Nesse trabalho, estudamos a equação de Schrödinger não-linear com uma função potencial delta atrativa. As soluções para essa equação tem uma componente localizada e uma dispersiva. Além de estudar o comportamento das soluções dessa equação em espaços de Sobolev clássicos, mostramos algumas propriedades do grupo unitário em espaços Lp, L2 com peso, Sobolev com peso e assim obtemos alguns resultados de boa colocação local e global das soluções. O ponto central desta tese é mostrarmos a existência de uma variedade invariante centro que irá consistir de órbitas periódicas no tempo. / In this work, we study the nonlinear Schrodinger equation with an attractive delta function potential.The solutions to this equation have a localized and a dispersive component. In addition to studying the behavior of solutions of this equation in classical Sobolev space, we show some properties for the unitary group in Lp, weighted L2 and Sobolev spaces and so we get some results of local and global well-posedness of solutions. The central theme this thesis is to show the existence of a center invariant manifold, which will consist of time-periodic orbits.

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L'objet de cette thèse est de proposer une étude numérique, fondée sur l'application de résultats de la théorie du contrôle optimal géométrique, des trajectoires spatiales du système Terre-Lune dans un contexte de poussée faible. Le mouvement du satellite est décrit par les équations du problème restreint des trois corps controlé. Nous nous concentrons sur la minimisation de la consommation énergétique et du temps de transfert. Les trajectoires optimales sont recherchées parmi les projections des courbes extrémales solutions du principe du maximum de Pontryagin et peuvent être calculées grâce à une méthode de tir. Ce procédé fait intervenir l'algorithme de Newton dont la convergence nécessite une initialisation précise. Nous surmontons cette difficulté au moyen de techniques homotopiques ou d'études géométriques du système de contrôle linéarisé. L'optimalité locale des trajectoires extrémales est ensuite vérifée en utilisant les conditions du second ordre liées au concept de point conjugué. Dans le cas du problème de minimisation de l'énergie, une technique de "recollement" de trajectoires optimales kepleriennes autour de la Terre et La Lune et d'une solution optimale de l'équation du mouvement linéarisée au voisinage du point d'équilibre L1 est également proposée pour approximer les transferts Terre-Lune à énergie minimale. / This PhD thesis provides a numerical study of space trajectories in the Earth-Moon system when low-thrust is applied. Our computations are based on fundamental results from geometric control theory. The spacecraft's motion is modelled by the equations of the controlled restricted three-body problem. We focus on minimizing energy cost and transfer time. Optimal trajectories are found among a set of extremal curves, solutions of the Pontryagin's maximum principle, which can be computed solving a shooting equation thanks to a Newton algorithm. In this framework, initial conditions are found using homotopic methods or studying the linearized control system. We check local optimality of the trajectories using the second order optimality conditions related to the concept of conjugate points. In the case of the energy minimization problem, we also describe the principle of approximating Earth-Moon optimal transfers by concatening optimal keplerian trajectories around The Earth and the Moon and an energy-minimal solution of the linearized system in the neighbourhood of the equilibrium point L1.