This dissertation investigates attitude transition maneuvers of a gyrosat-satellite between relative equilibria. The primary challenge in transitioning between relative equilibria is the proper adjustment of the system angular momentum so that upon completing the transition maneuver the gyrostat-satellite will satisfy all the requirements for a relative equilibrium. The system angular momentum is a function of the attitude trajectory taken during the transition maneuver. A new concept, the constant orbital momentum equilibrium trajectory or COMET, is introduced as a means to a straight-forward solution to a subset of the possible transitions between relative equilbria. COMETs are a class of paths in SO(3) that a gyrostat-satellite may travel along that maintain a constant system angular momentum. The primary contributions of this dissertation are the introduction and analysis of COMETs and their application to the problem of transitioning a gyrostat-satellite between two relative equilibria. The current work introduces, defines, and analyzes COMETs in detail. The requirements for a path in SO(3) to be a COMET are defined. It is shown via example that COMETs are closed-curves in SO(3). Visualizations of families of COMETs are presented and discussed in detail. A subset of COMETs are shown to contain critical points that represent isolated relative equilibrium attitudes or furcations of the COMET. The problem of transitioning between two relative equilibria is split into the sub-problems of transitioning between relative equilibria on the same COMET and transitioning between relative equilibria on different COMETs. For transitions between relative equilibria on the same COMET, an open-loop control law is developed that drives a gyrostat-satellite along the COMET until the target relative equilibrium is reached. For transitions between relative equilibria on different COMETs, an open-loop control law is developed that transfers a gyrostat-satellite from the initial relative equilibrium to a relative equilibrium that resides on the same COMET as the target relative equilbrium. Acquisition of the target relative equilibrium is then accomplished via the application of the open-loop control law for transitions between relative equilibria on the same COMET. The results of numeric simulations of gyrostat-satellites executing these transitions are presented. / Ph. D.

In this work we will study the linear stability of a relative equilibrium in the charged nbody problem. To do this, we will introduce the definition of spectral stability of relative equilibria and we will find conditions necessary to have such stability. We will start the work by showing relevant results in the theory of differentials equations, highlighting some important theorems, such as Existence and Uniqueness Theorem, theorems for linear stability, Floquet’s Theorem and Lyapunov Stability and Instability Theorems. We will do a concise study of Hamiltonian systems, in which we will provide results and definitions that will be of great utility during the dissertation. Among such definitions deserves attention the center configurations (C.C.), since we will show results relating them to the relative equilibria. We will introduce the concept of spectral stability and we will see propositions and theorems for the of n-body problem. An example will be displayed brings a particularity to the charged problem and that makes it very different from the classic n-body problem. Finally, we will apply the results obtained in the charged 3-body problem. / Neste trabalho faremos o estudo da estabilidade linear de um equilíbrio relativo no problema carregado de n-corpos. Para isso, introduziremos a definição de estabilidade espectral de um equilíbrio e encontraremos condições necessárias para termos tal estabilidade. Começaremos o trabalho mostrando resultados relevantes na teoria de equações diferenciais, dando destaque a alguns teoremas importantes, como por exemplo, Teorema de Existência e Unicidade, teoremas para estabilidade linear, Teorema de Floquet e Teoremas de Estabilidade e Instabilidade de Lyapunov. Será feito um estudo bastante conciso dos sistemas Hamiltonianos, no qual enunciaremos resultados e definições que serão de grande utilidade no decorrer da dissertação. Entre tais definições merece destaque a de configurações centrais (C.C.), uma vez que exibiremos resultados relacionando-as aos equilíbrios relativos. Introduziremos o conceito de estabilidade espectral e veremos proposições e teoremas para o problema carregado de n-corpos. Será exibido um exemplo que traz uma particularidade ao problema carregado e que o diferencia bastante do problema clássico de n-corpos. Por fim, faremos uma aplicação dos resultados obtidos ao problema carregado de 3-corpos. / São Cristóvão, SE

Dans cette thèse, nous étudions l'ensemble des solutions d'équations aux dérivées partielles résultant de modèles d'astrophysique et de biologie. Nous répondons aux questions de l'existence, mais aussi nous essayons de décrire le comportement de certaines familles de solutions lorsque les paramètres varient. Tout d'abord, nous étudions deux problèmes issus de l'astrophysique, pour lesquels nous montrons l'existence d'ensembles particuliers de solutions dépendant d'un paramètre à l'aide de la méthode de réduction de Lyapunov-Schmidt. Ensuite un argument de perturbation et le théorème du Point xe de Banach réduisent le problème original à un problème de dimension finie, et qui peut être résolu, habituellement, par des techniques variationnelles. Le reste de la thèse est consacré à l'étude du modèle Keller-Segel, qui décrit le mouvement d'amibes unicellulaires. Dans sa version plus simple, le modèle de Keller-Segel est un système parabolique-elliptique qui partage avec certains modèles gravitationnels la propriété que l'interaction est calculée au moyen d'une équation de Poisson / Newton attractive. Une différence majeure réside dans le fait que le modèle est défini dans un espace bidimensionnel, qui est expérimentalement consistant, tandis que les modèles de gravitationnels sont ordinairement posés en trois dimensions. Pour ce problème, les questions de l'existence sont bien connues, mais le comportement des solutions au cours de l'évolution dans le temps est encore un domaine actif de recherche. Ici nous étendre les propriétés déjà connues dans des régimes particuliers à un intervalle plus large du paramètre de masse, et nous donnons une estimation précise de la vitesse de convergence de la solution vers un profil donné quand le temps tend vers l'infini. Ce résultat est obtenu à l'aide de divers outils tels que des techniques de symétrisation et des inégalités fonctionnelles optimales. Les derniers chapitres traitent de résultats numériques et de calculs formels liés au modèle Keller-Segel / In this thesis we study the set of solutions of partial differential equations arising from models in astrophysics and biology. We answer the questions of existence but also we try to describe the behavior of some families of solutions when parameters vary. First we study two problems concerned with astrophysics, where we show the existence of particular sets of solutions depending on a parameter using the Lyapunov-Schmidt reduction method. Afterwards a perturbation argument and Banach's Fixed Point Theorem reduce the original problem to a finite-dimensional one, which can be solved, usually, by variational techniques. The rest of the thesis is de-voted to the study of the Keller-Segel model, which describes the motion of unicellular amoebae. In its simpler version, the Keller-Segel model is a parabolic-elliptic system which shares with some gravitational models the property that interaction is computed through an attractive Poisson / Newton equation. A major difference is the fact that it is set in a two-dimensional setting, which experimentally makes sense, while gravitational models are ordinarily three-dimensional. For this problem the existence issues are well known, but the behaviour of the solutions during the time evolution is still an active area of research. Here we extend properties already known in particular regimes to a broader range of the mass parameter, and we give a precise estimate of the convergence rate of the solution to a known profile as time goes to infinity. This result is achieved using various tools such as symmetrization techniques and optimal functional inequalities. The last chapters deal with numerical results and formal computations related to the Keller-Segel model