Dans cette thèse, nous nous intéressons a la résolution de problèmes de contrôle optimal associés a des équations de diffusion et onde fractionnaires en temps et a données incomplètes, ou les dérivées sont prises au sens de Riemann-Liouville. / In this thesis, we are interested in the résolution of optimal control problems associated to fractional diffusion-wave equations in time with incomplete data, and where derivatives are understood in Riemann-Liouville sense.

Gestion de la variation de la qualité des sols par les agriculteurs : enjeux pour la rentabilité et la durabilité des exploitations agricoles françaisesLa qualité des sols constitue un enjeu majeur en termes d’environnement et de préservation du potentiel agronomique et économique des exploitations agricoles. Les pratiques agricoles ont des impacts sur la qualité des sols, dont certains entraînent une dégradation des sols et mènent à une réduction de leur productivité. La thèse a pour objectif de caractériser et d'éclairer les enjeux économiques de moyen et long terme de la variation de la qualité des sols en mobilisant des outils théoriques, statistiques et empiriques. Nous tentons, en simulant différents scénarios économiques, d’identifier les leviers permettant de préserver le potentiel des sols. Nous utilisons un modèle dynamique de contrôle optimal où l'agent-agriculteur rationnel maximise son profit dans le temps sous contrainte de la dynamique de la qualité des sols.Il y a deux facteurs de production : les intrants productifs (tels les engrais minéraux azotés) et la qualité du sol, capturée par sa matière organique (MO). La qualité du sol est impactée par les intrants productifs utilisés par l'agriculteur, qui peut investir dans la qualité de ses sols via l'utilisation des résidus de culture, l'intensité de labour et les choix des rotations. Nos résultats montrent que l'investissement dans la qualité des sols fait partie d'une stratégie optimale de l'agriculteur qui, face à l'augmentation des prix des engrais et de l'énergie, substitue ainsi les fonctionnalités / Soil quality is a major issue for the environment and the preservation of the agronomic and economic potential of farms. Farming practices have substantial impacts on soil quality; some are detrimental and lead to a long-term decrease in productivity. The objective of this thesis is to characterize the mid-term and long-term economic issues related to soil quality changes using theoretical, statistical and empirical tools and to propose a dynamic bioeconomic model that highlights these issues. Using the simulations of different economic scenarios, we seek to identify the levers that make it possible to preserve the agronomic and economic potential of soil. The model used is a dynamic optimal control model where the rational agent-farmer with perfect information maximizes his profits over time under a soil quality dynamics constraint. We consider two production factors: productive inputs (such as mineral nitrogen fertilizers) and soil quality, captured by the quantity of soil organic mSoil quality is negatively impacted by the productive inputs used by the farmer, who can invest in his soil quality (crop residue use, tillage intensity, crop rotation choice). Our results show that soil quality investment is a component of the farmer's optimal strategy. The farmer substitutes the ecosystemic functionalities of his soil for chemical inputs in response to the increase in fertilizers and energy prices. However, the mitigated results of our empirical model in terms of SOM final values show the importance of considering a large range of farming practices. Our results offer a ba

