Este trabalho teve por objetivo investigar alguns aspectos dinâmicos de modelos de gravidade generalizada escalares-tensoriais e f(R), que pretendem resolver de modo mais natural o problema da existência da energia escura, que seria a componente do Universo responsável por sua expansão acelerada. Num espaço-tempo de Friedmann-Lemaître-Robertson-Walker com curvatura espacial nula foi possível escrever as equações de movimento de forma a se obter um sistema dinâmico com um número menor de variáveis e cujo espaço de fase foi estudado genericamente e esboçado para alguns modelos em particular. Em seguida, as regiões dinamicamente proibidas e os pontos fixos do espaço de fase foram analisados. Para os modelos f(R), apresentamos Lagrangianas e Hamiltonianas efetivas e deduzimos uma expressão geral para o parâmetro de equação de estado w. Discutimos ainda a equivalência entre os modelos f(R) e os escalares-tensoriais. Por fim, introduzimos o Princípio de Maupertuis-Jacobi, que permite relacionar a Lagrangiana de um sistema mecânico a uma métrica numa determinada variedade Riemanniana, para determinar singularidades que podem surgir nos modelos f(R), tanto numa métrica isotrópica como numa anisotrópica do tipo mais simples (Bianchi tipo I). Encontramos, de maneira mais direta, as mesmas singularidades já conhecidas através de métodos de análise dinâmica. / This work aims the investigation of some dynamical aspects of generalized gravity models, namely scalar-tensor and f(R) models. These models intend to solve in a more natural way the problem of the existence of the dark energy, which is supposedly the component of the Universe that causes its accelerated expansion. In a null spatial curvature Friedmann-Lemaître-Robertson-Walker spacetime, it has been possible to write the equations of movement in a fashion that allowed us to obtain a dynamical system with a reduced number of variables, whose phase space has been generically studied and depicted for some particular models. In sequence, the dynamically forbidden regions and the fixed points of the phase space have been analyzed. For f(R) models, we have presented effective Lagrangians and Hamiltonians and derived a general expression for the equation of state parameter w. Furthermore, we have discussed the equivalence between f(R) and scalar-tensor models. Finally, we have introduced the Maupertuis-Jacobi Principle, which allows one to relate the Lagrangian for a mechanical system to a metric in a certain Riemannian manifold, to determine singularities which may appear in f(R) models, in an isotropic metric as well as in an anisotropic one of the simplest kind (Bianchi type I). We have found, in a more direct way, the same singularities that arise by using dynamical analysis methods.

Cette thèse traite de l'identification des paramètres dynamiques des robots, en s'appuyant sur les méthodes d'identification en robotique, qui utilisent le modèle dynamique inverse, ou le modèle de puissance, ou le modèle d'énergie du robot. Ce travail revisite le modèle d'énergie en exploitant le caractère intégral des fonctions modulatrices appliquées au modèle de puissance du robot. En outre, les procédures d'intégration sont analysées dans le domaine fréquentiel, et certains groupes de fonctions modulatrices sont sélectionnés afin d'offrir un bon comportement de filtre passe-bas. Ensuite, l'introduction d'un différentiateur algèbrique récemment développé est proposé, nommé différentiateurs de Jacobi. L'analyse est effectuée dans le domaine temporel, et dans le domaine fréquenciel, ce qui met en évidence la propriété de filtrage passe bande et permet de sélectionner les paramètres des différentiateurs. Puis, ces différentiateurs sont appliqués avec succès à l'identification de robot, ce qui prouve leur bonne performance. Les comparaisons entre les différents modèles d'identification, les différenciateurs, les techniques des moindres carrés sont présentées et des conclusions sont tirées dans le domaine de l'identification de robot. / This thesis discusses the identification issues of the robot dynamic parameters. Starting with the well-known inverse dynamic identification model, power and energy identification models for robots, it extends the identification model from an energy point of view, by integrating modulating functions with robot power model. This new identification model avoids the computation of acceleration data. As well, the integration procedures are analyzed in frequency domain so that certain groups of modulating functions are selected in order to offer a good low-pass filtering property. Then, a recently developed high order algebraic differentiator is proposed and studied, named Jacobi differentiators. The analyses are done in both the time domain and in the frequency domain, which gives a clear clue about the differentiator filtering property and about how to select the differentiator parameters. Comparisons among different identification models, differentiators, least square techniques are presented and conclusions are drawn in the robot identification issues.

