Thèses sur le sujet « Hamiltonian Problem »

Dans la classe de complexité QMA – la généralisation quantique de la classe NP – un état quantique est fourni comme preuve à un algorithme de vérification pour l’aider à résoudre un problème. Cette classe de complexité a un problème complet naturel, le problème des Hamiltoniens locaux. Inspiré par la Physique de la matière condensée, ce problème concerne l’énergie de l’état fondamental d’un système quantique. Dans le cadre de cette thèse, nous étudions quelques problèmes liés à la classe QMA et au problème des Hamiltoniens locaux. Premièrement, nous étudions la différence de puissance si au lieu d’une preuve quantique, l’algorithme de vérification quantique reçoit une preuve classique. Nous proposons un cadre intermédiaire à ces deux cas, où la preuve consiste en un état quantique “plus simple” et nous arrivons à démontrer que ces états plus simples sont suffisants pour résoudre tous les problèmes dans QMA. À partir de ce résultat, nous obtenons un nouveau problème QMA-complet et nous étudions aussi la version de notre nouvelle classe de complexité avec erreur unilatérale. Ensuite, nous proposons le premier schéma de délégation vérifiable relativiste de calcul quantique. Dans ce cadre, un client classique délègue son calcul quantique à deux serveurs quantiques intriqués. Ces serveurs peuvent communiquer entre eux en respectant l’hypothèse que l’information ne peut pas être propagé plus vite que la vitesse de la lumière. Ce protocole a été conçu à partir d’un jeu non-local pour le problème des Hamiltoniens locaux avec deux prouveurs et un tour de communication. Dans ce jeu, les prouveurs exécutent des calculs quantiques de temps polynomiaux sur des copies de l’état fondamental du Hamiltonien. Finalement, nous étudions la conjecture PCP quantique, où l’on demande si tous les problèmes dans la classe QMA acceptent un système de preuves où l’algorithme de vérification a accès à un nombre constant de qubits de la preuve quantique. Notre première contribution consiste à étendre le modèle QPCP avec une preuve auxiliaire classique. Pour attaquer le problème, nous avons proposé une version plus faible de la conjecture QPCP pour ce nouveau système de preuves. Nous avons alors montré que cette nouvelle conjecture peut également être exprimée dans le contexte des problèmes des Hamiltoniens locaux et ainsi que dans lecadre de la maximisation de la probabilité de acceptation des jeux quantiques. Notre résultat montre la première équivalence entre un jeu multi-prouveur et une conjecture QPCP
In QMA, the quantum generalization of the complexity class NP, a quantum state is provided as a proof of a mathematical statement, and this quantum proof can be verified by a quantum algorithm. This complexity class has a very natural complete problem, the Local Hamiltonian problem. Inspired by Condensed Matters Physics, this problem concerns the groundstate energy of quantum systems. In this thesis, we study some problems related to QMA and to the Local Hamiltonian problem. First, we study the difference of power when classical or quantum proofs are provided to quantum verification algorithms. We propose an intermediate setting where the proof is a “simpler” quantum state, and we manage to prove that these simpler states are enough to solve all problems in QMA. From this result, we are able to present a new QMA-complete problem and we also study the one-sided error version of our new complexity class. Secondly, we propose the first relativistic verifiable delegation scheme for quantum computation. In this setting, a classical client delegates her quantumcomputation to two entangled servers who are allowed to communicate, but respecting the assumption that information cannot be propagated faster than speed of light. This protocol is achieved through a one-round two-prover game for the Local Hamiltonian problem where provers only need polynomial time quantum computation and access to copies of the groundstate of the Hamiltonian. Finally, we study the quantumPCP conjecture, which asks if all problems in QMA accept aproof systemwhere only a fewqubits of the proof are checked. Our result consists in proposing an extension of QPCP proof systems where the verifier is also provided an auxiliary classical proof. Based on this proof system, we propose a weaker version of QPCP conjecture. We then show that this new conjecture can be formulated as a Local Hamiltonian problem and also as a problem involving the maximum acceptance probability of multi-prover games. This is the first equivalence of a multi-prover game and some QPCP statement

The SHIRA method of Mehrmann and Watkins belongs among the structure preserving Krylov subspace methods for solving skew-Hamiltonian eigenvalue problems. It can also be applied to Hamiltonian eigenproblems by considering a suitable transformation. Structure induced shift-and-invert techniques are employed to steer the algorithm towards the interesting region of the spectrum. However, the shift cannot be altered in the middle of the computation without discarding the information that has been accumulated so far. This paper shows how SHIRA can be combined with ideas from Ruhe's Rational Krylov algorithm to yield a method that permits an adjustment of shift after every step of the computation, adding greatly to the flexibility of the algorithm. We call this new method rational SHIRA. A numerical example is presented to demonstrate its efficiency.

We begin this dissertation by presenting a brief introduction to the theory of solitons and integrability (plus some classical methods applied in this field) in Chapter 1, mainly using the Korteweg-de Vries equation as a typical model. At the end of this Chapter a mathematical framework of notations and terminologies is established for the whole dissertation. In Chapter 2, we first introduce two specific matrix spectral problems (with 3 potentials) associated with matrix Lie algebras $\mbox{sl}(2;\mathbb{R})$ and $\mbox{so}(3;\mathbb{R})$, respectively; and then we engender two soliton hierarchies. The computation and analysis of their Hamiltonian structures based on the trace identity affirms that the obtained hierarchies are Liouville integrable. This chapter shows the entire process of how a soliton hierarchy is engendered by starting from a proper matrix spectral problem. In Chapter 3, at first we elucidate the Gauge equivalence among three types $u$-linear Hamiltonian operators, and construct then the corresponding B\"acklund transformations among them explicitly. Next we derive the if-and-only-if conditions under which the linear coupling of the discussed u-linear operators and matrix differential operators with constant coefficients is still Hamiltonian. Very amazingly, the derived conditions show that the resulting Hamiltonian operators is truncated only up to the 3rd differential order. Finally, a few relevant examples of integrable hierarchies are illustrated. In Chapter, 4 we first present a generalized modified Korteweg-de Vries hierarchy. Then for one of the equations in this hierarchy, we build the associated Riemann-Hilbert problems with some equivalent spectral problems. Next, computation of soliton solutions is performed by reducing the Riemann-Hilbert problems to those with identity jump matrix, i.e., those correspond to reflectionless inverse scattering problems. Finally a special reduction of the original matrix spectral problem will be briefly discussed.

A restarted symplectic Lanczos method for the Hamiltonian eigenvalue problem is presented. The Lanczos vectors are constructed to form a symplectic basis. Breakdowns and near-breakdowns are overcome by inexpensive implicit restarts. The method is used to compute eigenvalues, eigenvectors and invariant subspaces of large and sparse Hamiltonian matrices and low rank approximations to the solution of continuous-time algebraic Riccati equations with large and sparse coefficient matrices.

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CAPES
In this work, we discuss the article The Existence and Stability of Equilibrium Points in the Robe Restricted Three-Body Probem due to Hallan and Rana. For this we present some basic definitions and results abut Hamiltonian systems such as equilibrium stability of linear Hamiltonian systems. We set out the restricted problem of the three bodies and show some classic results of the problem. Finally we present the Robe’s problem and discuss the main results using Hamiltonian systems theory.
Nesse trabalho, dissertamos sobre o artigo \The Existence and Stability of Equilibrium Points in the Robe Restricted Three-Body Probem" devido a Hallan e Rana. Para isso apresentamos definições e resultados básicos sobre sistemas Hamiltonianos tais como estabilidade de equilíbrios de sistemas Hamiltonianos lineares. Enunciamos o problema restrito dos três corpos e mostramos alguns resultados clássicos do problema. Por fim apresentamos o problema de Robe e discutimos os principais resultados usando a teoria de sistemas Hamiltonianos.

The Minimum Score Separation Problem (MSSP) is a combinatorial problem that has been introduced in JORS 55 as an open problem in the paper industry arising in conjunction with the cutting-stock problem. During the process of producing boxes, áat papers are prepared for folding by being scored with knives. The problem is to determine if and how a given production pattern of boxes can be arranged such that a certain minimum distance between the knives can be kept. While it was originally suggested to analyse the MSSP as a specific variant of a Generalized Travelling Salesman Problem, the thesis introduces the concept of twin-constrained Hamiltonian cycles and models the MSSP as the problem of finding a twin-constrained Hamiltonian path on a threshold graph (threshold graphs are a specific type of interval graphs). For a given undirected graph G(N,E) with an even node set N and edge set E, and a bijective function b on N that assigns to every node i in N a "twin node" b(i)6=i, we define a new graph G'(N,E') by adding the edges {i,b(i)} to E. The graph G is said to have a twin-constrained Hamiltonian path with respect to b if there exists a Hamiltonian path on G' in which every node has its twin node as its predecessor (or successor). We start with presenting some general Öndings for the construction of matchings, alternating paths, Hamiltonian paths and alternating cycles on threshold graphs. On this basis it is possible to develop criteria that allow for the construction of twin-constrained Hamiltonian paths on threshold graphs and lead to a heuristic that can quickly solve a large percentage of instances of the MSSP. The insights gained in this way can be generalized and lead to an (exact) polynomial time algorithm for the MSSP. Computational experiments for both the heuristic and the polynomial-time algorithm demonstrate the efficiency of our approach to the MSSP. Finally, possible extensions of the approach are presented.

We derive two hierarchies of 1+1 dimensional soliton-type integrable systems from two spectral problems associated with the Lie algebra of the special orthogonal Lie group SO(3,R). By using the trace identity, we formulate Hamiltonian structures for the resulting equations. Further, we show that each of these equations can be written in Hamiltonian form in two distinct ways, leading to the integrability of the equations in the sense of Liouville. We also present finite-dimensional Hamiltonian systems by means of symmetry constraints and discuss their integrability based on the existence of sufficiently many integrals of motion.