[pt] O presente projeto de pesquisa objetiva desenvolver um procedimento para determinação de trajetórias ótimas em pistas de corrida baseado em técnicas de otimização, considerando os limites geométricos da pista e as características dinâmicas do veículo. O veículo será representado por meio de um modelo simplificado de partícula orientada, mas que inclui as capacidades de tração, frenagem e aceleração normal típicas de um veículo terrestre de competição. Primeiramente, é determinada a trajetória de tempo mínimo para uma curva de 90 graus por meio da análise geométrica do problema e em seguida, é obtida a solução analítica geral para aplicação a qualquer ângulo. Em seguida, técnicas de otimização com restrição são empregadas de forma a obter a curva de menor tempo que concatena as trajetórias ótimas individuais de cada curva, previamente determinadas. São estudadas, ainda, as características dinâmicas de algumas curvas polinomiais para inferir aquela que melhor pode ser aplicada no processo de concatenação. A trajetória de menor tempo da pista de corrida obtida pelo procedimento de concatenação é apresentada e é feita uma análise das vantagens e desvantagens do método proposto. Como alternativa, é apresentada uma visão geral do problema de controle ótimo e é formulada a modelagem completa do problema de trajetória de mínimo tempo utilizando esta abordagem, incluindo as restrições dinâmicas do veículo e as restrições geométricas da pista. Algumas técnicas possíveis para solução do problema de controle ótimo são sugeridas. / [en] This work proposes a new procedure to determine the optimal trajectory on race tracks based on constrained optimization techniques, where the constraints are defined by means of the dynamic characteristics of the vehicle and the geometrical limits of the track. The vehicle is represented by an oriented particle with the capabilities of traction, braking and normal acceleration, which are typical in a competition vehicle. First, the minimum-time trajectory for a 90-degree curve is obtained through a geometrical analysis of the problem. The solution is then expanded to be applied to all angles. Starting from the individual minimum-time trajectory for each curve of the track, constrained optimization techniques are employed in order to obtain the shorter curve that concatenates these individual optimal trajectories. The dynamic characteristics of some polynomial curves are analyzed to infer the one that can best be applied in the concatenation process. The minimum-time trajectory for the race track obtained by the concatenation procedure is presented and the advantages and disadvantages of the proposed method are discussed. Alternatively, an overview of the optimal control problem is presented and a complete model of the minimum-time trajectory problem is developed using this approach, including the dynamic constraints of the vehicle and the geometric constraints of the track. Some possible methods for the solution of the optimal control problem are suggested.

Motion planning is central to the efficient operation and autonomy of robots in the industry. Generally, motion planning of industrial robots is treated in a two-step approach. First, a geometric path between the start and goal position is planned where the objective is to achieve as short path as possible together with avoiding obstacles. Alternatively, a pre-defined geometric path is provided by the end user. Second, the velocity profile along the geometric path is calculated accounting for system dynamics together with other constraints. This approach is computationally efficient, but yield sub-optimal solutions as the system dynamics is not considered in the first step when the geometric path is planned. In this thesis, an alternative to the two-step approach is investigated and a trajectory planner is designed and implemented which plans both the geometric path and the velocity profile simultaneously. The motion planning problem is formulated as an optimal control problem, which is solved by a direct collocation method where the trajectory is parametrised by splines, and the spline nodes and knots are used as optimization variables. The implemented trajectory planner is evaluated in simulations, where the planner is applied to a simple planar elbow robot and ABB's SCARA robot IRB 910SC. Trade-off between computation time and optimality is identified and the results indicate that the trajectory planner yields satisfactory solutions. On the other hand, the simulations indicate that it is not possible to apply the proposed method on a real robot in real-time applications without significant modifications in the implementation to decrease the computation time.

>Magister Scientiae - MSc / Bank asset management mainly involves profit maximization through invest- ment in loans giving high returns on loans, investment in securities for reducing risk and providing liquidity needs. In particular, commercial banks grant loans to creditors who pay high interest rates and are not likely to default on their loans. Furthermore, the banks purchase securities with high returns and low risk. In addition, the banks attempt to lower risk by diversifying their asset portfolio. The main categories of assets held by banks are loans, treasuries (bonds issued by the national treasury), reserves and intangible assets. In this mini-thesis, we solve an optimal asset allocation problem in banking under the mean-variance frame work. The dynamics of the different assets are modelled as geometric Brownian motions, and our optimization problem is of the mean- variance type. We assume the Basel II regulations on banking supervision. In this contribution, the bank funds are invested into loans and treasuries with the main objective being to obtain an optimal return on the bank asset port- folio given a certain risk level. There are two main approaches to portfolio optimization, which are the so called martingale method and Hamilton Jacobi Bellman method. We shall follow the latter. As is common in portfolio op- timization problems, we obtain an explicit solution for the value function in the Hamilton Jacobi Bellman equation. Our approach to the portfolio prob- lem is similar to the presentation in the paper [Hojgaard, B., Vigna, E., 2007. Mean-variance portfolio selection and efficient frontier for defined contribution pension schemes. ISSN 1399-2503. On-line version ISSN 1601-7811]. We pro- vide much more detail and we make the application to banking. We illustrate our findings by way of numerical simulations.