Cette thèse concerne les jeux différentiels à somme nulle et à deux joueurs avec information incomplète. La structure de l'information est liée à un signal que reçoivent les joueurs. Cette information est dite symétrique quand la connaissance du signal est la même pour les deux joueurs (le signal est public), et asymétrique quand les signaux reçus par les joueurs peuvent être différents (le signal est privé).Ces signaux sont révélés au cours du jeu. Dans plusieurs situations de tels jeux, il est montré dans cette thèse, l'existence d'une valeur du jeu et sa caractérisation comme unique solution d'une équation aux dérivées partielles.Un type de structure d'information concerne le cas symétrique où le signal est réduit à la connaissance par les joueurs de l'état du système au moment où celui-ci atteint une cible donnée (les données initiales inconnues sont alors révélées). Pour ce type du jeu, nous avons introduit des stratégies non anticipatives qui dépendent du signal et nous avons obtenu l'existence d'une valeur.Comme les fonctions valeurs sont en général irrégulières (seulement continues), un des points clefs de notre approche est de prouver des résultats d'unicité et des principes de comparaison pour des solutions de viscosité lipschitziennes de nouveaux types d'équation d'Hamilton-Jacobi-Isaacs associées aux jeux étudiés. / In this thesis we investigate two-person zero-sum differential games with incomplete information. The information structure is related to a signal communicated to the players during the game.In such games, the information is symmetric if both players receive the same signal (namely it is a public signal). Otherwise, if the players could receive different signals (i.e. they receive private signals), the information is asymmetric. We prove in this thesis the existence of value and the characterization of the value function by a partial differential equation for various types of such games.A particular type of such information structure is the symmetric case in which the players receive as their signal the current state of the dynamical system at the moment when the state of the dynamic hits a fixed target set (the unknown initial data are then revealed to both players). For this type of games, we introduce the notion of signal-depending non-anticipative strategies with delay and we prove the existence of value with such strategies.As the value functions are in general irregular (at most continuous), a crucial step of our approach is to prove the uniqueness results and the comparison principles for viscosity solutions of new types of Hamilton-Jacobi-Isaacs equation associated to the games studied in this thesis.

Este trabalho teve por objetivo investigar alguns aspectos dinâmicos de modelos de gravidade generalizada escalares-tensoriais e f(R), que pretendem resolver de modo mais natural o problema da existência da energia escura, que seria a componente do Universo responsável por sua expansão acelerada. Num espaço-tempo de Friedmann-Lemaître-Robertson-Walker com curvatura espacial nula foi possível escrever as equações de movimento de forma a se obter um sistema dinâmico com um número menor de variáveis e cujo espaço de fase foi estudado genericamente e esboçado para alguns modelos em particular. Em seguida, as regiões dinamicamente proibidas e os pontos fixos do espaço de fase foram analisados. Para os modelos f(R), apresentamos Lagrangianas e Hamiltonianas efetivas e deduzimos uma expressão geral para o parâmetro de equação de estado w. Discutimos ainda a equivalência entre os modelos f(R) e os escalares-tensoriais. Por fim, introduzimos o Princípio de Maupertuis-Jacobi, que permite relacionar a Lagrangiana de um sistema mecânico a uma