Neste trabalho, introduzimos os conceitos de controlabilidade e de observabilidade para equações diferenciais ordinárias generalizadas, apresentamos resultados inéditos sobre condições suficientes e necessárias para controlabilidade e para observabilidade para estas equações e também apresentaremos uma aplicação. Utilizando teoremas de correspondência entre equações diferenciais ordinárias generalizadas e outras equações diferenciais, traduzimos os resultados obtidos para os casos particulares de controlabilidade e observabilidade para equações diferenciais em medida e equações diferencias com impulsos. O fato de trabalharmos no ambiente das equações diferenciais ordinárias generalizadas permitiu que os resultados obtidos pudessem envolver funções com muitas descontinuidades e muito oscilantes, ou seja, de variação ilimitada. Os resultados novos apresentados aqui estão contidos no artigo [21] que se encontra em fase final de redação e será submetido à publicação em breve. / In this work, we introduce concepts of controllability and observability for generalized ordinary differential equations, we present new results on necessary and sufficient conditions for controllability and observability for these equations and we also present an application. Using theorems of correspondence between generalized ordinary differential equations and other differential equations, we translate the results obtained for the particular cases of controllability and observability for measure differential equations and differential equations with impulses. The fact that we work in the framework of generalized ordinary differential equations allows us to obtain results where the functions involved can have many discontinuities and be highly oscillating, that is, of unbounded variation. The new results presented here are contained in the preprint [21] which is under final revision and will soon be submitted for publication.

Made available in DSpace on 2016-04-27T16:57:35Z (GMT). No. of bitstreams: 1 Eliane Alves de Oliveira.pdf: 5276695 bytes, checksum: 3ae0043a45ace509116acacd60a2301b (MD5) Previous issue date: 2014-12-19 / This research had the target to check into teaching strategies that could favour students to learn about Ordinary Differential Equations and their applications inEngineering graduation courses. The study directed to the elaboration of a didactic engineering and was centered in the casting definition of these engineering components, having graphic, algebraic and numerical approaches which involved problem situations by means of the use of computational resources. The Theory of Didactical Situations, by Guy Brosseau and the Didactical Engineering, by Michèle Artigue compose the main theoretical-methodological inputs of the research. Sixteen students of the second year of Environmental Engineering and Production Engineering graduation courses of a higher graduation institution voluntarily joined the experiment. The GeoGebra Software was utilized for that. The data collection was made by using the following instruments: activities guide, initial and final knowledge tests and field diary. The results indicated that the software use favored the activities accomplishment and revealed the importance and productivity of arguments in pairs. The obtained data analysis enabled us to assert that the didactical engineering characteristics developed in this workfavouredthe construction of concepts of Ordinary Differential Equations by the students, attending the research aims / Esta pesquisa teve por objetivo investigar estratégias de ensino com vistas a favorecer a aprendizagem deestudantes acerca de Equações Diferenciais Ordinárias e suas aplicações em cursos de graduação em Engenharia. O estudo direcionou-se para a elaboração deuma engenharia didática, e centrou-se na definição do elenco de componentes dessa engenharia, tendo por alvo abordagens gráfica, algébrica e numérica, que envolvessem situações-problema, por meio da utilização de recursos computacionais.ATeoriadas Situações Didáticas de Guy Brousseau e a Engenharia Didática segundo Michèle Artigue compõem os aportesteórico-metodológicos principais da pesquisa. Dezesseis alunos do segundo ano de graduação em Engenharia Ambiental e Engenharia de Produção de uma Instituição de Ensino Superior participaram voluntariamente do experimento. Foi utilizado o software GeoGebra. A coleta de dados foi realizada por meio dos seguintes instrumentos:guias de atividades, teste inicial e final de conhecimentos e diário de campo. Os resultados indicaram que o uso do software favoreceua realização das atividades e revelaram a importância e a produtividade das discussões em dupla. A análise dos dados obtidos possibilitou afirmar que as características da engenharia didática desenvolvida no trabalho favorecerama construção de conceitos de Equações Diferenciais Ordinárias pelos alunos, atendendo os objetivos da pesquisa

