Ce travail s’intéresse à la phase de montée d’un aéronef civil. Les trajectoires minimisant le temps de montée ainsi que que celles minimisant la consommation de carburant sont étudiées au travers du contrôle optimal géométrique. La dynamique associée à la phase de montée possède un phénomène dit de perturbation singulière. Ce phénomène, présent dans les systèmes multi-échelle, rend difficile la résolution numérique du problème de contrôle associé. La réduction desystème hamiltonien, permettant de s’affranchir de la difficulté numérique introduite par la perturbation singulière, est étudiée d’un point de vue théorique puis numérique. Dans un second temps, le système réduit est étudié géométriquement. L’utilisation des outils du contrôle géométrique combinée à celui des synthèses à temps court permet de déterminer des familles de trajectoires localement temps-optimales pour des temps courts. Cette étude est complétée par une étude des trajectoires temps-optimales en présence de contraintes d’état. D’un point de vue plus numérique, les méthodes directes et indirectes sont utilisées pour résoudre les différents problèmes. Une synthèse locale est alors réalisée en partant des familles de trajectoires déterminées pour des temps courts. Une étude des trajectoires minimisant la consommation de carburant est également réalisée. / This work concerns the climbing phase of a civil aircraft. The trajectories which minimize the climbing time and the one which minimize the fuel consumption are studied throughout geometric optimal control. The climbing phase dynamics presents a characteristics called singular perturbation. This phenomena exists in multi-scale dynamics which makes the numerical study of the associated control problem difficult. Theoretically and numerically we study the reduction of hamiltonian system. This concept allows to remove the numerical complexity induced by the singular perturbation. Secondly, the reduced system is studied geometrically. Families of timeoptimal trajectories in small time are determined thanks to geometric control tools and small time synthesis. A study of time-optimal trajectories with active state constraints completes this work. From a more numerical point of view, direct and indirect methods are used to solve the climbing problems. A local synthesis for time-optimal trajectory is established starting from the families of trajectory determined in small time. A study of minimum fuel consumption trajectories is also realized.

Orientadores: Marcelo da Silva Montenegro, Massimo Grossi / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-20T13:53:15Z (GMT). No. of bitstreams: 1 Neves_SergioLeandroNascimento_D.pdf: 694748 bytes, checksum: 52d4109b562640e98c9a0a6098d9cb46 (MD5) Previous issue date: 2012 / Resumo: Neste trabalho, consideramos uma classe de problemas de Neumann com perturbação singular e fazemos um estudo do número de soluções do tipo "single peak" que se concentram em um mesmo ponto. Estudamos casos de concentração no interior e na fronteira do domínio. Obtemos um resultado de multiplicidade exata que relaciona o número de tais soluções com o número de zeros estáveis de um campo vetorial associado / Abstract: In this work, we consider a class of Neumann problems with singular perturbation and we study the number of single peak solutions which concentrate at the same point. We study concentration in the interior and at the boundary of the domain. We obtain an exact multiplicity result which relates the number of such solutions with the number of stable zeros of an associated vector field. / Doutorado / Matematica / Doutor em Matemática

Philosophiae Doctor - PhD / In this thesis, we consider some nonlinear differential models that govern unsteady magneto-hydrodynamic convective flow and mass transfer of viscous, incompressible,electrically conducting fluid past a porous plate with/without heat sources. The study focusses on the effect of a combination of a number of physical parameters (e.g., chemical reaction, suction, radiation, soret effect,thermophoresis and radiation absorption) which play vital role in these models.Non dimensionalization of these models gives us sets of differential equations. Reliable solutions of such differential equations can-not be obtained by standard numerical techniques. We therefore resorted to the use of the singular perturbation approaches. To proceed, each of these model problems is discretized in time by using a suitable time-stepping method and then by using a fitted operator finite difference method in spatial direction. The combined methods are then analyzed for stability and convergence. Aiming to study the robustness of the proposed numerical schemes with respect to change in the values of the key parame- ters, we present extensive numerical simulations for each of these models. Finally, we confirm theoretical results through a set of specificc numerical experiments.