métrica numa determinada variedade Riemanniana, para determinar singularidades que podem surgir nos modelos f(R), tanto numa métrica isotrópica como numa anisotrópica do tipo mais simples (Bianchi tipo I). Encontramos, de maneira mais direta, as mesmas singularidades já conhecidas através de métodos de análise dinâmica. / This work aims the investigation of some dynamical aspects of generalized gravity models, namely scalar-tensor and f(R) models. These models intend to solve in a more natural way the problem of the existence of the dark energy, which is supposedly the component of the Universe that causes its accelerated expansion. In a null spatial curvature Friedmann-Lemaître-Robertson-Walker spacetime, it has been possible to write the equations of movement in a fashion that allowed us to obtain a dynamical system with a reduced number of variables, whose phase space has been generically studied and depicted for some particular models. In sequence, the dynamically forbidden regions and the fixed points of the phase space have been analyzed. For f(R) models, we have presented effective Lagrangians and Hamiltonians and derived a general expression for the equation of state parameter w. Furthermore, we have discussed the equivalence between f(R) and scalar-tensor models. Finally, we have introduced the Maupertuis-Jacobi Principle, which allows one to relate the Lagrangian for a mechanical system to a metric in a certain Riemannian manifold, to determine singularities which may appear in f(R) models, in an isotropic metric as well as in an anisotropic one of the simplest kind (Bianchi type I). We have found, in a more direct way, the same singularities that arise by using dynamical analysis methods.

Orientador: Edmundo Capelas de Oliveira / Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-17T14:35:07Z (GMT). No. of bitstreams: 1 Rosa_EsterCristinaFontesdeAquino_M.pdf: 847998 bytes, checksum: d177526572b19cc1fdd5eeccdf511380 (MD5) Previous issue date: 2011 / Resumo: Este trabalho tem por objetivo modelar e resolver, matematicamente, um problema físico conhecido como pêndulo simples. Discutimos, como caso particular, as chamadas oscilações de pequena amplitude, isto é, uma aproximação que nos leva a mostrar que o período de oscilação é proporcional à raiz quadrada do quociente entre o comprimento do pêndulo e a aceleração da gravidade. Como vários outros problemas oriundos da Física, o pêndulo simples é representado através de equações diferenciais parciais. Assim, na busca de sua solução, aplicamos a metodologia de separação de variáveis que nos leva a um conjunto de equações ordinárias passíveis de simples integração. Escolhendo um sistema de coordenadas adequado, é conveniente usar o método de Hamilton-Jacobi, discutindo, antes, o problema do oscilador harmónico, apresentando, em seguida, o problema do pêndulo simples e impondo condições a fim de mostrar que as equações diferenciais associadas a esses dois sistemas são iguais, ou seja, suas soluções são equivalentes. Para tanto, estudamos o método de separação de variáveis associado às equações diferenciais parciais, lineares e de segunda ordem, com coeficientes constantes e três variáveis independentes, bem como a respectiva classificação quanto ao tipo. Posteriormente, estudamos as equações hipergeométricas, cujas soluções, as funções hipergeométricas. podem ser encontradas pelo método de Frobenius. Apresentamos o método de Hamilton-Jacobi, já mencionado, para o enfren-tamento do problema apresentado. Fizemos no capítulo final um apêndice sobre a função gama por sua presente importância no trato de funções hipergeométricas, em especial a integral elíptica completa de primeiro tipo que compõe a solução