A new method is presented for the numerical computation of the generalized eigen- values of real Hamiltonian or symplectic pencils and matrices. The method is strongly backward stable, i.e., it is numerically backward stable and preserves the structure (i.e., Hamiltonian or symplectic). In the case of a Hamiltonian matrix the method is closely related to the square reduced method of Van Loan, but in contrast to that method which may suffer from a loss of accuracy of order sqrt(epsilon), where epsilon is the machine precision, the new method computes the eigenvalues to full possible accuracy.

This thesis focuses on the design and implementation of an application involving the principles of DNA computing simulation for solving some selected problems. DNA computing represents an unconventional computing paradigm that is totally different from the concept of electronic computers. The main idea of DNA computing is to interpret the DNA as a medium for performing computation. Despite the fact, that DNA reactions are slower than operations performed on computers, they may provide some promising features in the future. The DNA operations are based on two important aspects: massive parallelism and principle of complementarity. There are many important problems for which there is no algorithm that would be able to solve the problem in a polynomial time using conventional computers. Therefore, the solutions of such problems are searched by exploring the entire state space. In this case the massive parallelism of the DNA operations becomes very important in order to reduce the complexity of finding a solution.

After reviewing Dirac's canonical quantization program and the canonical quantization of General Relativity, we study the problem of time in the context of a quantum minisuperspace cosmological model: a Friedmann-Lemaître-Robertson-Walker spacetime coupled minimally to a scalar field. We explore different methods to include time and evolution in our formalism. We begin by discussing the possibility to identify a dynamical time variable before quantization. Such a time variable is constructed as a function of the phase space variables and leads to a multiple choice problem for the evolution of our quantum system. We then explore the connection between the Born-Oppenheimer (BO) approach to the problem of time and gauge fixing. We find that by choosing a particular gauge we can recover the Born-Oppenheimer approach ansatz both in the classical and in the quantum theory. In the latter, the result of the BO approach is recovered by performing a phase transformation in the Wheeler-DeWitt equation and requiring that the resulting Schrödinger-like equation is unitary.

Hamilton-Jacobi theory provides a powerful method for extracting the equations of motion out of some given systems in classical mechanics. On occasion it allows some systems to be solved by the method of separation of variables. If a system with n degrees of freedom has 2n - 1 constants of the motion that are polynomial in the momenta, then that system is called superintegrable. Such a system can usually be solved in multiple coordinate systems if the constants of the motion are quadratic in the momenta. All superintegrable two dimensional Hamiltonians of the form H = (p_x)sup2 + (p_y)sup2 + V(x,y), with constants that are quadratic in the momenta were classified by Kalnins et al [5], and the coordinate systems in which they separate were found. We discuss Hamilton-Jacobi theory and its development from a classical viewpoint, as well as superintegrability. We then proceed to use the theory to find equations of motion for some of the superintegrable Hamiltonians from Kalnins et al [5]. We also discuss some of the properties of the Poisson algebra of those systems, and examine the orbits.

Cette thèse porte sur l’étude des systèmes de lois de conservation couplés par une interface mobile. Un modèle dynamique d’un procédé d’extrusion obtenu à partir des bilans de masse, de taux d’humidité et d’énergie est proposé. Ce modèle exprime le transport de la matière et de la chaleur dans une extrudeuse par des systèmes d’équations hyperboliques définis sur deux domaines complémentaires variant dans le temps. L’évolution des domaines est dictée par une Equation aux Dérivées Ordinaires (EDO) issue du bilan de masse total dans une extrudeuse. Par le principe des applications contractantes l’existence et l’unicité de la solution pour cette classe de système sont prouvées. Le problème de stabilisation de l’interface mobile est aussi abordé en utilisation le formalisme des systèmes à retard. La méthode des caractéristiques permet de représenter le système composé des équations issues du bilan de masse par un système à retard sur l’entrée. Au moyen d’un contrôleur prédictif la position de l’interface est stabilisée autour d’un point équilibre. La dernière partie de ce travail est dédiée à l’étude des systèmes Hamiltoniens à ports frontière couplés par une interface mobile. Ces systèmes augmentés de variables couleur qui sont des fonctions caractéristiques du domaine peuvent s’exprimer comme des systèmes Hamiltoniens à ports frontière
This thesis is devoted to the analysis of Partial Differential Equations (PDEs) which are coupled through a moving interface. The motion of the interface obeys to an Ordinary Differential Equation (ODE) which arises from a conservation law. The first part of this thesis concerns the modelling of an extrusion process based on mass, moisture content and energy balances. These balances laws express heat and homogeneous material transport in an extruder by hyperbolic PDEs which are defined in complementary time-varying domains. The evolution of the coupled domains is given by an ODE which is derived from the conservation of mass in an extruder. In the second part of the manuscript, a mathematical analysis has been performed in order to prove the existence and the uniqueness of solution for such class of systems by mean of contraction mapping principle. The third part of the thesis concerns the transformation of an extrusion process mass balance equations into a particular input delay system framework using characteristics method. Then, the stabilization of the moving interface by a predictor-based controller has been proposed. Finally, an extension of the analysis of moving interface problems to a particular class of systems of conservations laws has been developed. Port-Hamiltonian formulation of systems of two conservation laws defined on two complementary time-varying intervals has been studied. It has been shown that the coupled system is a port-Hamiltonian system augmented with two variables being the characteristic functions of the two spatial domains

In this thesis we explored some aspects of the dynamics around the Earth-Moon L1 and L2 points in the context of two Restricted Four Body Problems: the Bicircular Problem (BCP) and the Quasi-bicircular Problem (QBCP). Both the BCP and QBCP model the dynamics of a massless particle moving under the influence of the Sun, Earth, and Moon. Although these two models focus on the same system, it is relevant to study both because their behavior around the L2 is qualitatively different.Ðhese two models can be written in the Hamiltonian formalism as periodic time-dependent perturbations of the RTBP. To study these Hamiltonians, we used numerical tools tailored to these type of models to get an insight on the phase space. These two techniques are the reduction to the center manifold, and the computation and continuation of 2D tori.ßor the BCP, the analysis focused around the L2 point. The results obtained showed that the reduction to the center manifold, and the non-autonomous normal form computed in this thesis do not provide useful information about the neutral motion around L2. The approach taken was to compute families of 2D tori, and explore any connections and their stability. As a summary of this effort we identified a total of six families of 2D tori: two Lyapunov-type planar quasi-periodic orbits, and four vertical. One of the vertical families was obtained by direct continuation of Halo orbits from the RTBP. This showed that the family of Halo orbits from the RTBP survive in the BCP, with the understanding that this new family is Cantorian. It was also shown that one of the other vertical families is Halo-like. Hence, members of this family may be potential candidates for future space missions. However, these tori are hyperbolic, as opposed the ones coming directly from the RTBP Halo obits, which are partially elliptic. It was also shown that this family of Halo-like tori comes from a family of quasi-periodic orbits in the RTBP that are resonant with the frequency of the Sun. Hence, these family of Halo-like orbits in the BCP have their counterparts in the RTBP.ßor the QBCP, the focus of the analyses was there Earth-Moon L1 and L2 points. In this model, the reduction to the center manifold provided relevant qualitative information about the dynamics around L1 and L2. The main takeaway was that L1 and L2 had a similar qualitative behavior. In both cases there were two families of quasi-periodic Lyapunov orbits, one planar and one vertical. It was also shown that the quasi-periodic planar Lyapunov family underwent a (quasi-periodic) pitchfork bifurcation, giving rise to two families of quasi-periodic orbits with an out-of-plane component. Between them, there was a family of Lissajous quasi-periodic orbits, with three basic frequencies. Qualitatively, the phase space of the center manifold, as constructed in this thesis, resembled the phase space of the center manifold of the RTBP around L1 and L2. In the QBCP we also continued families of invariant 2D tori, and for both L1 and L2. In these cases, the quasi-periodic planar and vertical families were continued. The bifurcations of the quasi-periodic planar Lyapunov were identified. A conclusion from this numerical experiment was that the family of out-of-plane orbits born from the bifurcation seemed not to be the RTBP Halo counterparts in the QBCP. The RTBP Halo orbits do survive in the QBCP, but do not seem to be connected to the quasi-periodic planar Lyapunov family. ßinally, and also in the context of the BCP and the QBCP, numerical simulations to study transfers from a parking orbit around the Earth to a Halo orbit around the Earth-Moon L2 point were studied. The main conclusion is that the invariant manifolds of the target orbits studied intersect with potential parking orbits around the Earth. The relevance of this result is that it shows that there are one-maneuver transfers from a vicinity of the Earth to Earth-Moon L2 Halo orbits. This is not case when using the RTBP as reference model. Experiments were done for both the BCP and the QBCP, and in all cases is it was shown that the total cost in terms of ∆V and transfer time is comparable to other techniques requiring two or more maneuvers.
En aquesta tesi explorem alguns aspectes de la dinàmica al voltant dels punts L1 i L2 Terra-Lluna en el context de dos problemes restringits de quatre cossos: el problema bicircular (PBC) i el problema quasi-circular (PQBC). Tant el PBC com el PQBC modelen la dinàmica d’una partícula sense massa que es mou sota la influència del Sol, la Terra i la Lluna. Tot i que aquests dos models es centren en el mateix sistema, és rellevant estudiar-los tots dos perquè el seu comportament al voltant de la L2 és qualitativament diferent. Aquests dos models es poden escriure en el formalisme hamiltonià com a pertorbacions periòdiques del problema restringit dels tres cossos (PRTC) dependents del temps. Per estudiar aquests hamiltonians, utlitzem eines numèriques adaptades a aquest tipus de models per obtenir una idea de l’espai de fases. Aquestes dues tècniques són la reducció a la variety central i el càlcul i la continuació de tors 2D. Per al PBC, l'anàlisi es centra al voltant del punt L2. Els resultats obtinguts mostren que la reducció a la varietat central i la forma normal no autònoma calculada en aquesta tesi no proporcionen informació útil sobre el moviment neutre al voltant de L2. L'enfocament adoptat es calcular famílies de tors 2D, i explorar les seves connexions i estabilitat. Com a resum d’aquest esforç, s'identifiquen un total de sis famílies de tors 2D: dues famílies d'òrbites quasi-periòdiques planes tipus Lyapunov i quatre verticals. Una de les famílies verticals s'obté per continuació directa de les òrbites Halo del PRTC. Això demostra que la família de les òrbites Halo del PRTC sobreviuen al PBC, entenent que aquesta nova família és cantoriana. També es demostra que una de les altres famílies verticals és semblant a les Halo. Per tant, els membres d’aquesta família poden ser candidats potencials per a futures missions espacials. No obstant això, aquests tors són hiperbòlics, a diferència dels que provenen directament de les Halo del PRTC, que són parcialment el·líptics. També es mostra que aquesta família de tors semblants a les Halo prové d’una família d’òrbites quasi-periòdiques del PRTC que són ressonants amb la freqüència del Sol. Per tant, aquestes famílies d’òrbites semblants a les Halo al PBC tenen els seus homòlegs al PRTC. Per al PQBC, el focus de les anàlisis es troba en els punts L1 i L2 Terra-Lluna. En aquest model, la reducció a la varietat central proporciona informació qualitativa rellevant sobre la dinàmica al voltant de L1 i L2. El principal resultat és que L1 i L2 tenen un comportament qualitatiu similar. En ambdós casos hi ha dues famílies d’òrbites quasi-periòdiques tipus Lyapunov, una plana i una vertical. També es demostra que la família plana quasi-periòdica tipus Lyapunov sobrevé una bifurcació tipus pitchfork (quasi-periòdica), donant lloc a dues famílies d’òrbites quasi periòdiques amb un component vertical. Entre ells, hi havia una família d’òrbites quasi-periòdiques tipos Lissajous, amb tres freqüències bàsiques. Qualitativament, l’espai de fase de la varietat central, tal com es construeix en aquesta tesi, s’assembla a l’espai de fase del la varietat central del PRTC al voltant de L1 i L2. Al PQBC també es continuen famílies de tors 2D invariants, tant per a L1 com per a L2. En aquests casos, es continuen les famílies planes i verticals quasi-periòdiques. Durant aquest procés es troben bifurcacions a les families d'òrbites quasi-periòdiques planes. Una conclusió d’aquest experiment numèric és que la família d’òrbites amb component vertical nascudes de la bifurcació no són les contraparts de les Halo de PRTC. Les òrbites Halo de PRTC sobreviuen en el PQBC, però no semblen estar connectades a la família plana quasi-periòdica. Finalment, i també en el context del PBC i el PQBC, es s'estudien simulacions numèriques per estudiar les transferències des d’una òrbita d’estacionària al voltant de la Terra fins a una òrbita Halo al voltant del punt L2 Terra-Lluna. La principal conclusió és que les varietats invariants de les òrbites objectiu estudiades passen molt a prop de la Terra. La rellevància d’aquest resultat és que mostra que hi ha transferències d’una maniobra des de d'una òrbita al voltat de Terra a les òrbites L2 Halo Terra-Lluna. No és així quan s’utilitza el PRTC com a model de referència. Es fan experiments tant per al BCP com per al QBCP, i en tots els casos es demostra que el cost total en termes de ∆V i temps de transferència és comparable a altres tècniques que requereixen dues o més maniobres.