Le cancer est une maladie caractérisée par la croissance incontrôlée des cellules. Le nombre de personnes atteintes par le carcinome hépatocellulaire (CHC) est en progression croissante. Les traitements utilisés jusqu'à présent par les médecins tels que la chimioembolisation transartérielle (TACE) et la radioembolisation transartérielle (TARE) présentent des limitations à cause des effets secondaires causés sur les tissus sains. En vue d'atteindre un meilleur contrôle tumoral avec le minimum de complications des tissus sains, les approches microrobotiques peuvent apporter des solutions au problème du ciblage thérapeutique. Une solution consiste à contrôler la direction de transporteurs thérapeutiques (bolus magnétiques), composés de microparticules magnétiques et d’agents anti-cancéreux, à l’intérieur des vaisseaux sanguins vers la zone tumorale. Des champs magnétiques extérieurs sont alors utilisés pour propulser, guider et naviguer une flottille de bolus magnétiques au travers du réseau artériel. Cette thèse propose donc une méthodologie globale à mettre en place afin de rendre les procédures locorégionales transartérielles robotisées plus ciblées et plus localisées. Dans un premier temps, nous avons optimisé la quantité de médicament à injecter sous forme de bolus magnétiques. Ensuite, nous nous sommes intéressés à l'optimisation de la structure du bolus en vue d’assurer d’une part, la navigation optimale à l’intérieur des vaisseaux et d’autre part, d’offrir la possibilité d’embarquer une quantité d’agents thérapeutiques plus importante. La navigation des bolus délivrés par un cathéter vers la zone ciblée (tumeur) est assurée grâce au développement et à l'implémentation d’une loi de commande optimale. La validation de l'injection et de la navigation des bolus magnétiques a été réalisée sur une plateforme magnétique robotisée développée dans le cadre de cette thèse. / Cancer is a disease characterized by an uncontrolled cell growth. The number of people with hepatocellular carcinoma (HCC) is growing constantly. The treatments used by doctors until nowadays such as transarterial chemoembolization (TACE) and transarterial Radioembolization (TARE) have limitations because of the side effects caused to healthy tissues. In order to achieve best tumor control with minimal complications on healthy tissues, microrobotics technology can provide solutions to the problem of therapeutic targeting. One solution is to control the direction of the therapeutic carriers (magnetic bolus), composed of magnetic microparticles and anti-cancer agents, inside the blood vessels to the tumor area (target). External magnetic fields are then used to propel, steer and navigate a magnetic bolus fleet through the arterial system. This thesis offers a global methodology to implement in order to make the robotic transarterial locoregional procedures more targeted and localized. First, we have optimized the amount of drug to be injected as magnetic boluses. Then, we have carried out the optimization of the magnetic bolus structure in order to ensure firstly, the optimal navigation inside the vessels and secondly, to offer the possibility of carrying a larger amount of therapeutic agents. The navigation of boluses delivered by the catheter to the target area (tumor) is ensured through the development and implementation of the optimal control law. The validation of the injection and navigation magnetic bolus are performed on a magnetic microrobotic platform.

Esta dissertação tem por objetivo a implementação de um código para simular problemas hidrodinâmicos, bem como a possibilidade de controlar as elevações de onda resultantes numa determinada região por meio de uma vazão ótima controlada dentro do sistema estudado. O algoritmo implementado é baseado nas equações de águas rasas, as quais são aplicáveis em situações onde a altura d’água é de ordem muito menor do que as dimensões do sistema, que é discretizado espacial e temporalmente pelo Método dos Elementos Finitos e pelo método CBS (Characteristic Based-Split), respectivamente. O método de controle consiste na busca de uma curva de vazão de controle ótima que minimize a função objetivo, a qual compara os valores de altura d’água que se deseja encontrar em uma região especificada com os calculados pela simulação numérica. Para isso, utiliza-se um algoritmo evolutivo SCE-UA (Shuffled Complex Evolution – University of Arizona), que busca otimizar parâmetros de geração das curvas de vazão de controle, podendo estas serem modeladas por NURBS (Non- Uniform Rational B-Splines), que são capazes de encontrar a solução ótima, ou modeladas com curvas de forma triangular (linear) ou parabólica (quadrática) que apresentam uma solução aproximada de fácil implementação. Por fim, várias aplicações são realizadas, tanto para a simples simulação, quanto para o controle de problemas hidrodinâmicos, a fim de validar os algoritmos desenvolvidos e os resultados obtidos mostraram que os objetivos foram alcançados, encontrando uma forma eficiente de se fazer o controle de enchentes. / Implementation of a computational code for the numerical simulation of hydrodynamic problems as well as the ability to control the resulting wave elevations in a specific area, using an optimal flow controlled within the studied system are the aims of this work. The implemented algorithm is based on the shallow waters equations, which are applicable in situations where the water height is much smaller than the system dimensions, and are spatial and temporally discretized by the Finite Element Method and the CBS method (Caractheristic Based-Split), respectively. The control method consists in finding an optimal control flow curve that minimizes the objective function, which compares the objective value of water elevations in a specified region with those calculated by numerical simulation. An evolutionary algorithm called SCE-UA (Shuffled Complex Evolution - University of Arizona), which looks for optimize parameters of control flow curves generation, is used. These curves may be modeled by NURBS (Non-Uniform Rational B-Splines) which are able to find the optimal solution, or by curves of triangular (linear) or parabolic quadratic forms, which are an approximate solution easy to implement. Finally, several applications are performed for both simulation and control of hydrodynamic problems in order to validate the developed algorithms, and the results showed that the aims of this work were reached, finding an efficient way to control floods.