Dans cette thèse, on s'est attaché à résoudre un certain nombre de problèmes qui apparaissent lorsqu'on traite des problèmes concrets de contrôle: phénomènes à plusieurs échelles de temps, discontinuités de la commande lors du basculement d'un correcteur à un autre, nécessité de concevoir un nombre limité de correcteurs différents malgré une gamme très importante des produits traités. Pour illustrer concrètement les résultats obtenus, nous nous sommes appuyés sur un exemple industriel concret, le contrôle de guidage de bande durant le processus de laminage dans un laminoir à chaud. D'abord, nous proposons une solution convexe au problème de commande optimale linéaire quadratique pour les systèmes linéaires à deux échelles de temps en temps discret. Ensuite, nous établissons des conditions suffisantes, formulées sous la forme d'inégalités matricielles linéaires, qui permettent de vérifier la stabilité d'un système à commutation à deux échelles de temps et de synthétiser des correcteurs stabilisants. Nous proposons aussi dans ce travail une méthode pour minimiser les discontinuités sur la commande dans le cadre des systèmes à commutation. Dans le contexte du contrôle de guidage de bande pour un laminoir à chaud, nous ne pouvons pas négliger l'influence des paramètres incertains, qui sont dus principalement au fait que ce genre de système traite une gamme de produits très large. Donc, dans la synthèse du correcteur, nous prenons en compte ces variations en divisant l'ensemble des produits en plusieurs familles et en synthétisant un correcteur différent pour chaque famille / This Ph.D. thesis deals with a certain number of problems arising in practical implementation of control systems: multi time scale phenomena, sudden modifications on the system dynamics, discontinuities on the control signal due to controller switchings, the need of design a limited number of controllers in spite of a wide variation on the physical parameters. In order to illustrate the validity of the obtained results, we resort to a real problem concerning the steel production framework, the robust steering control of a hot strip finishing mill. First, a convex solution of the linear quadratic control design for discrete two time scale systems is proposed. Hence, we address the stability problem of two time scale switched systems. We show that stability of the slow and fast switched subsystems under arbitrary switching rules does not imply the stability of the corresponding two time scale switched system in the singular perturbation form. An additional constraint, independent of the value of the singular parameter and of the switching rule, is provided in terms of linear matrix inequalities. We also introduce a bumpless transfer method for switched systems aiming at reducing the discontinuities on the control signal. Dwell time conditions assessing the asymptotic stability of the closed loop switched system are established. The practical contribution of this thesis, the robust steering control design, exploits most of previous results. The objective is to guarantee the stability of the hot strip mill system and improve the quality of the rolled products

Neste trabalho, estuda-se o problema de estabilidade de tensão do ponto de vista dinâmico, enfocando a análise nas instabilidades originadas por grandes perturbações. A modelagem utilizada nestes estudos envolve dinâmicas que possuem múltiplas escalas de tempo, mas na prática, os problemas de instabilidade de tensão estão geralmente associados às dinâmicas de longo prazo. Nesse sentido, o método de análise QSS fornece muitas vantagens do ponto de vista computacional, reduzindo significativamente o tempo de processamento, mediante a substituição das equações de dinâmica rápida por suas correspondentes equações de equilíbrio. As contribuições deste trabalho são duas: a primeira consiste no estudo da importância da região de estabilidade das dinâmicas lentas do SEP e sua relação com a região de estabilidade do sistema original. A segunda consiste em estudar a aplicação do método QSS no SEP oleando para as condições existentes na literatura e analisando principalmente as desvantagens de sua aplicação. Os sistemas de potência apresentados na literatura são utilizados como exemplos. Com base nestas simulações e na teoria existente da análise QSS, são estudadas algumas condições sob as quais o método QSS é válido. A teoria de região de estabilidade para estes sistemas é explorada para fornecer indicativos de margem de estabilidade e controle preventivo. / In this work, we study the problem of voltage stability of the dynamic point of view, focusing the analysis on the instability caused by large disturbances. The modeling used in these studies involves dynamics that have multiple time scales, but in practice, the problems of voltage instability are generally associated with long-term scale. Thus, the QSS method of analysis provides many advantages in terms of computational resource, significantly reducing the processing time, by replacing the equations by their corresponding fast dynamic equilibrium equations. The contributions of this work are two: the first is to study the importance of the stability region of the slow dynamics of the power system and its relation to the stability region of the original system. The second is to study the application of the QSS method in power systems considering the current theory in the literature and analyzing the main disadvantages of its application. The power system presented in the literature are used as examples. Based on these simulations and the existing theory of the QSS method, we study some conditions under which this method is valid. The theory of stability region for these systems is exploited to provide indicative of margin stability and preventive control.