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Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)
O objetivo da Tese é investigar a aplicabilidade e propor extensões do método do hamiltoniano termodinamicamente equivalente (MHTE) para sistemas de muitos corpos descritos por uma teoria de campos. Historicamente, o MHTE tem sua origem na teoria quântica de muitos corpos para descrever o fenômeno da supercondutividade. O método consiste na observação de que o hamiltoniano de um sistema pode ser diagonalizado exatamente através de uma transformação unitária quando um número finito de momentos transferidos que contribuem para a interação é levado em conta no limite termodinâmico. Essa transformação unitária depende explicitamente de funções de gap que podem ser determinadas através do método variacional de Gibbs. Na presente Tese, extensões do método são feitas visando aplicações em sistemas de muitos corpos em diferentes situações, tais como: transições de fase estáaticas, evolução temporal de parâmetros de ordem descrita por equações dinâmicas estocásticas do tipo Ginzburg-Landau-Langevin (GLL), teorias quânticas de campos escalares relativísticos e teorias de muitos corpos para sistemas fermiônicos não relativísticos. Mostra-se, em particular, que o MHTE é um esquema de aproximação sistemático e controlável que permite incorporar acoplamentos de componentes de Fourier de parâmetros de ordem além do modo zero, da mesma forma que em teorias quânticas relativísticas ou não relativísticas ele incorpora correlações não perturbativas entre as partículas além daquelas levadas em conta pelas tradicionais aproximações de campo médio. Métodos são desenvolvidos para obtermos soluções numéricas explícitas com o objetivo de avaliar a aplicabilidade do MHTE em alguns casos específicos. Particular atenção é dedicada ao controle de divergências de Rayleigh-Jeans nas simulações numéricas de equações de GLL
The general objective of the Thesis is to apply the Method of the Thermodynamically Equivalent Hamiltonian (MTEH) to many-body systems described by a field theory. Historically, the MTEH has its origins in the quantum theory of manybody systems to describe the phenomenon of superconductivity. The method is based on the observation that the Hamiltonian of the system can be diagonalized exactly with a unitary transformation when a finite number of transfer momenta of the interaction are taken into account in the thermodynamic limit. This unitary transformation depends explicitly on gap functions that can be determined with the use of the Gibbs variational principle. In the present Thesis, extensions of the method are made envisaging applications in many-body systems in different situations, like: static phase transitions, time evolution of order parameters described by dynamic stochastic Ginzburg-Landau-Langevin equations, relativistic quantum scalar field theories, and many-body theories for nonrelativistic fermionic systems. It is shown that the MTEH is a systematic and controllable approximation scheme that in the theory of phase transitions allows to incorporate Fourier modes of the order parameter beyond the zero mode, in the same way that in the relativistic and nonrelativistic theories it incorporates particle nonperturbative correlations beyond those taken into account by the traditional mean field approximation. Methods are developed to obtain explicit numerical solutions with the aim to assess the applicability of the MTEH in specific situations. Particular attention is devoted to the control of Rayleigh-Jeans ultraviolet divergences in the numerical simulations of Ginzburg-Landau-Langevin equations

Orientador: Gastão Inácio Krein
Banca: Marcus Benghi Pinto
Banca: Ney Lemke
Banca: Sandra dos Santos Padula
Banca: Yogiro Hama
Resumo: O objetivo da Tese é investigar a aplicabilidade e propor extensões do método do hamiltoniano termodinamicamente equivalente (MHTE) para sistemas de muitos corpos descritos por uma teoria de campos. Historicamente, o MHTE tem sua origem na teoria quântica de muitos corpos para descrever o fenômeno da supercondutividade. O método consiste na observação de que o hamiltoniano de um sistema pode ser diagonalizado exatamente através de uma transformação unitária quando um número finito de momentos transferidos que contribuem para a interação é levado em conta no limite termodinâmico. Essa transformação unitária depende explicitamente de funções de gap que podem ser determinadas através do método variacional de Gibbs. Na presente Tese, extensões do método são feitas visando aplicações em sistemas de muitos corpos em diferentes situações, tais como: transições de fase estáaticas, evolução temporal de parâmetros de ordem descrita por equações dinâmicas estocásticas do tipo Ginzburg-Landau-Langevin (GLL), teorias quânticas de campos escalares relativísticos e teorias de muitos corpos para sistemas fermiônicos não relativísticos. Mostra-se, em particular, que o MHTE é um esquema de aproximação sistemático e controlável que permite incorporar acoplamentos de componentes de Fourier de parâmetros de ordem além do modo zero, da mesma forma que em teorias quânticas relativísticas ou não relativísticas ele incorpora correlações não perturbativas entre as partículas além daquelas levadas em conta pelas tradicionais aproximações de campo médio. Métodos são desenvolvidos para obtermos soluções numéricas explícitas com o objetivo de avaliar a aplicabilidade do MHTE em alguns casos específicos. Particular atenção é dedicada ao controle de divergências de Rayleigh-Jeans nas simulações numéricas de equações de GLL
Abstract: The general objective of the Thesis is to apply the Method of the Thermodynamically Equivalent Hamiltonian (MTEH) to many-body systems described by a field theory. Historically, the MTEH has its origins in the quantum theory of manybody systems to describe the phenomenon of superconductivity. The method is based on the observation that the Hamiltonian of the system can be diagonalized exactly with a unitary transformation when a finite number of transfer momenta of the interaction are taken into account in the thermodynamic limit. This unitary transformation depends explicitly on gap functions that can be determined with the use of the Gibbs variational principle. In the present Thesis, extensions of the method are made envisaging applications in many-body systems in different situations, like: static phase transitions, time evolution of order parameters described by dynamic stochastic Ginzburg-Landau-Langevin equations, relativistic quantum scalar field theories, and many-body theories for nonrelativistic fermionic systems. It is shown that the MTEH is a systematic and controllable approximation scheme that in the theory of phase transitions allows to incorporate Fourier modes of the order parameter beyond the zero mode, in the same way that in the relativistic and nonrelativistic theories it incorporates particle nonperturbative correlations beyond those taken into account by the traditional mean field approximation. Methods are developed to obtain explicit numerical solutions with the aim to assess the applicability of the MTEH in specific situations. Particular attention is devoted to the control of Rayleigh-Jeans ultraviolet divergences in the numerical simulations of Ginzburg-Landau-Langevin equations
Doutor

When the eigenvalues of a given eigenvalue problem are symmetric with respect to the real and the imaginary axes, we speak about a Hamiltonian eigenvalue symmetry or a Hamiltonian structure of the spectrum. This property can be exploited for an efficient computation of the eigenvalues. For some elliptic boundary value problems it is known that the derived eigenvalue problems have this Hamiltonian symmetry. Without having a specific application in mind, we trace the question, under which assumptions the spectrum of a given quadratic eigenvalue problem possesses the Hamiltonian structure.