Este trabalho apresenta uma proposta para o controle da suspensão veicular utilizando o amortecedor magneto-reológico, sendo o controle proposto composto pela associação de duas estratégias de controle, o controle ótimo e o controle fuzzy. O Controle ótimo é utilizado para determinar a força a ser utilizada pelo amortecedor magneto-reológico, e o controle fuzzy é utilizado para determinar a corrente elétrica, a ser utilizada no amortecedor magento-reológico e é obtido considerando o modelo de Mandani. Para o controle fuzzy, são consideradas duas entradas, a velocidade de deslocamento do pistão do amortecedor e a força prevista pelo controle ótimo, e uma saída, a corrente elétrica [A]. Para demonstrar a eficiência do controle proposto são consideradas simulações computacionais, utilizando um modelo matemático não-linear de um quarto de veículo. A análise do desempenho do controle é realizada, considerando excitações provocadas por irregularidades na pista, as irregularidades são representadas por entradas tipo degrau, impulso e senoidal. As simulações computacionais são realizadas, utilizando o Matlab® e o Simulink. Os resultados das simulações demonstram que o controle proposto aumenta a segurança do veículo e melhora sua dirigibilidade, reduzindo o deslocamento vertical do conjunto eixo e roda e o espaço de trabalho do amortecedor, quando comparado como o sistema passivo. Também contribui com o conforto dos passageiros, reduzindo as oscilações da carroceria, mantendo os níveis de aceleração abaixo dos considerados desconfortáveis pela norma BS 6841, 1987. Para verificar o comportamento do controle proposto, diante de incertezas, são realizadas simulações computacionais, considerando a possibilidade de erros paramétricos. As simulações, considerando os erros paramétricos, demonstram que o controle ótimo, mesmo quando sujeito a incertezas, permanece sendo estável e ótimo. / This work presents a proposal for control of vehicular suspension using the magneto-rheological damper, the proposed control is composed by association of two control strategy, the optimal control and the fuzzy control. The optimal control is used to determine the power to be applied by the magneto-rheological damper, and the fuzzy control is used to determine the electric current to be used in the magneto-rheological damper and is obtained considering the Mandani's model. For the fuzzy control two inputs are considered, the velocity of the piston's damper and the force provided by the optimal control, and one output, the electric current [A]. To demonstrate the efficiency of the proposed control, computational simulations are considered using a nonlinear mathematical model for a quarter-car. The performance of the control is analyzed considering excitements provoked by irregularities in the track, the irregularities are represented by entrances step type, pulse and sinusoidal. The computational simulations are performed using the Matlab® and the Simulink. The results of simulations show that the proposed control increases the vehicle security and improves the drive ability by reducing the vertical wheel displacement and the workspace to be used by the damper when compared to the passive system. It also helps with the comfort of passengers, reducing the bodywork oscillations, maintaining levels of accelerating below considered uncomfortable by standard BS 6841, 1987. To verify the behavior of the proposed control, in the face of uncertainty, computational simulations are carried out, considering the possibility of parametric errors. The simulations, show that the Optimal Control, even when subject to uncertainties, remains stable and optimal.