exata do período do pêndulo simples / Abstract: This work aims to present and solve, mathematically, the physics problem that is called simple pendulum. We reasoned, as an specific case, the so called low amplitude oscillation, that is, a convenient approximation that make us show that the period of oscillation is proportional to the quotient square root between the pendulum length and the gravity acceleration. Like several other problems arising from the physics, we are going to broach it through partial differential equations. Thus, in the search of its solution, we made use of the variable separation methodology that leads us to a body of ordinary equations susceptible of simple integration. Choosing an appropriate coordinate system, it is convenient to use the method Hamilton-Jacobi, arguing, first, the problem of the harmonic oscillator, with, then the problem of sf simple pendulum and imposing conditions to show that the differential equations associated with these two systems are equal, that is, their solutions are equivalent. With the purpose of reaching the objectives, we studied the variable separation method associated with partial differential equations, linear and of second order, with constant coefficient and three independent variables, as well as the respective classification about the type. Afterwards, we studied the hypergeometrical equations whose solutions, the hypergeometrical functions, are found by the Frobenius method. Introducing the Hamilton-Jacobi method, already mentioned, for addressing the problem presented. We made an appendix in the final chapter on the gamma function by its present importance in dealing with hypergeometric functions, in particular the elliptic integral of first kind consists of the exact period of sf simple pendulum / Mestrado / Fisica-Matematica / Mestre em Matemática

Neste trabalho buscamos caracterizar o espectro de uma classe de operadores bloco--Jacobi limitados definidos em $l^2(\\Lambda,\\mathbb{C}^L)$ ($\\Lambda: \\mathbb{Z}_+\\times\\{0,1,\\ldots,L-1\\}$ representa uma faixa de largura $L\\ge 2$ no semi--plano $\\mathbb{Z}_+^2$) e sujeitos a perturbações esparsas (no sentido que as distâncias entre as ``barreiras\'\' crescem geometricamente à medida que estas se afastam da origem) distribuídas aleatoriamente. Tais operadores são construídos a partir da soma de Kronecker de matrizes de Jacobi $J$, cada qual atuando em uma direção do espaço. Demonstramos, por meio da bloco--diagonalização do operador, que %o estudo de suas principais propriedades espectrais dependem da %se limita à caracterização da ``medida de mistura\'\' $\\frac{1}{L}\\sum_{j=0}^{L-1}\\mu_j$, $\\mu_j$ a medida espectral associada à matriz de Jacobi $J^j=J+2\\cos(2\\pi j/L)I $. Para tanto, buscamos primeiramente caracterizar cada uma das medidas $\\mu_j$, explorando e aperfeiçoando algumas técnicas bastante conhecidas no estudo de operadores esparsos unidimensionais. Demonstramos, por exemplo, que a seqüência de ângulos de Prüfer (variáveis que, juntamente com os raios de Prüfer, parametrizam as soluções da equação de autovalores) é uniformemente distribuída no intervalo $[0,\\pi)$, o %que %resultado que nos permite determinar o comportamento assintótico médio das soluções da equação de autovalores. Tal resultado, aliado às técnicas desenvolvidas por Marchetti \\textit{et. al.