In this thesis we bring together two areas of mathematics; Hamiltonian dynamics and data assimilation. We construct a four dimensional variational (4d Var) data assimilation scheme for two Hamiltonian systems. This is to reflect the Hamiltonian behaviour observed in the atmosphere. We know, for example, that potential vorticity is conserved in atmospheric models. However, current data assimilation schemes do not explicitly include such physical relationships. In this thesis, by considering the two and three body problems, we demonstrate how such characteristic behaviour can be included in the data assimilation schemes. In our 4d Var schemes we add a weak constraint that imposes the conservation of the Hamiltonian, the total energy, at the initial time. This is effectively imposing an energy constraint from one data assimilation window to the next. Our results imply that these weak constraints affect the underlying geometry of the resulting data assimilation solution. We also demonstrate that this constraint reduces the error on this solution and the forecast. By imposing this constraint we are including additional information to the system. Due to the additional term in the cost function gradient, the analysis can only change in such a way as to satisfy this weak constraint. This thesis therefore demonstrates that the inclusion of similar weak con-straints, perhaps using the conservation of potential vorticity, could improve the analysis and forecast for atmospheric models.

This diploma thesis deals with evolutionary algorithms used for travelling salesman problem (TSP). In the first section, there are theoretical foundations of a graph theory and computational complexity theory. Next section contains a description of chosen optimization algorithms. The aim of the diploma thesis is to implement an application that solve TSP using evolutionary algorithms.

We explore algorithmic aspects of two known combinatorial problems, Graph Colouring and Hamiltonian Cycle, by examining properties of their solution space. One can model the set of solutions of a combinatorial problem $P$ by the solution graph $R(P)$, where vertices are solutions of $P$ and there is an edge between two vertices, when the two corresponding solutions satisfy an adjacency reconfiguration rule. For example, we can define the reconfiguration rule for graph colouring to be that two solutions are adjacent when they differ in colour in exactly one vertex. The exploration of the properties of the solution graph $R(P)$ can give rise to interesting questions. The connectivity of $R(P)$ is the most prominent question in this research area. This is reasonable, since the main motivation for modelling combinatorial solutions as a graph is to be able to transform one into the other in a stepwise fashion, by following paths between solutions in the graph. Connectivity questions can be made binary, that is expressed as decision problems which accept a 'yes' or 'no' answer. For example, given two specific solutions, is there a path between them? Is the graph of solutions $R(P)$ connected? In this thesis, we first show that the diameter of the solution graph $R_{l}(G)$ of vertex $l$-colourings of k-colourable chordal and chordal bipartite graphs $G$ is $O(n^2)$, where $l > k$ and n is the number of vertices of $G$. Then, we formulate a decision problem on the connectivity of the graph colouring solution graph, where we allow extra colours to be used in order to enforce a path between two colourings with no path between them. We give some results for general instances and we also explore what kind of graphs pose a challenge to determine the complexity of the problem for general instances. Finally, we give a linear algorithm which decides whether there is a path between two solutions of the Hamiltonian Cycle Problem for graphs of maximum degree five, and thus providing insights towards the complexity classification of the decision problem.

Les problèmes de contrôle optimal se rencontrent dans l'industrie aérospatiale et dans la mécanique. Leur étude conduit à des difficultés mathématiques importantes. Dans cette thèse, nous nous intéressons aux conditions d'optimalité pour certains problèmes de contrôle avec des contraintes exprimées en termes d'inclusions différentielles. Nous considérons aussi des problèmes de contrôle associés aux modèles mathématiques issus de la Mécanique du Contact. Cette thèse est structurée en deux parties et six chapitres. La première partie, contenant les Chapitres 1, 2 et 3, représente un résumé de nos résultats, en Français. Nous y présentons les problèmes étudiés, les hypothèses sur les données, les notations utilisées ainsi que l’énoncé des principaux résultats. Les démonstrations sont omises. La deuxième partie du manuscrit représente la partie principale de la thèse. Elle contient les Chapitres 4, 5 and 6, chacun ayant fait l'objet d'une publication (parue ou soumise) dans une revue internationale avec comité de lecture.Nous y présentons nos principaux résultats, accompagnés des démonstrations et des références bibliographiques
Optimal control problems arise in aerospace industry and in mechanics. They are challenging and involve important mathematical difficulties. In this thesis, we are interested to derive optimality conditions for optimal control problems with constraints under the form of differential inclusions. We also consider optimal control problems in the study of some boundary value problems arising in Contact Mechanics. The thesis is structured in two parts and six chapters. Part I represents an abstract of the main results, in French. It contains Chapters 1, 2 and 3. Here we present the problems we study together with the assumptions on the data, the notation and the statement of the main results. The proofs of these results are omitted, since them are presented in Part II of the manuscript.Part II represents the main part of the thesis. It contains Chapters 4, 5 and 6. Each of these chapters made the object of a paper published (or submitted) in an international journal. Here we present our main results, together with the corresponding proofs and bibliographical references

Neste trabalho consideramos sistemas elípticos de tipo hamiltoniano, envolvendo o operador Laplaciano, com uma parte linear dependendo de dois parâmetros e uma perturbação sublinear. Obtemos a existência de pelo menos duas soluções quando a parte linear está perto da ressonância (este fenômeno é chamado de quase ressonância). Mostramos também a existência de uma terceira solução, quando a quase ressonância é em relação ao primeiro autovalor do operador Laplaciano. No caso ressonante obtemos resultados análogos, adicionando mais uma perturbação sublinear. Os sistemas estão associados a funcionais fortemente indefinidos, e as soluções são obtidas através do Teorema de Ponto de Sela e aproximação de Galerkin.
In this work we consider elliptic systems of hamiltonian type, involving the Laplacian operator, a linear part depending on two parameters and a sublinear perturbation. We obtain the existence of at least two solutions when the linear part is near resonance (this phenomenon is called almost-resonance). We also show the existence of a third solution when the almost-resonance is with respect to the first eigenvalue of the Laplacian operator. In the resonant case, we obtain similar results, with an additional sublinear term. These systems are associated with strongly indefinite functionals, and the solutions are obtained by Saddle Point Theorem and Galerkin approximation.

Thesis: M. Eng., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2017.
This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.
Cataloged from student-submitted PDF version of thesis.
Includes bibliographical references (pages 123-124).
In this thesis, we analyze the computational complexity of several problems related to the Hamiltonian Cycle problem. We begin by introducing a new problem, which we call Tree-Residue Vertex-Breaking (TRVB). Given a multigraph G some of whose vertices are marked "breakable," TRVB asks whether it is possible to convert G into a tree via a sequence of applications of the vertex-breaking operation: disconnecting the edges at a degree-G breakable vertex by replacing that vertex with G degree-1 vertices. We consider the special cases of TRVB with any combination of the following additional constraints: G must be planar, G must be a simple graph, the degree of every breakable vertex must belong to an allowed list G, and the degree of every unbreakable vertex must belong to an allowed list G. We fully characterize these variants of TRVB as polynomially solvable or NP-complete. The TRVB problem is useful when analyzing the complexity of what could be called single-traversal problems, where some space (i.e., a configuration graph or a grid) must be traversed in a single path or cycle subject to local constraints. When proving such a problem NP-hard, a reduction from TRVB can often be used as a simpler alternative to reducing from a hard variant of Hamiltonian Cycle. Next, we analyze several variants of the Hamiltonian Cycle problem whose complexity was left open in a 2007 paper by Arkin et al [3]. That paper is a systematic study of the complexity of the Hamiltonian Cycle problem on square, triangular, or hexagonal grid graphs, restricted to polygonal, thin, super-thin, degree-bounded, or solid grid graphs. The authors solved many combinations of these problems, proving them either polynomially solvable or NP-complete, but left three combinations open. We prove two of these unsolved combinations to be NP-complete: Hamiltonian Cycle in Square Polygonal Grid Graphs and Hamiltonian Cycle in Hexagonal Thin Grid Graphs. We also consider a new restriction, where the grid graph is both thin and polygonal, and prove that the Hamiltonian Cycle problem then becomes polynomially solvable for square, triangular, and hexagonal grid graphs. Several of these results are shown by application of the TRVB results, demonstrating the usefulness of that problem. Finally, we apply the Square Grid Graph Hamiltonian Cycle problem to close a longstanding open problem: we prove that optimally solving an n x n x n Rubik's Cube is NP-complete. This improves the previous result that optimally solving an n x n x n Rubik's Cube with missing stickers is NP-complete. We prove this result first for the simpler case of the Rubik's Square -- an n x n x 1 generalization of the Rubik's Cube -- and then proceed with a similar but more complicated proof for the Rubik's Cube case.
by Mikhail Rudoy.
M. Eng.

On presente une approche du probleme du centre qui utilise la theorie des formes normales de germes de champs de vecteurs polynomiaux a l'origine dans r 2 n. Un de nos points de vue est l'effectivite, au sens ou on utilise le calcul formel pour obtenir et calculer les coefficients de la forme normale de birkhoff. On calcule explicitement les premiers coefficients de la forme normale d'un champ de vecteurs hamiltonien polynomial homogene, et un debut d'etude des champs hamiltoniens non homogenes est entreprise, au sens ou, la aussi, on calcule completement les premiers coefficients des formes normales mises en jeu. Notre travail permet d'attacher des invariants algebriques aux champs de vecteurs, et donc d'ebaucher une classification de ces champs. De plus, les equations algebriques provenant des ces invariants completement explicites dans certains cas, en particulier pour le cas hamiltonien, permettent de generer des conditions pour la linearisation des champs polynomiaux au voisinage de l'origine, qui est un point critique de type elliptique. Le probleme de la linearisation, ou encore du probleme du centre, est ainsi aborde par le biais des varietes algebriques definies par les equations associees aux champs de vecteurs etudies.