Partial differential equations are the chief means of providing mathematical models in science, engineering and other fields. Optimal control of partial differential equations (PDEs) has tremendous applications in engineering and science, such as shape optimization, image processing, fluid dynamics, and chemical processes. In this thesis, we develop and analyze several efficient numerical methods for the optimal control problems governed by elliptic PDE, parabolic PDE, and wave PDE, respectively. The thesis consists of six chapters. In Chapter 1, we briefly introduce a few motivating applications and summarize some theoretical and computational foundations of our following developed approaches. In Chapter 2, we establish a new multigrid algorithm to accelerate the semi-smooth Newton method that is applied to the first-order necessary optimality system arising from semi-linear control-constrained elliptic optimal control problems. Under suitable assumptions, the discretized Jacobian matrix is proved to have a uniformly bounded inverse with respect to mesh size. Different from current available approaches, a new strategy that leads to a robust multigrid solver is employed to define the coarse grid operator. Numerical simulations are provided to illustrate the efficiency of the proposed method, which shows to be computationally more efficient than the popular full approximation storage (FAS) multigrid method. In particular, our proposed approach achieves a mesh-independent convergence and its performance is highly robust with respect to the regularization parameter. In Chaper 3, we present a new second-order leapfrog finite difference scheme in time for solving the first-order necessary optimality system of the linear parabolic optimal control problems. The new leapfrog scheme is shown to be unconditionally stable and it provides a second-order accuracy, while the classical leapfrog scheme usually is well-known to be unstable. A mathematical proof for the convergence of the proposed scheme is provided under a suitable norm. Moreover, the proposed leapfrog scheme gives a favorable structure that leads to an effective implementation of a fast solver under the multigrid framework. Numerical examples show that the proposed scheme significantly outperforms the widely used second-order backward time differentiation approach, and the resultant fast solver demonstrates a mesh-independent convergence as well as a linear time complexity. In Chapter 4, we develop a new semi-smooth Newton multigrid algorithm for solving the discretized first-order necessary optimality system that characterizes the optimal solution of semi-linear parabolic PDE optimal control problems with control constraints. A new leapfrog discretization scheme in time associated with the standard five-point stencil in space is established to achieve a second-order accuracy. The convergence (or unconditional stability) of the proposed scheme is proved when time-periodic solutions are considered. Moreover, the derived well-structured discretized Jacobian matrices greatly facilitate the development of an effective smoother in our multigrid algorithm. Numerical simulations are provided to illustrate the effectiveness of the proposed method, which validates the second-order accuracy in solution approximations as well as the optimal linear complexity of computing time. In Chapter 5, we offer a new implicit finite difference scheme in time for solving the first-order necessary optimality system arising in optimal control of wave equations. With a five-point central finite difference scheme in space, the full discretization is proved to be unconditionally convergent with a second-order accuracy, which is not restricted by the classical Courant-Friedrichs-Lewy (CFL) stability condition on the spatial and temporal step sizes. Moreover, based on its advantageous developed structure, an efficient preconditioned Krylov subspace method is provided and analyzed for solving the discretized sparse linear system. Numerical examples are presented to confirm our theoretical conclusions and demonstrate the promising performance of proposed preconditioned iterative solver. Finally, brief summaries and future research perspectives are given in Chapter 6.

Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro / Neste trabalho, será considerado um problema de controle ótimo quadrático para a equação do calor em domínios retangulares com condição de fronteira do tipo Dirichlet é nos quais, a função de controle (dependente apenas no tempo) constitui um termo de fonte. Uma caracterização da solução ótima é obtida na forma de uma equação linear em um espaço de funções reais definidas no intervalo de tempo considerado. Em seguida, utiliza-se uma sequência de projeções em subespaços de dimensão finita para obter aproximações para o controle ótimo, o cada uma das quais pode ser gerada por um sistema linear de dimensão finita. A sequência de soluções aproximadas assim obtidas converge para a solução ótima do problema original. Finalmente, são apresentados resultados numéricos para domínios espaciais de dimensão 1. / In this work, a quadratic optimal control problem will be considered for the heat equation in rectangular domains with Dirichlet type boundary conditions in which the control function (depending only on time) constitutes a source term. A characterization of the solution is obtained in the form of a linear equation in a real function space defined in a considered time interval. Then, a sequence of projections in finite dimensional subspaces is used to obtain approximations for the optimal control, each of them can be generated by a finite dimension linear system. The sequence of approximate solutions obtained in this way converges to an optimal solution of the original problem. Finally, numerical results are presented for spatial domains of 1 dimension.