Estuda-se o espalhamento de doenças contagiosas utilizando modelos suscetível-infectado-recuperado (SIR) representados por equações diferenciais ordinárias (EDOs) e por autômatos celulares probabilistas (ACPs) conectados por redes aleatórias. Cada indivíduo (célula) do reticulado do ACP sofre a influência de outros, sendo que a probabilidade de ocorrer interação com os mais próximos é maior. Efetuam-se simulações para investigar como a propagação da doença é afetada pela topologia de acoplamento da população. Comparam-se os resultados numéricos obtidos com o modelo baseado em ACPs aleatoriamente conectados com os resultados obtidos com o modelo descrito por EDOs. Conclui-se que considerar a estrutura topológica da população pode dificultar a caracterização da doença, a partir da observação da evolução temporal do número de infectados. Conclui-se também que isolar alguns infectados causa o mesmo efeito do que isolar muitos suscetíveis. Além disso, analisa-se uma estratégia de vacinação com base em teoria dos jogos. Nesse jogo, o governo tenta minimizar os gastos para controlar a epidemia. Como resultado, o governo realiza campanhas quase-periódicas de vacinação. / The spreading of contagious diseases is studied by using susceptible-infected-recovered (SIR) models represented by ordinary differential equations (ODE) and by probabilistic cellular automata (PCA) connected by random networks. Each individual (cell) of the PCA lattice experiences the influence of others, where the probability of occurring interaction with the nearest ones is higher. Simulations for investigating how the disease propagation is affected by the coupling topology of the population are performed. The numerical results obtained with the model based on randomly connected PCA are compared to the results obtained with the model described by ODE. It is concluded that considering the topological structure of the population can pose difficulties for characterizing the disease, from the observation of the time evolution of the number of infected individuals. It is also concluded that isolating a few infected subjects can cause the same effect than isolating many susceptible individuals. Furthermore, a vaccination strategy based on game theory is analyzed. In this game, the government tries to minimize the expenses for controlling the epidemic. As consequence, the government implements quasi-periodic vaccination campaigns.

Neste trabalho, estabelecemos a teoria de dicotomias para equações diferenciais ordinárias generalizadas, introduzindo os conceitos de dicotomias para essas equações generalizadas, estudando as suas propriedades e propondo resultados novos. Investigamos condições para a existência de soluções limitadas e condições para a existência de dicotomia exponencial. Utilizando teoremas de correspondência entre equações diferenciais ordinárias generalizadas e outras equações, traduzimos os resultados obtidos para os casos particulares de dicotomias para equações diferenciais em medida e para equações diferenciais com impulsos. O fato de trabalharmos no ambiente das equações diferenciais ordinárias generalizadas faz com que os resultados obtidos para os casos particulares possam envolver funções com muitas descontinuidades e de variação ilimitada. / In this work we establish the theory of dichotomies for generalized ordinary dierential equations, introducing the concepts of dichotomies for these equations, studying their properties and proposing new results. We investigate conditions of existence of exponential dichotomies and bounded solutions. Using correspondence theorems between generalized ordinary dierential equations and other equations, we translate the obtained results to the particular cases of dichotomies for measure dierential equations and for impulsive dierential equations. The fact that we work in the framework of generalized ordinary dierential equations allows us to obtain results for the particular cases where the functions involved can have many discontinuities and be of unbounded variation.

Neste trabalho, estabelecemos a teoria de dicotomias para equações diferenciais ordinárias generalizadas, introduzindo os conceitos de dicotomias para essas equações generalizadas, estudando as suas propriedades e propondo resultados novos. Investigamos condições para a existência de soluções limitadas e condições para a existência de dicotomia exponencial. Utilizando teoremas de correspondência entre equações diferenciais ordinárias generalizadas e outras equações, traduzimos os resultados obtidos para os casos particulares de dicotomias para equações diferenciais em medida e para equações diferenciais com impulsos. O fato de trabalharmos no ambiente das equações diferenciais ordinárias generalizadas faz com que os resultados obtidos para os casos particulares possam envolver funções com muitas descontinuidades e de variação ilimitada. / In this work we establish the theory of dichotomies for generalized ordinary dierential equations, introducing the concepts of dichotomies for these equations, studying their properties and proposing new results. We investigate conditions of existence of exponential dichotomies and bounded solutions. Using correspondence theorems between generalized ordinary dierential equations and other equations, we translate the obtained results to the particular cases of dichotomies for measure dierential equations and for impulsive dierential equations. The fact that we work in the framework of generalized ordinary dierential equations allows us to obtain results for the particular cases where the functions involved can have many discontinuities and be of unbounded variation.