In this thesis, we focus on the following topics: supereulerian graphs, hamiltonian line graphs, fault-tolerant Hamiltonian laceability of Cayley graphs generated by transposition trees, and several extremal problems on the (minimum and/or maximum) size of graphs under a given graph property. The thesis includes six chapters. The first one is to introduce definitions and summary the main results of the thesis, and in the last chapter we introduce the furture research of the thesis. The main studies in Chapters 2 - 5 are as follows. In Chapter 2, we explore conditions for a graph to be supereulerian.In Section 1 of Chapter 2, we characterize the graphs with minimum degree at least 2 and matching number at most 3. By using the characterization, we strengthen the result in [93] and we also address a conjecture in the paper.In Section 2 of Chapter 2, we prove that if $d(x)+d(y)\geq n-1-p(n)$ for any edge $xy\in E(G)$, then $G$ is collapsible except for several special graphs, where $p(n)=0$ for $n$ even and $p(n)=1$ for $n$ odd. As a corollary, a characterization for graphs satisfying $d(x)+d(y)\geq n-1-p(n)$ for any edge $xy\in E(G)$ to be supereulerian is obtained. This result extends the result in [21].In Section 3 of Chapter 2, we focus on a conjecture posed by Chen and Lai [Conjecture~8.6 of [33]] that every 3-edge connected and essentially 6-edge connected graph is collapsible. We find a kind of sufficient conditions for a 3-edge connected graph to be collapsible.In Chapter 3, we mainly consider the hamiltonicity of 3-connected line graphs.In the first section of Chapter 3, we give several conditions for a line graph to be hamiltonian, especially we show that every 3-connected, essentially 11-connected line graph is hamilton- connected which strengthens the result in [91].In the second section of Chapter 3, we show that every 3-connected, essentially 10-connected line graph is hamiltonian-connected.In the third section of Chapter 3, we show that 3-connected, essentially 4-connected line graph of a graph with at most 9 vertices of degree 3 is hamiltonian. Moreover, if $G$ has 10 vertices of degree 3 and its line graph is not hamiltonian, then $G$ can be contractible to the Petersen graph.In Chapter 4, we consider edge fault-tolerant hamiltonicity of Cayley graphs generated by transposition trees. We first show that for any $F\subseteq E(Cay(B:S_{n}))$, if $|F|\leq n-3$ and $n\geq4$, then there exists a hamiltonian path in $Cay(B:S_{n})-F$ between every pair of vertices which are in different partite sets. Furthermore, we strengthen the above result in the second section by showing that $Cay(S_n,B)-F$ is bipancyclic if $Cay(S_n,B)$ is not a star graph, $n\geq4$ and $|F|\leq n-3$.In Chapter 5, we consider several extremal problems on the size of graphs.In Section 1 of Chapter 5, we bounds the size of the subgraph induced by $m$ vertices of hypercubes. We show that a subgraph induced by $m$ (denote $m$ by $\sum\limits_{i=0}^ {s}2^{t_i}$, $t_0=[\log_2m]$ and $t_i= [\log_2({m-\sum\limits_{r=0}^{i-1}2 ^{t_r}})]$ for $i\geq1$) vertices of an $n$-cube (hypercube) has at most $\sum\limits_{i=0}^{s}t_i2^{t_i-1} +\sum\limits_{i=0}^{s} i\cdot2^{t_i}$ edges. As its applications, we determine the $m$-extra edge-connectivity of hypercubes for $m\leq2^{[\frac{n}2]}$ and $g$-extra edge-connectivity of the folded hypercube for $g\leq n$.In Section 2 of Chapter 5, we partially study the minimum size of graphs with a given minimum degree and a given edge degree. As an application, we characterize some kinds of minimumrestricted edge connected graphs.In Section 3 of Chapter 5, we consider the minimum size of graphs satisfying Ore-condition.

Dans ce travail, l'étude porte sur l'existence, et le comportement asymptotique, de la solution d'un problème d'évolution non classique qui entre dans la catégorie des problèmes dits problèmes de perturbations singulières, avec des conditions aux bords non usuelles. Ce problème régit les petites oscillations d'un système élastique constitué de deux milieux en interaction. Par exemple, une membrane élastique rectangulaire, fixée sur trois de ses cotés, le quatrième coté étant solidaire d'une tige élastique. On suppose que l'inertie de la membrane est négligeable par rapport à celle de la tige et l'on cherche alors un développement asymptotique du mouvement lorsque l'inertie de la membrane tend vers zéro. En d'autres termes, on cherche comment les vibrations de la membrane perturbent celles de la tige. On étudié également le comportement asymptotique des éléments propres d'un operateur perturbé qui intervient dans la formulation variationnelle du problème décrit ci-dessus. On montre qu'il existe deux groupes de valeurs caractéristiques tel que l'un tend vers les valeurs propres d'un problème classique (problème de Dirichlet) et l'autre vers les valeurs propres d'un problème non classique (les conditions aux bords sont données par des équations aux dérivées partielles)

Orientador: Claudina Izepe Rodrigues
Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica
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Resumo: Neste trabalho, o objetivo foi o de estudar o problema da reconstrução para torneios com o quociente simples normal. Com este intuito, introduzimos e desenvolvemos no capítulo 1 diversos conceitos, tais como, o quociente de um torneio e mostramos que torneios hipomorfos tais que ambos sejam não simples possuem o mesmo quociente simples. No capítulo 2 introduzimos os conceitos de ciclo minimais e característico. Ao final mostramos que a existência de quociente simples normal é uma propriedade hipomorfa para torneios de ordem superior ou igual a 7. No capítulo 3 demonstramos que os torneios hamiltonianos de ordem maior ou igual a 4 que têm quociente simples normal são reconstrutíveis, se excluirmos um torneio de ordem 5 e dois de ordem 6. Além disso, no início deste capítulo verificamos que os torneios exibidos por Stockmeyer são realmente contra exemplos da conjectura da reconstrução , a qual diz que se dois torneios têm as mesmas cartas são isomorfos. E finalmente apresentamos uma análise das relações entre as classes dos torneios reconstrutíveis atualmente conhecidos(1999).
Abstract: In this work, the objective was to study the reconstruction problem for tournaments with simple normal quotient. With this intention, we introduced and developed in chapter one few concepts, so as, quotient of a tournaments which are not both simple have the same simple quotient. In chapter two we introduce the concepts of minimal and characteristic cycles, and ending this topic we show that the existence of a normal simple quotient is a hipomorphic property for tournaments of order seven or higher. In third chapter we show that hamiltonian tournaments of order four or higher which have normal simple quotient are reconstructible, if we exclude an order five and two of order six tournaments. Moreover, in the beginning of this chapter we check that tour- naments showed by Stockmeyer, be really counterexamples of reconstruction conjecture, which says that if two tournaments with the same cards are isomorphic. Finally we pre-sent and analyse the relation between the reconstruction classes atually known (1999).
Mestrado
Mestre em Matemática

2013 - 2014
The subject of this thesis is the analysis and development of new numerical methods for Ordinary Di erential Equations (ODEs). This studies are motivated by the fundamental role that ODEs play in applied mathematics and applied sciences in general. In particular, as is well known, ODEs are successfully used to describe phenomena evolving in time, but it is often very di cult or even impossible to nd a solution in closed form, since a general formula for the exact solution has never been found, apart from special cases. The most important cases in the applications are systems of ODEs, whose exact solution is even harder to nd; then the role played by numerical integrators for ODEs is fundamental to many applied scientists. It is probably impossible to count all the scienti c papers that made use of numerical integrators during the last century and this is enough to recognize the importance of them in the progress of modern science. Moreover, in modern research, models keep getting more complicated, in order to catch more and more peculiarities of the physical systems they describe, thus it is crucial to keep improving numerical integrator's e ciency and accuracy. The rst, simpler and most famous numerical integrator was introduced by Euler in 1768 and it is nowadays still used very often in many situations, especially in educational settings because of its immediacy, but also in the practical integration of simple and well-behaved systems of ODEs. Since that time, many mathematicians and applied scientists devoted their time to the research of new and more e cient methods (in terms of accuracy and computational cost). The development of numerical integrators followed both the scienti c interests and the technological progress of the ages during whom they were developed. In XIX century, when most of the calculations were executed by hand or at most with mechanical calculators, Adams and Bashfort introduced the rst linear multistep methods (1855) and the rst Runge- Kutta methods appeared (1895-1905) due to the early works of Carl Runge and Martin Kutta. Both multistep and Runge-Kutta methods generated an incredible amount of research and of great results, providing a great understanding of them and making them very reliable in the numerical integration of a large number of practical problems. It was only with the advent of the rst electronic computers that the computational cost started to be a less crucial problem and the research e orts started to move towards the development of problem-oriented methods. It is probably possible to say that the rst class of problems that needed an ad-hoc numerical treatment was that of sti problems. These problems require highly stable numerical integrators (see Section ??) or, in the worst cases, a reformulation of the problem itself. Crucial contributions to the theory of numerical integrators for ODEs were given in the XX century by J.C. Butcher, who developed a theory of order for Runge-Kutta methods based on rooted trees and introduced the family of General Linear Methods together with K. Burrage, that uni ed all the known families of methods for rst order ODEs under a single formulation. General Linear Methods are multistagemultivalue methods that combine the characteristics of Runge-Kutta and Linear Multistep integrators... [edited by Author]
XIII n.s.