} em \\cite{MarWre} e a uma adaptação dos critérios de Last e Simon \\cite{LS} para operadores esparsos, nos permitem demonstrar a existência de uma transição aguda (pontual) entre os espectros singular--contínuo e puramente pontual. Empregamos em seguida os resultados de Jitomirskaya e Last presentes em \\cite{JitLast} e obtemos a dimensão Hausdorff exata associada à medida $\\mu_j$, dada por $\\alpha_j=1+\\frac{4(1-p^2)^2}{p^2(4- (\\lambda-2\\cos(2\\pi j/L))^2)}$ ($\\lambda\\in[-2,2]$), recuperando um resultado análogo obtido por Zlato\\v s em \\cite{Zla}. Por fim, adaptamos tais resultados à situação da medida de mistura associada à matriz bloco--Jacobi, obtendo $\\alpha=\\min_{j\\in\\mathcal{I}(\\lambda)}\\alpha_j$, $\\mathcal{I}(\\lambda):\\{m \\in\\{0,1,\\ldots,L-1\\}:\\lambda\\in[-2+2\\cos(2\\pi j/L),2+2\\cos(2\\pi j/L)]\\}$, como sua dimensão Hausdorff exata. Estudamos modelos idênticos com esparsidades sub e super-geométricas, obtendo na primeira situação um espectro puramente pontual (de dimensão Hausdorff nula) e na segunda um espectro puramente singular--contínuo (de dimensão Hausdorff 1). Finalmente, verificamos a existência de transição entre os espectros puramente pontual e singular--contínuo em um modelo com esparsidade super-geométrica cuja dimensão Hausdorff associada à medida espectral é nula. / In this work we attempt to caracterize the spectrum of a class of limited block--Jacobi operators defined in $l^2(\\Lambda,\\mathbb{C}^L)$ ($\\Lambda: \\mathbb{Z}_+\\times\\{0,1,\\ldots,L-1\\}$ represents a strip of width $L\\ge 2$ on the semi--plane $\\mathbb{Z}_+^2$) subject to a sparse perturbation (which means that the distance between the ``barries\'\' grow geometrically with their distance to the origin) randomly distributed. Such operators are defined as Kronecker sums of unidimensional Jacobi matrices $J$, each one acting in different directions of the space. We prove, by means of a block--diagonalization of the operator, that %the study of its most relevant spectral properties depend on %is related to the caracterization of the ``mixture measure\'\' $\\frac{1}{L}\\sum_{j=0}^{L-1}\\mu_j$, $\\mu_j$ the spectral measure of the Jacobi matrix $J^j=J+2\\cos(2\\pi j/L)I$. For this, we must characterize at first each one of the measures $\\mu_j$, exploiting and improving some well known techniques developed in the study of unidimensional sparse operators. We prove, for instance, that the sequence of Prüfer angles (variables which parametrize the solutions of the eigenvalue equation) are uniform distributed on the interval $[0,\\pi)$, a result which gives us condition to determine the average asymptotic behavior of the solutions of the eigenvalue equation. Such result, in association with the techniques developed by Marchetti \\textit{et. al.} in \\cite{MarWre} and with an adaptation of Last--Simon \\cite{LS} criteria for sparse operator, permit us to prove the existence of a sharp transition between singular continuous and pure point spectra. Following on, we use the results from Jitomirskaya--Last of \\cite{JitLast} and obtain the exact Hausdorff dimension of the measure $\\mu_j$, given by $\\alpha_j=1+\\frac{4(1-p^2)^2}{p^2(4-(\\lambda-2\\cos(2\\pi j/L))^2)}$ ($\\lambda\\in[- 2,2]$), recovering an analogous result due to Zlato\\v s in \\cite{Zla}. At last, we adapt these results to the mixture measure of the block--Jacobi matrix, obtaining $\\alpha=\\min_{j\\in\\mathcal{I}(\\lambda)}\\alpha_j$, $\\mathcal{I}(\\lambda):\\{m \\in\\{0,1,\\ldots,L-1\\}:\\lambda\\in[-2+2\\cos(2\\pi j/L),2+2\\cos(2\\pi j/L)]\\}$, as its exact Hausdorff dimension. We study as well identical models with sub and super geometric sparsities conditions, obtaining a pure point spectrum (with null Hausdorff dimension) in the first case, and a purely singular continuous spectrum (such that its Hausdorff dimension is 1) in the second. Finally, we prove the existence of a transition between pure point and singular continuous spectra in a model with sub--geometric sparsity whose Hausdorff dimension related to the spectral measure is null.

