados, destacando-se, por exemplo, as questões físicas e dinâmicas envolvidas, os principais tipos e metodologias de controle no contexto quântico, bem como aplicações existentes e potenciais de Controle Quântico, muitas das quais situando-se na vanguarda da Ciência e da Tecnologia. Segue-se uma exposição do arcabouço teórico básico e do formalismo padrão da Mecânica Quântica, tendo em vista prover os elementos necessários à compreensão de sistemas quânticos, sua dinâmica e seu controle. O conceito de Controlabilidade é, então, apresentado no contexto de Sistemas Quânticos. Em seqüência, os fundamentos do Controle Quântico Ótimo são desenvolvidos como uma extensão da Teoria Clássica de Controle Ótimo, apresentando-se exemplos de aplicações. Ao problema da transferência de estados quânticos para um estado-alvo em tempo mínimo é devotada especial atenção, dada sua grande relevância em aplicações tecnológicas de ponta, como em Computação Quântica e Processamento de Informação Quântica. A partir de limitações físicas que são inerentes a qualquer sistema quântico, no tocante ao tempo mínimo necessário para que ocorra uma transição de estados, propõem-se Fatores de Mérito para quantificar a eficiência dos controles quânticos ótimos que minimizam o tempo de transferência de estados. Exemplos de aplicação, estudos teóricos e estudos de casos são levados a cabo para a definição dos Fatores de Mérito associados. Este trabalho termina com estudos relativos a uma possível formulação da Teoria de Controle Quântico Ótimo em termos de Integrais de Trajetória para o tratamento de sistemas quânticos contínuos, em especial, o controle espaço-temporal de partículas quânticas. Um possível emprego do Efeito Aharonov-Bohm é também discutido como estratégia de Controle Quântico. / Firstly, the fundamental concepts and the basic issues concerning the Control of Quantum Systems are presented, highlighting, for example, related physical and dynamical questions, the main control types and methodologies in the quantum context, as well as current and potential applications of Quantum Control, many of them situated on the avant-garde of Science and Technology. Then follows an exposition of the basic theoretical framework and the standard formalism of Quantum Mechanics, whose aim is to provide the necessary elements for understanding quantum systems, quantum dynamics and control. The concept of Controlability is then presented in the context of Quantum Systems. Subsequently, the fundamental concepts of Quantum Optimal Control are developed as an extension of the Classical Optimal Control Theory, featuring some examples of application. To the problem of transfering quantum states to a certain target state at minimal time a special attention is devoted, having in mind its great relevance in state-of-art technological applications, e.g., Quantum Computation and Quantum Information Processing. From physical limitations that are inherent to any quantum systems, regarding the minimal time necessary to perform a state transition, one proposes Figures of Merit in order to quantify the efficiency of optimal quantum controls which minimize the state transfer time. Examples of applications, theoretical studies and case studies are carried out in order to define the associated Figures of Merit. This work ends with studies concerning a possible formulation of Optimal Quantum Control Theory in terms of Path Integrals for handling continuous quantum systems, particularly, the space-time control of quantum particles. A possible use of the Aharonov-Bohm Effect is also discussed as a Quantum Control strategy.

Este trabalho trata do planejamento de trajetórias que minimizam as perdas elétricas no KA\'I yxo, um robô escalador de árvores que tem por finalidade realizar monitoramento ambiental em florestas através da coleta de diferentes tipos de dados. Como essa aplicação requer que o robô permaneça em ambientes remotos, o estudo de técnicas que reduzam as perdas de energia a fim de que se aumente o tempo em operação do robô se mostra relevante, sendo a minimização das perdas elétricas uma contribuição importante nesse sentido. Estruturalmente, o KA\'I yxo consiste em um robô bípede com duas garras e quatro ligamentos interconectados por três juntas rotacionais. Além disso, seu mecanismo de andadura foi biologicamente inspirado na forma de locomoção observada em lagartas mede-palmos, o que permitiu tratar o robô como um manipulador industrial, cuja base é o ligamento associado à garra engastada e cujo efetuador é o ligamento associado à garra livre. Com isso, quando conveniente, o robô foi tratado em dois casos, conforme a garra que se encontra engastada. Inicialmente, realizou-se a modelagem matemática do robô, obtendo-se as equações cinemáticas direta e inversa, e dinâmicas, bem como o modelo das juntas segundo a abordagem do controle independente por junta. Posteriormente, formulou-se um problema de controle ótimo, solucionado através de um método numérico que o transformou em um problema de programação quadrática, que por sua vez foi resolvido iterativamente. Por fim, as trajetórias ótimas planejadas foram implementadas no robô real e, como forma de validação, as novas perdas elétricas foram comparadas com as das trajetórias anteriormente executadas pelo robô, determinando-se a correspondente economia de energia. / This work deals with the minimum-energy trajectory