Ce travail aborde dans trois parties trois problemes de la theorie des graphes. Dans la premiere partie, cycles hamiltoniens et graphes planaires, nous developpons une procedure de construction de certaines familles de graphes planaires 3-connexes, hamiltoniens et non hamiltoniens. Ces graphes echappent aux caracterisations donnees par les theoremes de tutte et whitney d'une part (conditions suffisantes) et par le theoreme de grinberg d'autre part (conditions necessaires). Cette procedure s'avere utile pour la production d'exemples de graphes en vue de tester des heuristiques pour trouver des chemins hamiltoniens, ce qui est un probleme np-complet. Dans la deuxieme partie, generalisations du delta-sous-graphe pour les multigraphes, nous nous proposons d'etendre aux multigraphes des concepts deja utilises pour les colorations d'aretes d'un graphe simple. Dans la troisieme partie, pavages de figures par des dominos, nous developpons une procedure d'obtention systematique de tous les pavages d'une figure pavable, en mettant en evidence la structure de treillis formee par l'ensemble de ces pavages. Pour cela, nous commencons par exposer, dans le chapitre 7, des resultats specifiques obtenus a partir de la presentation par thurston, en 1990, d'un algorithme lineaire pour la pavage d'une figure plane par dominos, qui utilise l'approche de la theorie des groupes. Nous privilegions, cependant, l'approche de fournier, dont l'etude de l'algorithme de thurston met en evidence le rapport entre la pavabilite d'une figure et la condition de hall pour les couplages

This thesis deals with non-linear non-quadratic optimal control problems in an autonomous system and a related iterative numerical method, the Kleinman-Newton method, for solving the problem. The thesis proves the local convergence of Kleinman-Newton method using the contraction mapping theorem and then describes how this Kleinman-Newton method may be used to numerically solve for the optimal control and the corresponding solution. In order to show the proof and the related numerical work, it is necessary to review some of earlier work in the beginning of Chapter 1 [Zhang], and to introduce the Kleinman-Newton method at the end of the chapter. In Chapter 2 we will demonstrate the proof. In Chapter 3 we will show the related numerical work and results.
Ph. D.

Ensemble and variational techniques have gained wide popularity as the two main approaches for solving data assimilation and inverse problems. The majority of the methods in these two approaches are derived (at least implicitly) under the assumption that the underlying probability distributions are Gaussian. It is well accepted, however, that the Gaussianity assumption is too restrictive when applied to large nonlinear models, nonlinear observation operators, and large levels of uncertainty. This work develops a family of fully non-Gaussian data assimilation algorithms that work by directly sampling the posterior distribution. The sampling strategy is based on a Hybrid/Hamiltonian Monte Carlo (HMC) approach that can handle non-normal probability distributions. The first algorithm proposed in this work is the "HMC sampling filter", an ensemble-based data assimilation algorithm for solving the sequential filtering problem. Unlike traditional ensemble-based filters, such as the ensemble Kalman filter and the maximum likelihood ensemble filter, the proposed sampling filter naturally accommodates non-Gaussian errors and nonlinear model dynamics, as well as nonlinear observations. To test the capabilities of the HMC sampling filter numerical experiments are carried out using the Lorenz-96 model and observation operators with different levels of nonlinearity and differentiability. The filter is also tested with shallow water model on the sphere with linear observation operator. Numerical results show that the sampling filter performs well even in highly nonlinear situations where the traditional filters diverge. Next, the HMC sampling approach is extended to the four-dimensional case, where several observations are assimilated simultaneously, resulting in the second member of the proposed family of algorithms. The new algorithm, named "HMC sampling smoother", is an ensemble-based smoother for four-dimensional data assimilation that works by sampling from the posterior probability density of the solution at the initial time. The sampling smoother naturally accommodates non-Gaussian errors and nonlinear model dynamics and observation operators, and provides a full description of the posterior distribution. Numerical experiments for this algorithm are carried out using a shallow water model on the sphere with observation operators of different levels of nonlinearity. The numerical results demonstrate the advantages of the proposed method compared to the traditional variational and ensemble-based smoothing methods. The HMC sampling smoother, in its original formulation, is computationally expensive due to the innate requirement of running the forward and adjoint models repeatedly. The proposed family of algorithms proceeds by developing computationally efficient versions of the HMC sampling smoother based on reduced-order approximations of the underlying model dynamics. The reduced-order HMC sampling smoothers, developed as extensions to the original HMC smoother, are tested numerically using the shallow-water equations model in Cartesian coordinates. The results reveal that the reduced-order versions of the smoother are capable of accurately capturing the posterior probability density, while being significantly faster than the original full order formulation. In the presence of nonlinear model dynamics, nonlinear observation operator, or non-Gaussian errors, the prior distribution in the sequential data assimilation framework is not analytically tractable. In the original formulation of the HMC sampling filter, the prior distribution is approximated by a Gaussian distribution whose parameters are inferred from the ensemble of forecasts. The Gaussian prior assumption in the original HMC filter is relaxed. Specifically, a clustering step is introduced after the forecast phase of the filter, and the prior density function is estimated by fitting a Gaussian Mixture Model (GMM) to the prior ensemble. The base filter developed following this strategy is named cluster HMC sampling filter (ClHMC ). A multi-chain version of the ClHMC filter, namely MC-ClHMC , is also proposed to guarantee that samples are taken from the vicinities of all probability modes of the formulated posterior. These methodologies are tested using a quasi-geostrophic (QG) model with double-gyre wind forcing and bi-harmonic friction. Numerical results demonstrate the usefulness of using GMMs to relax the Gaussian prior assumption in the HMC filtering paradigm. To provide a unified platform for data assimilation research, a flexible and a highly-extensible testing suite, named DATeS , is developed and described in this work. The core of DATeS is implemented in Python to enable for Object-Oriented capabilities. The main components, such as the models, the data assimilation algorithms, the linear algebra solvers, and the time discretization routines are independent of each other, such as to offer maximum flexibility to configure data assimilation studies.
Ph. D.

Arnold a démontré l'existence de solutions quasipériodiques dans le problème planétaire à trois corps plan, sous réserve que la masse de deux des corps, les planètes, soit petite par rapport à celle du troisième, le Soleil. Cette condition de petitesse dépend de façon cachée de la largeur d'analyticité de l'hamiltonien du problème, dans des coordonnées transcendantes. Hénon ex- plicita un rapport de masses minimal nécessaire à l'application du théorème de Arnold. L'objectif de cette thèse sera de donner une condition suffisante sur les rapports de masses. Une première partie de mon travail consiste à estimer cette largeur d'analyticité, ce qui passe par l'étude précise de l'équation de Kepler dans le complexe, ainsi que celle des singularités complexes de la fonction perturbatrice. Une deuxième partie consiste à mettre l'hamiltonien sous forme normale, dans l'optique d'une application du théorème KAM (du nom de Kolmogorov-Arnold-Moser). Il est nécessaire d'étudier le hamiltonien séculaire pour le mettre sous une forme normale adéquate. On peut alors quantifier la non-dégénérescence de l'hamiltonien séculaire, ainsi qu'estimer la perturbation. Enfin, il faut démontrer une version quantitative fine du théorème KAM, inspirée de Pöschel, avec des constantes explicites. A l'issue de ce travail, il est montré que le théorème KAM peut être appliqué pour des rapports de masses entre planètes et étoile de l'ordre de 10^(-85)
Arnold showed the existence of quasi-periodic solutions in the plane planetary three-body prob- lem, provided that the mass of two of the bodies, the planets, is small compared to the mass of the third one, the Sun. This smallness condition depends in a sensitive way on the analyticity widths of the Hamiltonian of the three-body problem, expressed with the help of some tran- scendental coordinates. Hénon gave a minimal ratio of masses necessary to the application of Arnold’s theorem. The main objective of this thesis is to determine a sufficient condition on this ratio. A first part of this work consists in estimating these analyticity widths, which requires a precise study of the complex Kepler equation, as well as the complex singularities of the disturb- ing function. A second part consists in reworking the Hamiltonian to put it under normal form, in order to apply the KAM theorem (KAM standing for Kolmogorov-Arnold-Moser). In this aim, it is essential to work with the secular Hamiltonian to put it under a suitable normal form. We can then quantify the non-degeneracy of the secular Hamiltonian, as well as estimate the perturbation. Finally, it is necessary to derive a quantitative version of the KAM theorem, in order to identify the hypotheses necessary for its application to the plane three-body problem. After this work, it is shown that the KAM theorem can be applied for a ratio of masses that is close to 10^(−85) between the planets and the star

Etude approfondie sur la theorie des hamiltoniens effectifs et analyse de leurs applications aux methodes de diabatisation et au traitement des collisions reactives. Propositions pour le calcul des valeurs propres de l'hamiltonien par des methodes de perturbation ou iteratives, pour l'emploi d'hamiltoniens effectifs dans le calcul des surfaces quasi diabatiques et le traitement general des collisions reactives. Application a la reaction cs+h::(2) etudiee par les methodes ab initio pour la geometrie colineaire : surfaces de potentiel, sections efficaces et mecanismes predominants