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Cette thèse se compose de deux parties principales. Dans la première partie, le Chapitre 3 est consacré à l'étude du comportement limite d'un système contrôlé singulièrement perturbé avec deux variables d'état qui sont faiblement couplées. Afin de prouver notre résultat d'approximation, nous utilisons la méthode de moyennisation et un résultat récent sur le contrôle nonexpansif. La principale nouveauté de notre approche est de permettre la dynamique limite de dépendre de l'état initial du système rapide. Notons que dans la littérature, le comportement limite d'un tel système a été généralement traité dans des conditions qui garantissent que la limite est indépendante de l'état initial du système rapide. Dans le Chapitre 4, nous généralisons les résultats du Chapitre 3 supposant une condition de nonexpansivité plus générale. De plus, nous considérons un exemple ou la nouvelle condition de nonexpansivité est satisfaite, mais pas la condition de nonexpansivité du Chapitre 3. Dans la deuxième partie de la thèse, le Chapitre 5 porte sur les représentations stables des Hamiltoniens convexes associant à un Hamiltonien donné des fonctions correspondant au problème de Bolza en controle optimal. Dans le Chapitre 6 nous étudions également la stabilité des solutions des équations d'Hamilton-Jacobi-Bellman sous contraintes d'état en exploitant la stabilité des fonctions valeur d'une famille de problèmes de contrôle optimal de Bolza sous contraintes d'état. Nous montrons que sous des hypothèses appropriées, la fonction valeur est la solution unique d'équation d'Hamilton-Jacobi-Bellman et que les solutions sont stables par rapport à l'Hamiltonien et les contraintes d'état. / This thesis consists of two main parts. In the first part, Chapter 3 is devoted to the investigation of the limit behavior of a singularly perturbed control system with two state variables which are weakly coupled. In order to prove our approximation result we use the so called averaging method and a recent result on nonexpansive control. The main novelty of our averaging approach lies in the fact that the limit dynamic may depend on the initial condition of the fast system. In the literature, the investigation of the limit behavior of such systems has been usually addressed under conditions that ensure that the limit dynamic is independent from the initial condition of the fast system. In Chapter 4, we generalise the results of Chapter 3 by considering a more general nonexpansivity condition. Moreover, we consider an example where the new nonexpansity condition is satisfied but the nonexpansivity condition of Chapter 3 does not hold true. The second part deals with Hamilton-Jacobi equations under state constraints. Chapter 5 focuses on the stable representation of convex Hamiltonians by functions describing a Bolza optimal control problem. In Chapter 6 we investigate stability of solutions of Hamilton-Jacobi-Bellman equations under state constraints by studying stability of value functions of a suitable family of Bolza optimal control problems under state constraints. We show that under suitable assumptions, the value function is a unique viscosity solution to Hamilton-Jacobi-Bellman equation and that solutions are stable with respect to Hamiltonians and state constraints.

Inthis degree project, a solution on a coarse grid is recovered by fitting apartial differential equation to a few known data points. The PDE to consideris the heat equation and the Dupire’s equation with their synthetic data,including synthetic data from the Black-Scholes formula. The approach to fit aPDE is by optimal control to derive discrete approximations to regularized Hamiltoncharacteristic equations to which discrete stepping schemes, and parameters forsmoothness, are examined. By non-parametric numerical implementation thedervied method is tested and then a few suggestions on possible improvementsare given / I detta examensarbete återskapas en lösning på ett glest rutnät genom att anpassa en partiell differentialekvation till några givna datapunkter. De partiella differentialekvationer med deras motsvarande syntetiska data som betraktas är värmeledningsekvationen och Dupires ekvation inklusive syntetiska data från Black-Scholes formel. Tillvägagångssättet att anpassa en PDE är att med hjälp av optimal styrning härleda diskreta approximationer på ett system av regulariserade Hamilton karakteristiska ekvationer till vilka olika diskreta stegmetoder och parametrar för släthet undersöks. Med en icke-parametrisk numerisk implementation prövas den härledda metoden och slutligen föreslås möjliga förbättringar till metoden.

This thesis aims to study a connection between indecomposable elements in the cubic fields and the Jacobi-Perron algorithm (JPA). JPA is a multidimensional generalization of the usual continued fractions algorithm. We work in the family of Ennola's cubic fields and we examine how the indecomposable elements are related to elements originating from this algorithm and whether some of these elements generate all indecomposable elements in the fields. We formulate conjectures on how to determine which elements will generate the indecomposable elements. We also prove some necessary conditions that have to hold for elements originating from this algorithm to generate indecomposable elements. 1