planning, related to the electrical losses, in KA\'I yxo, a tree-climbing robot that aims to perform environmental monitoring in forests through the collection of different types of data. As this application requires that the robot remains in remote environments, the study of techniques that reduce energy losses in order to increase the operation time of the robot is shown to be relevant, and the minimization of the electrical losses is an important contribution in this sense. Structurally, KA\'I yxo consists of a biped robot with two claws and four links interconnected by three revolute joints. In addition, its gait mechanism was biologically inspired in the form of locomotion observed in caterpillars, allowing to treat the robot as an industrial manipulator, which base is the link associated with the fixed claw and which end-effector is the link associated with the free claw. In consequence, when convenient, the robot was treated in two cases, according to the claw that is fixed. Initially, the mathematical model of the robot was developed, being obtained the forward and inverse kinematic and dynamic equations, as well as the model of the joints according to the independent joint control approach. Subsequently, an optimal control problem was formulated, which was solved through a numerical method that turned it into a quadratic programming problem, which in turn was solved iteratively. Finally, the planned optimal trajectories were implemented in the real robot and, as a form of validation, the new electrical losses were compared with those of the trajectories previously executed by the robot, being determined the corresponding energy saving.

Esta dissertação aborda o problema de redução do consumo de combustível para veículos. Com esse objetivo, realiza-se o levantamento de um modelo estocástico e de seus parâmetros, o desenvolvimento de um controlador para o veículo, e análise dos resultados. O problema considera a interação com o trânsito de outros veículos, que limita a aplicação de resultados antes disponíveis. Para isto, propõe-se modelar a dinâmica do problema de maneira aproximada, usando sistemas com saltos markovianos, e levantar as probabilidades de transição dos estados da cadeia através de um modelo mais completo para o trânsito no percurso. / This dissertation deals with control of vehicles aiming at the fuel consumption optimization, taking into account the interference of traffic. Stochastic interferences like this and other real world phenomena prevents us from directly applying available results. We propose to employ a relatively simple system with Markov jumping parameters as a model for the vehicle subject to traffic interference, and to obtain the transition probabilities from a separate model for the traffic. This dissertation presents the model identification, the solution of the new problem using dynamic programming, and simulation of the obtained control.

Le syndrome d'apnée du sommeil (SAS) est une maladie multifactorielle caractérisée par des épisodes récurrents de pauses respiratoires ou des réductions significatives de l'amplitude respiratoire pendant le sommeil. Ces épisodes peuvent provoquer des réactions cardiorespiratoires aiguës; délétères à long terme. Plusieurs thérapies ont été proposées, étant la pression positive continue des voies respiratoires (CPAP) le traitement de référence. Malgré ces excellents résultats chez les patients symptomatiques, le taux de refus initial est de 15% et une adhésion à long terme est difficile à atteindre. Par conséquent, le développement de méthodes de traitement non invasives, avec une meilleure acceptabilité, reste d’une importance majeure. Dans ce contexte, l’hypothèse qui sous-tend ce travail est qu’une stimulation kinesthésique contrôlée, délivrée au cours de la phase précoce de l’apnée, peut réduire la durée des événements respiratoires et, par la suite, limiter les désaturations d’oxygène associées, par une activation contrôlée du réflexe de sursaut. La première partie de ce manuscrit est consacrée à la description d'un nouveau système (PASITHEA) de surveillance en temps réel et de neuromodulation thérapeutique, qui fonctionne comme un dispositif polyvalent de diagnostic et de traitement de SAS par stimulation kinesthésique. Les principales contributions de cette thèse se concentrent sur les aspects du traitement du signal et du contrôle de ce système, ainsi que sur l'électronique associée. Une autre contribution est liée à l'évaluation de ces méthodes et dispositifs par des protocoles cliniques spécifiques. Dans une deuxième partie, nous proposons une première méthode de contrôle On/Off optimale pour délivrer la stimulation, en utilisant comme variable de contrôle la sortie d'un détecteur d'événements respiratoires en temps réel. Lors de la détection d'un événement, une stratégie de stimulation unique avec amplitude de stimulation constante est appliquée, cette dernière a été mise en œuvre dans le cadre d'un premier protocole clinique dédié à l'évaluation de la réponse du patient au traitement. Les résultats ont montré que 75% des patients répondaient correctement au traitement en termes de durées des épisodes respiratoires. De plus, des diminutions significatives de la variabilité du SaO2 ont également été constatées lors de la mise en œuvre d'une nouvelle méthode d'analyse aiguë. Puisque nous avons supposé qu'une sélection inappropriée des