Un área importante de la Matemática Aplicada es el Análisis Matricial dado que muchos problemas pueden reformularse en términos de matrices y de así facilitar su resolución. El problema de valor propio inverso consiste en la reconstrucción de una matriz a partir de datos espectrales dados. Este tipo de problemas se presenta en diferentes áreas de la ingeniería y surge en numerosas aplicaciones. En esta tesis se resuelve el problema de valor propio inverso para tres tipos específicos de matrices. Los problemas de valores propios inversos han sido estudiados tanto desde los puntos de vista teórico, numérico como del de las aplicaciones. Un problema de valor propio inverso adecuadamente planteado debe satisfacer restricciones referidas a los datos espectrales y a la estructura deseada. Dada una matriz X y una matriz diagonal D, se buscan soluciones de la ecuación AX = XD siendo A una matriz con una determinada estructura. A partir de estas restricciones sobre la matriz A surgen una variedad de problemas de valores propios inversos. El problema para el caso de una matriz A hermítica y reflexiva o antireflexiva con respecto a una matriz J tripotente y hermítica ha sido resuelto por L. Lebtahi y N. Thome. En el Capítulo 2 de esta memoria se extiende este trabajo al caso de una matriz A hermítica y reflexiva con respecto a una matriz J {k +1}-potente y normal. En el Teorema 2.2.1 se dan las condiciones bajo las cuales el problema tiene solución y se proporciona la forma explícita de la solución general. Además, si el conjunto de soluciones del problema de valor propio inverso es no vacío, se resuelve el problema de Procrustes asociado. Las matrices Hamiltonianas y antiHamiltonianas aparecen en la resolución de importantes problemas de la Teoría de Sistemas y Control. El problema de valor propio inverso para matrices hermíticas y Hamiltonianas generalizadas fue analizado por Z. Zhang, X. Hu y L. Zang. Más tarde, Z. Bai consideró el caso de matrices hermíticas y antiHamiltonianas generalizadas. En ambos casos se estudió el problema de valor propio inverso y el problema optimización. Una extensión de las matrices Hamiltonianas son las matrices J-Hamiltonianas, y corresponden a una de las aportaciones originales que se realizan en esta memoria. En los Capítulos 3 y 4 de esta tesis se estudian el problema de valor propio inverso para matrices normales J-Hamiltonianas y para normales J-antiHamiltonianas. Para la resolución del caso de las matrices normales y J-Hamiltonianas se presentan cuatro métodos diferentes. Los dos primeros métodos son generales, dan condiciones para que el problema tenga solución. El tercer método se formaliza en el Teorema 3.2.2 que proporciona las condiciones bajo las cuales el problema tiene solución y se presentan infinitas soluciones del mismo. Todas las soluciones se obtienen con el último método. El principal resultado se da en el Teorema 3.2.3. Una sección completa está dedicada a la resolución del problema de optimización de Procrustes asociado. La organización de esta tesis es la siguiente: Capítulo 1 contiene una introducción al problema de valor propio inverso y al problema de Procrustes. En el Capítulo 2 se estudia el problema de valor propio inverso para una matriz hermítica y reflexiva con respecto a una matriz normal {k + 1}-potente, así como también el problema de optimización de Procrustes asociado. Además, se propone un algoritmo que resuelve el problema de Procrustes y se da un ejemplo que muestra el funcionamiento del mismo. El problema de valor propio inverso para una matriz normal y J-Hamiltoniana se resuelve en el Capítulo 3 usando distintos métodos y además se considera el problema de optimización de Procrustes asociado. Se propone un algoritmo que sirve para calcular la solución del problema de optimización y se presentan algunos ejemplos. En el Capítulo 4, en base a los resultados obtenidos en el Capítulo 3, se aborda el problema
An important area of Applied Mathematics is Matrix Analysis due to the fact that many problems can be reformulated in terms of matrices and, in this way, their resolution is facilitated. The inverse eigenvalue problem consists of the reconstruction of a matrix from given spectral data. This type of problems occurs in different engineering areas and arises in numerous applications. In this thesis the inverse eigenvalue problem for three specific sets of matrices is solved. Inverse eigenvalue problems have been studied from theoretical and numerical points of view as well as from their applications. An inverse eigenvalue problem properly posed must satisfy constraints referring to the spectral data and to the desirable structure. Given a matrix X and a diagonal matrix D, solutions of the equation AX = XD are searched, where A is a matrix with a prescribed structure. Based on these restrictions on matrix A, a variety of inverse eigenvalue problems arise. L. Lebtahi and N. Thome solved the problem for the case of a matrix A hermitian and reflexive or antireflexive with respect to a matrix J tripotent and hermitian. In Chapter 2 of this tesis, the results are extended to the case of a matrix A hermitian and reflexive with respect to a matrix J {k+1}-potent and normal. Theorem 2.2.1 provides conditions under which the problem has a solution and the explicit form of the general solution is given. In addition, in case of the set of solutions of the inverse eigenvalue problem is not empty, the associated Procrustes problem is solved. Hamiltonian and skewHamiltonian matrices appear in the resolution of important problems of Systems and Control Theory. The inverse eigenvalue problem for hermitian and generalized Hamiltonian matrices was analyzed by Z. Zhang, X. Hu and L. Zang. Afterwards, the case of hermitian and skewHamiltonian generalized matrices by Z. Bai was considered. In both cases, the inverse eigenvalue problem and the best approximation problem were studied. An extension of the Hamiltonian matrices are the J-Hamiltonian matrices, and it is one of the original contributions of this work. In Chapters 3 and Chapter 4 of this thesis the inverse eigenvalue the respective problems for normal J-Hamiltonian matrices and for normal J-skewHamiltonian matrices are studied. For the resolution of the normal J-Hamiltonian matrices case, four methods are presented. The first two methods are general and they give conditions under which the problem is solvable. The third method is formalized in the Theorem 3.2.2. It provides the conditions under which the problem has a solution and the infinite solutions are presented. The last method states the form of all the solutions. The main result is established in the Theorem 3.2.3. A complete section is dedicated to solve the associated optimization Procrustes problem in case of the problem admits solution. Below, a summary of the organization of this thesis and a brief description of its four chapters are presented. Chapter 1 contains an introduction to the inverse eigenvalue problem, the Procrustes problem, and some other ones studied in the literature. In Chapter 2, the inverse eigenvalue problem for a hermitian reflexive matrix with respect to a normal {k + 1}-potent matrix is studied, as well as the associated optimization Procrustes problem. In addition, an algorithm that solves the Procrustes problem is designed and an example that shows the performance of the algorithm is given. The inverse eigenvalue problem for a normal J-Hamiltonian matrix is investigated in Chapter 3 by using several methods and the associated optimization Procrustes problem is considered. An algorithm that allows us to calculate the solution of the optimization problem is proposed and some examples are provided. In Chapter 4, based on the results obtained in Chapter 3, the inverse eigenvalue problem for normal J-skewHamiltonian matrices is addressed.
Una àrea important de la Matemàtica és l'Anàlisi Matricial ja que molts problemas poden reformular-se en termes de matrius i així facilitar la seua resolució. El problema de valor propi invers consisteix en la reconstrucció d'una matriu a partir de dades espectrals donades. Aquest tipus de problemes es presenta a diferents àrees de l'enginyeria i sorgeix a nombroses aplicacions. Els problemes de valors propis inversos han estat estudiats des dels punts de vista teòric, numèric com també del de les aplicacions. A aquesta tesi es resol el problema per a tres tipus específics de matrius. En diversos casos, per tal de que el problema de valor propi tingui sentit, és necessari imposar una estructura específica a la matriu. Un problema de valor propi invers adequadament plantejat ha de satisfer dues restriccions: la referida a les dades espectrals i la restricció estructural desitjada. Donada una matriu X i una matriu diagonal D, es busquen solucions de l'equació AX = XD sent A una matriu amb una determinada estructura. A partir d'aquestes restriccions sobre la matriu A sorgeixen una varietat de problemes de valors propis inversos. El problema pel cas d'una matriu A hermítica i reflexiva o antireflexiva respecte d'una matriu J tripotent i hermítica ha sigut resolt per L. Lebtahi i N. Thome. Al Capítol 2 d'aquesta memòria s'estén este treball esmentat pel cas d'una matriu A hermítica reflexiva respecte d'una matriu J {k+1}-potent I normal. Al Teorema 2.2.1 es donen les condicions sota les quals el problema té solució i es proporciona la forma explícita de la solució general. A més, en el cas de que el conjunt de solucions del problema sigui no buit, es resol el problema de Procrustes associat. Les matrius Hamiltonianes i antiHamiltonianes apareixen en la resolució d'importants problemes de la Teoria de Sistemes i Control. El problema de valor propi invers per a matrius hermítiques i Hamiltonianes generalitzades va ser analitzat per Z. Zhang, X. Hu i L. Zang i posteriorment va ser considerat el cas de matrius hermítiques i antiHamiltonianes generalitzades per Z. Bai. En ambdós casos no només s'estudia el problema de valor propi invers i el problema de trobar la millor aproximació. Una extensió de les matrius Hamiltonianes són les matrius J-Hamiltonianes, i correspon a una de les aportacions originals que es realitzen a aquesta memòria. Als Capítols 3 i 4 s'estudien el problema de valor propi invers per a matrius normals J-Hamiltonianes i per a normals J-antiHamiltonianes. Per a la resolució del cas de les matrius normals J-Hamiltonianes es presenten quatre mètodes diferents. Els dos primers mètodes són generals i donen condicions per a que el problema tingui solución. El tercer mètode queda formalitzat al Teorema 3.2.2 que proporciona les condicions sota les quals el problema té solució i es presenten infinites solucions del mateix. Totes les solucions s'obtenen amb l'últim mètode. El principal resultat es dona al Teorema 3.2.3. Una secció completa està dedicada a la resolució del problema de Procrustes associat. L'organització d'aquesta tesi es la següent. El Capítol 1 conté una introducció al problema de valor propi invers i al problema de Procrustes. Al Capítol 2 s'estudia el problema de valor propi invers per a una matriu hermítica reflexiva respecte d'una matriu normal {k + 1}-potent, així com també el problema d'optimització de Procrustes associat. A més, es proposa un algoritme que resol el problema de Procrustes i es dona un exemple que mostra el funcionament del mateix. El problema de valor propi invers per a una matriu normal J-Hamiltoniana es resol al Capítol 3 fent servir diferents mètodes i a més es considera el problema d'optimització de Procrustes associat. Es proposa un algoritme que serveix per a calcular la solució del problema d'optimització i es presenten alguns exemples. Al Capítol 4, en funció dels resultats obtinguts al Capítol 3, s'aborda e
Gigola, SV. (2018). Optimización del problema de valor propio inverso para matrices estructuradas [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/106367
TESIS

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O conteúdo desta Tese está relacionado a versão da conjectura de Smale sobre a trivialidade do centralizador para certas classes de fluxos e ações de Rn. A conjectura de Smale estabelece que a maioria dos sistemas dinâmicos tem centralizador trivial, significando que toda a dinâmica que comuta com a original é um reescalonamento temporal da mesma. Neste trabalho, mostramos a trivialidade do centralizador para as seguintes classes de sistemas dinâmicos: (i) conjunto aberto de campos de classe C1 com singularidades hiperbólicas não-ressonantes e que satisfazem a Komuro-expansividade, os quais contém o atrator de Lorenz clássico como caso particular; (ii) conjunto Baire residual de campos conservativos de classe C1; (iii) conjunto Baire residual de campos hamiltonianos de classe C1. Além disso, provamos que o conjunto das ações de Rd localmente livres, expansivas e de classe C1 têm centralizador quase-trivial. Em particular, obtivemos os seguintes: (i) Rd-ações Anosov transitivas em variedades compactas têm centralizador quase-trivial; (ii) caracterização de sub-ações expansivas de ações de Rd.