Cette thèse porte sur l'étude de problèmes de contrôle optimal sur des réseaux (c'est-à-dire des ensembles constitués de sous-régions reliées entre elles par des jonctions), pour lesquels on autorise différentes dynamiques et différents coûts instantanés dans chaque sous-région du réseau. Comme dans les cas plus classiques, on aimerait pouvoir caractériser la fonction valeur d'un tel problème de contrôle par le biais d'une équation de Hamilton-Jacobi-Bellman. Cependant, les singularités géométriques du domaine, ainsi que les discontinuités des données ne nous permettent pas d'appliquer la théorie classique des solutions de viscosité. Dans la première partie de cette thèse nous prouvons que les fonctions valeurs de problèmes de contrôle optimal définis sur des réseaux 1-dimensionnel sont caractérisées par de telles équations. Dans la seconde partie les résultats précédents sont étendus au cas de problèmes de contrôle définis sur une jonction 2-dimensionnelle. Enfin, dans une dernière partie, nous utilisons les résultats obtenus précédemment pour traiter un problème de perturbation singulière impliquant des problèmes de contrôle optimal dans le plan pour lesquels les dynamiques et les coûts instantanés peuvent être discontinus à travers une frontière oscillante. / This thesis focuses on the study of optimal control problems defined on networks (i.e. sets consisting of sub-regions connected together through junctions), where different dynamics and different running costs are allowed in each sub-region of the network. As in classical cases, we would like to characterize the value function of such an optimal control problem through an Hamilton-Jacobi-Bellman equation. However, the geometrical singularities of the domain and the data discontinuities do not allow us to apply the classical theory of viscosity solutions. In the first part of this thesis, we prove this kind of characterization for the value functions of optimal control problems defined on 1-dimensional networks. In the second part, the previous results are extended to the case of control problems defined on a 2-dimensional junction. Finally, in the last part, we use the results obtained previously to treat a singular perturbation problem involving optimal control problems in the plane for which the dynamics and running costs can be discontinuous through an oscillating border.