This thesis presents a real-time autonomous guidance and control method for a quadrotor in a GPS-denied environment. The quadrotor autonomously seeks a destination while it avoids obstacles whose shape and position are initially unknown. We implement the obstacle avoidance and destination seeking methods using off-the-shelf sensors, including a vision-sensing camera. The vision-sensing camera detects the positions of points on the surface of obstacles. We use this obstacle position data and a potential-field method to generate velocity commands. We present a backstepping controller that uses the velocity commands to generate the quadrotor's control inputs. In indoor experiments, we demonstrate that the guidance and control methods provide the quadrotor with sufficient autonomy to fly point to point, while avoiding obstacles.

Un système est défini par les entités et leurs interrelations dans un environnement qui est déterminé par une limite arbitraire. Les systèmes complexes présentent un comportement émergent sans un contrôleur central. Les systèmes concrets désignent ceux qui sont observables dans la réalité. Un modèle nous permet de comprendre, de contrôler et de prédire le comportement du système. Un modèle différentiel à partir d'un système pourrait être compris comme une sorte de loi physique sous-jacent représenté par l'un ou d'un ensemble d'équations différentielles. Ce travail vise à étudier et mettre en œuvre des méthodes pour effectuer la modélisation des systèmes automatisée par l'ordinateur. Cette thèse pourrait être divisée en trois étapes principales, ainsi: (1) le développement d'un solveur numérique automatisé par l'ordinateur pour les équations différentielles linéaires, partielles ou ordinaires, sur la base de la formulation de matrice pour une personnalisation propre de la méthode Ritz-Galerkin; (2) la proposition d'un schème de score d'adaptation qui bénéficie du solveur numérique développé pour guider l'évolution des modèles différentiels pour les systèmes complexes concrets; (3) une implémentation préliminaire d'une application de programmation génétique pour effectuer la modélisation des systèmes automatisée par l'ordinateur. Dans la première étape, il est montré comment le solveur proposé utilise les polynômes de Jacobi orthogonaux comme base complète pour la méthode de Galerkin et comment le solveur traite des conditions auxiliaires de plusieurs types. Solutions à approximations polynomiales sont ensuite réalisés pour plusieurs types des équations différentielles partielles linéaires, y compris les problèmes hyperboliques, paraboliques et elliptiques. Dans la deuxième étape, le schème de score d'adaptation proposé est conçu pour exploiter certaines caractéristiques du solveur proposé et d'effectuer l'approximation polynômiale par morceaux afin d'évaluer les individus différentiels à partir d'une population fournie par l'algorithme évolutionnaire. Enfin, une mise en œuvre préliminaire d'une application GP est présentée et certaines questions sont discutées afin de permettre une meilleure compréhension de la modélisation des systèmes automatisée par l'ordinateur. Indications pour certains sujets prometteurs pour la continuation de futures recherches sont également abordées dans ce travail, y compris la façon d'étendre ce travail à certaines classes d'équations différentielles partielles non-linéaires. / A system is defined by its entities and their interrelations in an environment which is determined by an arbitrary boundary. Complex systems exhibit emergent behaviour without a central controller. Concrete systems designate the ones observable in reality. A model allows us to understand, to control and to predict behaviour of the system. A differential model from a system could be understood as some sort of underlying physical law depicted by either one or a set of differential equations. This work aims to investigate and implement methods to perform computer-automated system modelling. This thesis could be divided into three main stages: (1) developments of a computer-automated numerical solver for linear differential equations, partial or ordinary, based on the matrix formulation for an own customization of the Ritz-Galerkin method; (2) proposition of a fitness evaluation scheme which benefits from the developed numerical solver to guide evolution of differential models for concrete complex systems; (3) preliminary implementations of a genetic programming application to perform computer-automated system modelling. In the first stage, it is shown how the proposed solver uses Jacobi orthogonal polynomials as a complete basis for the Galerkin method and how the solver deals with auxiliary conditions of several types. Polynomial approximate solutions are achieved for several types of linear partial differential equations, including hyperbolic, parabolic and elliptic problems. In the second stage, the proposed fitness evaluation scheme is developed to exploit some characteristics from the proposed solver and to perform piecewise polynomial approximations in order to evaluate differential individuals from a given evolutionary algorithm population. Finally, a preliminary implementation of a genetic programming application is presented and some issues are discussed to enable a better understanding of computer-automated system modelling. Indications for some promising subjects for future continuation researches are also addressed here, as how to expand this work to some classes of non-linear partial differential equations.