Dans l'optique de produire de l'énergie à travers les réactions de fusion, nous sommes amenés à étudier des phénomènes physiques qui ont lieux dans les tokamaks. Les instabilités qui existent dans les tokamaks peuvent fortement dégrader le confinement et ont un impacte sur le fonctionnement de futurs réactions à fusion. Des mesures révèlent un fort transport radial. Même si ce transport radial est en partie est une conséquence des collisions, l'instabilité d'interchange est la source dominante à ce transport puisque le type de plasmas nous intéressant sont faiblement collisionnels. Dans la limite non collisionnelle, la description hamiltonienne permet de décrire le système dynamique des particules du plasmas dans un champ électromagnétique. Nous donnons de l'importance à cette description afin de pouvoir accéder aux outils hamiltoniens.Nous travaillons sur la modélisation et le contrôle hamiltonien du transport radial. Après avoir écrit le modèle hamiltonien des particules d'un plasma magnétisé, nous introduisons les réductions de ce modèle lagrangien en modèles eulériens réduits afin de s'adapter à certains calculs numériques et théoriques. Ces réductions donnent lieux aux équations fluides hamiltonien. Cependant, nous montrons que ces réductions peuvent faire perdre la propriété hamiltonienne. En particulier pour obtenir un modèle ayant la température des ions (puisqu'elle n'est pas négligeable au centre du plasma), nous montrons la procédure conservant la propriété hamiltonienne à partir du modèle sans température des ions.Quant à l'étude du transport radial, nous appliquons une des propriétés hamiltoniennes (le contrôle) afin de créer une barrière de transport par des perturbations du système. Nous étudions de manière idéale l'effet du contrôle à travers la dynamique lagrangienne des traceurs appelés particules test. Nous faisons particulièrement des efforts dans la prise en compte des contraintes numériques et expérimentales. Nous montrons notamment la robustesse du contrôle lors de l'application des perturbations par des sondes de Langmuir.Finalement, nous étudions l'application du contrôle dans un modèle eulérien décrivant la rétroaction du plasmas (à travers la densité et le potentiel électrique) lorsque nous appliquons les perturbations. Cette étape permet de prendre en compte le couplage du système plasma-perturbations. En utilisant un code fluide permettant de décrire le plasma de bord lors de perturbations générées par des sondes de Langmuir. Nous développons un algorithme permettant de calculer le contrôle en tout temps en fonction du potentiel électrique. Nous montrons alors que la valeur moyenne du potentiel électrique joue un rôle important pour l'application du contrôle dans un modèle fluide
In order to produce energy through fusion reactions, we are led to study of physical phenomena that occur in tokamaks. The instabilities that exist in tokamaks can significantly degrade the confinement and have an impact on the operation of future fusion reactors. Measurements reveal a strong radial transport. Although this is partly a consequence of collisions, the interchange instability is the dominant source to transport since the type of plasmas that interest us are weakly collisional. Within non collisional limit, the Hamiltonian description used to describe the dynamical system of charged particles in an electromagnetic field. We give importance to this description in order to access the Hamiltonian tools.We are working on modeling and control Hamiltonian of radial transport. After writing the Hamiltonian model of particles in a magnetized plasma, we introduced some reductions from Lagrangian models to Eulerian reduced models in order to accommodate some theoretical and numerical calculations. These places give the Hamiltonian fluid equations. However, we show that these reductions may lose the Hamiltonian property. In particular for a model with the ion temperature (not neglected at the center of the plasma), we show the procedure preserving the Hamiltonian property from the model without ion temperature.As for the study of radial transport, we apply one of the Hamiltonian properties (the control) to create a transport barrier by perturbations of the system. We are looking ideally the effect of control through the Lagrangian dynamics of tracers called test particles. We make particular efforts in the consideration of numerical and experimental constraints. We show the robustness of control when applying perturbations by Langmuir probes.Finally, we study the application of control in an Eulerian model describing the feedback of plasma (through the density and the electric potential) when we apply the perturbations. This step allows to take into account the coupling of the system plasma-perturbations. We use a numerical code to describe the plasma at the edge during perturbations generated by Langmuir probes. We develop an algorithm to calculate the control at all times depending on the electric potential. Finally we show that the average value of electric potential plays an important role in the implementation of control in a fluid model

As equações do modelo bidimensional de veículos autônomos subaquáticos fornecem um exemplo de sistema de controle não linear com o qual podemos ilustrar propriedades da teoria de controle ótimo. Apresentamos, sistematicamente, como os conceitos de formalismo hamiltoniano e teoria de Lie aparecem de forma natural neste contexto. Para tanto, estudamos brevemente o Princípio do Máximo de Pontryagin e discutimos características de sistemas afins. Tratamos com cuidado do Fenômeno Fuller, fornecendo critérios para decidir quando ele está ou não presente em junções, utilizando para isso uma linguagem algébrica. Apresentamos uma abordagem numérica para tratar problemas de controle ótimo e finalizamos com a aplicação dos resultados ao modelo bidimensional de veículo autônomo subaquático.
The equations of the two-dimensional model for autonomous underwater vehicles provide an example of a nonlinear control system which illustrates properties of optimal control theory. We present, systematically, how the concepts of the Hamiltonian formalism and the Lie theory naturally appear in this context. For this purpose, we briefly study the Pontryagin\'s Maximum Principle and discuss features of affine systems. We treat carefully the Fuller Phenomenon, providing criteria to detect its presence at junctions with an algebraic notation. We present a numerical approach to treat optimal control problems and we conclude with an application of the results in the bidimesional model of autonomous underwater vehicle.

In questo elaborato si trattano le soluzioni possibili del Problema dei Tre Corpi nell'ambito della Meccanica Celeste. Nella prima parte viene proposta una trattazione matematica del problema legata alla Meccanica Analitica; nella seconda parte si confrontano i risultati trovati con esempi reali presenti nel Sistema Solare.

Dans les systèmes dynamiques l'étude du voisinage et de la stabilité d'une solution périodique commence habituellement par l'étude du premier ordre, c'est-à-dire l'étude du système variationnel linéarisé. Ce premier pas conduit souvent soit à la stabilité exponentielle soit à l'instabilité exponentielle mais il peut aussi, assez fréquemment, conduire à des cas critiques ou le plus grand exposant caractéristique de Liapounov est nul. Il est alors nécessaire de considérer les termes d'ordre élevé. Ce sont surtout les problèmes hamiltoniens qui conduisent à des cas critiques et l'étude des termes d'ordre élevé y commence par une série de simplifications présentées dans les chapitres I et II. Ces simplifications conduisent au théorème de quasi-résonance et aux notions commodes qui y sont associées: quasi-intégrales, résonances positives etc. . . Qui permettent une classification générale des types de stabilité et d'instabilité. Les chapitres III et IV appliquent ces résultats théoriques aux mouvements de Lagrange du problème des 3 corps. Les résultats diffèrent beaucoup selon les cas étudiés : le cas du problème restreint circulaire est entièrement traité (cas plan) ou presque entièrement (cas tri-dimensionnel). Dans les cas non restreint (3 masses quelconques) et/ou non circulaire (3 masses en mouvement elliptiques), l'étude fournit seulement les résultats principaux : zones critiques, résonances d'ordre 3. Le cas elliptique tri-dimensionnel possède une résonance générale d'ordre 4 qui menace de détruire la stabilité dans une grande part des zones critiques
In the dynamical systems the study of the vicinity and the stability of a periodic solution begins usually by the “first-order study” of the variational system. The first step leads either to the exponential stability or to the exponential instability or to the “critical case” in which the largest Liapounov characteristic exponent is zero. In this third case it becomes necessary to consider the higher order terms. Most critical cases appear in Hamiltonian, problems and the study of large order terms begins by several simplifications that are presented in chapters I and II. These simplifications lead to the near –resonance theorem and to the adjacent useful notions : quasi-integrals, positive resonances etc … that allow a general classification of the types of stability and instability. The chapters III and IV apply these theoretical results to the Lagrangian motions of the 3-body problem. The results are very different according the case of interest. The restricted circular problem is entirely solved (planar case) or almost entirely solved (three-dimensional case)

Nous mettons en évidence une condition géométrico-dynamique minimale créant de l'hyperbolicité au voisinage d'un tore homocline transverse partiellement hyperbolique dans un système Hamiltonien presque intégrable à trois degrés de liberté. On en déduit une généralisation du théorème de dynamique symbolique d'Easton. Nous donnons ensuite une estimation optimale du temps de diffusion d'Arnold le long d'une chaîne de transition dans les systèmes Hamiltoniens initialement hyperboliques à trois degrés de liberté en utilisant une chaîne d'orbites périodiques hyperboliques sous-jacente.
Nous décrivons ensuite géométriquement à partir d'un système Hamiltonien presque intégrable à trois degrés de liberté à deux paramètres dû à Chirikov, un mécanisme de diffusion mettant en jeu un réseau de plans résonnants parallèles et voisins et un plan résonnant transversal au réseau. Ainsi, nous montrons qu'en dessous d'un certain seuil atteint par le paramètre prépondérant, on peut construire une orbite de transition dérivant en action à travers ce réseau modulationnel. Un des scénarii envisagés, le mécanisme de diffusion modulationnelle, basé sur l'existence de connexions hétéroclines entre tores partiellement hyperboliques issus de deux plans résonnants distincts est valide lorsqu'une condition de chevauchement est vérifiée.
Nous étudions enfin le modèle bidimensionnel décrivant un écoulement laminaire avec convection mixte entre deux plaques planes puis dans un tube vertical. Avec des conditions aux bords réduites, nous montrons via le théorème de la variété centrale qu'il existe dans le premier cas une bifurcation de pitchfork pour une valeur critique du nombre de Rayleigh.