patients pourrait expliquer l'absence de réponse observée chez 25% des patients. Nous avons proposé une méthode pour différencier les patients qui pourraient bénéficier de cette thérapie, basée sur l'estimation d'indices de variabilité cardiaque. Les résultats de ces analyses ont montré que l'efficacité de cette thérapie semble corrélée à un système nerveux autonome fonctionnel. Enfin, une méthode améliorée de contrôle en boucle fermée, intégrant des correcteurs proportionnels-dérivés (PD) couplés et simultanés a été proposée afin de modifier de façon adaptative l’amplitude de stimulation kinesthésique délivrée au patient par le système thérapeutique, en utilisant comme variables de contrôle des signaux physiologiques enregistrés en temps réel. Un deuxième protocole clinique visant à valider l'algorithme de contrôle de la stimulation kinesthésique adaptative spécifique au patient a été initié. Plusieurs améliorations ont été effectuées à la première version du système afin de permettre l'intégration du contrôleur proposé. Les résultats préliminaires de cette étude ont validé le fonctionnement de notre contrôleur et ont montré que notre système était capable de fournir une stimulation kinesthésique adaptative en fonction des réponses propres au patient. Une autre phase de cette étude, mettant en œuvre le contrôleur avec un ensemble des paramètres de contrôle présélectionnés, est actuellement en cours. / Sleep apnea syndrome (SAS) is a multifactorial disease characterized by recurrent episodes of breathing pauses or significant reductions in respiratory amplitude during sleep. These episodes may provoke acute cardiorespiratory responses along with alterations of the sleep structure, which may be deleterious in the long term. Several therapies have been proposed for the treatment of SAS, being continuous positive airway pressure the gold standard treatment. Despite its excellent results in symptomatic patients, there is a 15% initial refusal rate and long term adherence is difficult to achieve in minimally symptomatic patients. Therefore, the development of non-invasive SAS treatment methods, with improved acceptability, is of major importance. The objective of this PhD thesis is to propose new signal processing and control methods of non-invasive neuromodulation for the treatment of SAS. The hypothesis underlying this work is that bursts of kinesthetic stimulation delivered during the early phase of apneas or hypopneas may elicit a controlled startle response that can activate sub-cortical centers controlling upper airways muscles and the autonomic nervous system, stopping respiratory events without generating a cortical arousal. In this context, the first part of this manuscript is dedicated to the description of a novel real-time monitoring and therapeutic neuromodulation system, which functions as a multi-purpose device for SAS diagnosis and treatment through kinesthetic stimulation. This system has been developed in the framework of an ANR TecSan project led by our laboratory, with the participation of Sorin CRM SAS. The main contributions in this thesis are focused on the signal processing and control aspects of this system, as well as the electronics associated. Another contribution is related to the evaluation of these methods and devices through specific clinical protocols. In a second part, we propose a first optimal On/Off control method for delivering kinesthetic stimulation, using as control variable the output of a real-time respiratory event detector. A unique stimulation strategy where a constant stimulation amplitude is applied upon event detention was implemented in a first clinical protocol, dedicated to assessing the patient response to therapy. Results showed that 75% of the patients responded correctly to therapy, showing statistically significant reductions in respiratory event durations. Also, significant decreases in the SaO2 variability were also found when implementing a novel acute analysis method. Since we hypothesized that inappropriate patient selection could explain the observed lack of response in 25% of patients, we proposed a method to differentiate patients who could benefit from this therapy based on the estimation of complexity-based indexes of heart rate variability. Results of these analyses showed that the effectiveness of this therapy seems correlated to a functional autonomic nervous system. Finally, an improved closed-loop control method integrating concurrent, coupled proportional-derivative (PD) controllers in order to adaptively change the kinesthetic stimulation was proposed. It uses as control variables three physiological signals recorded in real-time: Nasal pressure, oxygen saturation and the electrocardiogram signal. A second clinical protocol with the main objective of validating the control algorithm for patient-specific adaptive kinesthetic stimulation was launched. Several improvements to the first version of the system were developed to allow the integration of the proposed controller. Preliminary results from the first phase of this study validated the proposed controller operation and showed that the controller was able to provide adaptive kinesthetic stimulation in function of the patient-specific responses. A second phase of this study implementing the proposed controller and the set of the selected control parameters from the first phase is currently ongoing.