Neste trabalho, estuda-se o problema de estabilidade de tensão do ponto de vista dinâmico, enfocando a análise nas instabilidades originadas por grandes perturbações. A modelagem utilizada nestes estudos envolve dinâmicas que possuem múltiplas escalas de tempo, mas na prática, os problemas de instabilidade de tensão estão geralmente associados às dinâmicas de longo prazo. Nesse sentido, o método de análise QSS fornece muitas vantagens do ponto de vista computacional, reduzindo significativamente o tempo de processamento, mediante a substituição das equações de dinâmica rápida por suas correspondentes equações de equilíbrio. As contribuições deste trabalho são duas: a primeira consiste no estudo da importância da região de estabilidade das dinâmicas lentas do SEP e sua relação com a região de estabilidade do sistema original. A segunda consiste em estudar a aplicação do método QSS no SEP oleando para as condições existentes na literatura e analisando principalmente as desvantagens de sua aplicação. Os sistemas de potência apresentados na literatura são utilizados como exemplos. Com base nestas simulações e na teoria existente da análise QSS, são estudadas algumas condições sob as quais o método QSS é válido. A teoria de região de estabilidade para estes sistemas é explorada para fornecer indicativos de margem de estabilidade e controle preventivo. / In this work, we study the problem of voltage stability of the dynamic point of view, focusing the analysis on the instability caused by large disturbances. The modeling used in these studies involves dynamics that have multiple time scales, but in practice, the problems of voltage instability are generally associated with long-term scale. Thus, the QSS method of analysis provides many advantages in terms of computational resource, significantly reducing the processing time, by replacing the equations by their corresponding fast dynamic equilibrium equations. The contributions of this work are two: the first is to study the importance of the stability region of the slow dynamics of the power system and its relation to the stability region of the original system. The second is to study the application of the QSS method in power systems considering the current theory in the literature and analyzing the main disadvantages of its application. The power system presented in the literature are used as examples. Based on these simulations and the existing theory of the QSS method, we study some conditions under which this method is valid. The theory of stability region for these systems is exploited to provide indicative of margin stability and preventive control.

<p>In recent years, there has been a great interest towards the higher order numerical methods for singularly perturbed problems. As compared to their lower order counterparts, they provide better accuracy with fewer mesh points. Construction and/or implementation of direct higher order methods is usually very complicated. Thus a natural choice is to use some convergence acceleration techniques, e.g., Richardson extrapolation, defect correction, etc. In this thesis, we will consider various classes of problems described by singularly perturbed ordinary and partial differential equations. For these problems, we design some novel numerical methods and attempt to increase their accuracy as well as the order of convergence. We also do the same for existing numerical methods in some instances. We &macr / nd that, even though the Richardson extrapolation technique always improves the accuracy, it does not perform equally well when applied to different methods for certain classes of problems. Moreover, while in some cases it improves the order of convergence, in other cases it does not. These issues are discussed in this thesis for linear and nonlinear singularly perturbed ODEs as well as PDEs. Extrapolation techniques are analyzed thoroughly in all the cases, whereas the limitations of the defect correction approach for certain problems is indicated at the end of the thesis</p>

There is a growing interest in the dynamics of the co-infection of these two diseases. In this thesis, firstly we focus on studying the effect of a distributed delay representing the incubation period for the malaria parasite in the mosquito vector to possibly reduce the initial transmission and prevalence of malaria. This model can be regarded as a generalization of SEI models (with a class for the latently infected mosquitoes) and SI models with a discrete delay for the incubation period in mosquitoes. We study the possibility of occurrence of backward bifurcation. We then extend these ideas to study a full model of HIV and malaria co-infection. To get further inside into the dynamics of the model, we use the geometric singular perturbation theory to couple the fast and slow models from the full model. Finally, since the governing models are very complex, they cannot be solved analytically and hence we develop and analyze a special class of numerical methods to solve them.

Options are a special type of derivative securities because their values are derived from the value of some underlying security. Most options can be grouped into either of the two categories: European options which can be exercised only on the expiration date, and American options which can be exercised on or before the expiration date. American options are much harder to deal with than European ones. The reason being the optimal exercise policy of these options which led to free boundary problems. Ever since the seminal work of Black and Scholes [J. Pol. Econ. 81(3) (1973), 637-659], the differential equation approach in pricing options has attracted many researchers. Recently, numerical singular perturbation techniques have been used extensively for solving many differential equation models of sciences and engineering. In this thesis, we explore some of those methods which are based on spline approximations to solve the option pricing problems. We show a systematic construction and analysis of these methods to solve some European option problems and then extend the approach to solve problems of pricing American options as well as some exotic options. Proposed methods are analyzed for stability and convergence. Thorough numerical results are presented and compared with those seen in the literature.