<p>In computational science it is common to describe dynamic systems by mathematical models in forms of differential or integral equations. These models may contain parameters that have to be computed for the model to be complete. For the special type of ordinary differential equations studied in this thesis, the resulting parameter estimation problem is a separable nonlinear least squares problem with equality constraints. This problem can be solved by iteration, but due to complicated computations of derivatives and the existence of several local minima, so called short-cut methods may be an alternative. These methods are based on simplified versions of the original problem. An algorithm, called the modified Kaufman algorithm, is proposed and it takes the separability into account. Moreover, different kinds of discretizations and formulations of the optimization problem are discussed as well as the effect of ill-conditioning.</p><p>Computation of parameters often includes as a part solution of linear system of equations <i>Ax = b</i>. The corresponding pseudoinverse solution depends on the properties of the matrix <i>A</i> and vector <i>b</i>. The singular value decomposition of <i>A</i> can then be used to construct error propagation matrices and by use of these it is possible to investigate how changes in the input data affect the solution <i>x</i>. Theoretical error bounds based on condition numbers indicate the worst case but the use of experimental error analysis makes it possible to also have information about the effect of a more limited amount of perturbations and in that sense be more realistic. It is shown how the effect of perturbations can be analyzed by a semi-experimental analysis. The analysis combines the theory of the error propagation matrices with an experimental error analysis based on randomly generated perturbations that takes the structure of <i>A</i> into account</p>

Several targeting algorithms are developed and analyzed for possible future use onboard a spacecraft. Each targeter is designed to determine the appropriate propulsive burn for translunar injection to obtain desired orbital parameters upon arrival at the moon. Primary design objectives are to minimize the computational requirements for each algorithm but also to ensure reasonable accuracy, so that the algorithm’s errors do not force the craft to conduct large mid-course corrections. Several levels of accuracy for dynamical models are explored, the convergence range and speed of each algorithm are compared, and the possible benefits of the Broyden and trust-region targeters are evaluated. These targeters provide a proof of concept for the feasibility of a translunar injection targeting algorithm. Anticipating some future improvements, these algorithms could serve as a viable alternative to uploading ground-based targeting solutions and bypass the problems of delays and disruptions in communication, enabling the craft to conduct a translunar injection burn autonomously.

In computational science it is common to describe dynamic systems by mathematical models in forms of differential or integral equations. These models may contain parameters that have to be computed for the model to be complete. For the special type of ordinary differential equations studied in this thesis, the resulting parameter estimation problem is a separable nonlinear least squares problem with equality constraints. This problem can be solved by iteration, but due to complicated computations of derivatives and the existence of several local minima, so called short-cut methods may be an alternative. These methods are based on simplified versions of the original problem. An algorithm, called the modified Kaufman algorithm, is proposed and it takes the separability into account. Moreover, different kinds of discretizations and formulations of the optimization problem are discussed as well as the effect of ill-conditioning. Computation of parameters often includes as a part solution of linear system of equations Ax = b. The corresponding pseudoinverse solution depends on the properties of the matrix A and vector b. The singular value decomposition of A can then be used to construct error propagation matrices and by use of these it is possible to investigate how changes in the input data affect the solution x. Theoretical error bounds based on condition numbers indicate the worst case but the use of experimental error analysis makes it possible to also have information about the effect of a more limited amount of perturbations and in that sense be more realistic. It is shown how the effect of perturbations can be analyzed by a semi-experimental analysis. The analysis combines the theory of the error propagation matrices with an experimental error analysis based on randomly generated perturbations that takes the structure of A into account