Este estudo considera o problema de controle ótimo multi-período de média-variância para sistemas em tempo discreto com saltos markovianos e ruídos multiplicativos. Inicialmente considera-se um critério de desempenho formado por uma combinação linear da variância nal e valor esperado da saída do sistema. É apresentada uma solução analítica na obtenção da estratégia ótima para este problema. Em seguida são considerados os casos onde os critérios de desempenho são minimizar a variância nal sujeito a uma restrição no valor esperado ou maximizar o valor esperado nal sujeito a uma restrição na variância nal da saída do sistema. As estratégias ótimas de controle são obtidas de um conjunto de equações de diferenças acopladas de Riccati. Os resultados obtidos neste estudo generalizam resultados anteriores da literatura para o problema de controle ótimo com saldos markovianos e ruídos multiplicativos, apresentando condições explícitas e sucientes para a otimalidade da estratégia de controle. São apresentados modelos e simulações numéricas em otimização de carteiras de investimento e estratégias de gestão de ALM (asset liabilities management). / This thesis focuses on the stochastic optimal control problem of discrete-time linear systems subject to Markov jumps and multiplicative noise under three kinds of performance criterions related to the nal value of the expectation and variance of the output. In the first problem it is desired to minimize the nal variance of the output subject to a restriction on its nal expectation, in the second one it is desired to maximize the nal expectation of the output subject to a restriction on its nal variance, and in the third one it is considered a performance criterion composed by a linear combination of the nal variance and expectation of the output of the system. The optimal control strategies are obtained from a set of interconnected Riccati dierence equations and explicit sufficient conditions are presented for the existence of an optimal control strategy for these problems, generalizing previous results in the literature. Numerical simulations of investment portfolios and asset liabilities management models for pension funds with regime switching are presented.

Cette thèse se concentre sur la commande optimale des systèmes de complémentarité linéaire (notés LCS). Les LCS sont des systèmes dynamiques définis par des équations différentielles algébriques (ÉDA), où une des variables est définie par un problème de complémentarité linéaire.Ces systèmes se retrouvent dans la modélisation de nombreux phénomènes, tels que les équilibres dynamiques de Nash, les systèmes dynamiques hybrides ou encore la modélisation de circuits électriques. Les propriétés des solutions à ces ÉDA dépendent essentiellement de propriétés que doit vérifier la matrice D présente dans la complémentarité.Ces contraintes de complémentarité posent des problèmes à deux niveaux. Premièrement, l’analyse de ces systèmes dynamiques font souvent appel à des outils pointus, et leurs études laissent encore des questions non résolues. Deuxièmement, la commande optimale pour ces systèmes pose des soucis à cause d’une part de présence éventuelle de l’état dans les contraintes, et d’autre part une violation assurée des qualifications des contraintes qui sont une hypothèse récurrente des problèmes d’optimisation.La recherche de ce manuscrit se concentre sur la commande optimale de ces systèmes. On s’intéresse principalement à la commande quadratique (minimisation d’une fonctionnelle quadratique en l’état et la commande), et à la commande temps minimal. Les résultats se concentrent sur deux pans: d’un côté, on opère une approche analytique du problème afin de trouver des conditions nécessaires d’optimalité (si possible, on démontre qu’elles sont suffisantes) ; dans un deuxième temps, une approche numérique est effectuée, avec le soucis d’obtenir des résultats numériques précis de manière rapide. / This thesis focuses on the optimal control of Linear Complementarity Systems (LCS). LCS are dynamical systems defined through Differential Algebraic Equations (DAE), where one of the variable is defined by a Linear Complementarity Problem.These systems can be found in the modeling of various phenomena, as Nash equilibria, hybrid dynamical systems or modeling of electrical circuits. Properties of the solution to these DAE essentially depend on properties that the matrix D in the complementarity must meet. These complementarity constraints induce two different challenges. First, the analysis of these dynamical systems often use state of the art tools, and their study still has some unansweredquestions. Second, the optimal control of these systems causes troubles due to on one hand the presence of the state in the constraints, on the other hand the violation of Constraint Qualifications, that are a recurring hypothesis for optimisation problems.The research presented in this manuscript focuses on the optimal control of these systems. We mainly focus on the quadratic optimal control problem (minimisation of a quadratic functional involving the state and the control), and the minimal time control. The results present two different aspects: first, we start with an analytical approach in order to find necessary conditions of optimality (if possible, these conditions are proved to be sufficient); secondly, a numerical approach is tackled, with the aim of getting precise results with a reduced computational time.