This thesis outlines a novel approach to make ill-posed inverse source problem well-posed exploiting inhomogeneous media. More precisely, we use Maxwell fish-eye lens to make scattered field emanating from distinct regions of an object of interest more directive and concentrated onto distinct regions of observation. The object of interest in this thesis is a thin slab placed conformally to the Maxwell fish-eye lens. Focused Green’s function of the background medium results in diagonal dominance of the matrix to be inverted for inverse problem solution. Hence, the problem becomes well-posed. We have studied one-dimensional variation of a very thin dielectric slab of interest having conformal shape to the lens. This method has been tested solving the forward problem using both Mie series and using COMSOL. Most common techniques for solving inverse problem are full non-linear inversion techniques, such as: distorted Born iterative method (DBIM) and contrast source inversion (CSI). DBIM needs to be regularized at every iteration. In some cases, it converges to a solution, and, in some cases, it does not. Diffraction tomography does not utilize regularization. It is a technique under Born approximation. It eliminates ill-posedness, but it works only for small contrast. Our proposed method works for high contrast and also provides well-posedness. In this thesis, our objective is to demonstrate inverse source problem and inverse scattering problem are not inherently ill-posed. They are ill-posed because conventional techniques usually use homogeneous or non-focusing background medium. These mediums do not support separation of scattered field. Utilization of background medium for scattered field separation casts the inverse problem in well-posed form.

<p> The inverse heat conduction problem also frequently referred as the sideways heat equation, in short SHE, is considered as a mathematical model for a real application, where it is desirable for someone to determine the temperature on the surface of a body. Since the surface itself is inaccessible for measurements, one is restricted to use temperature data from the interior measurements. From a  mathematical point of view, the entire situation leads to a non-characteristic Cauchy problem, where by using recorded temperature one can solve a well-posed nonlinear problem in the finite region for computing heat flux, and consequently obtain the Cauchy data [u, u_x]. Further by using these data and by performing an appropriate method, e.g. a space marching method, one can eventually achieve the desired temperature at x = 0.</p><p>The problem is severely ill-posed in the sense that the solution does not depend continuously on the data. The problem solved by two different methods, and for both cases we stabilize the computations by replacing the time derivative in the heat equation by a bounded operator. The first one, a spectral method based on finite Fourier space is illustrated to supply an analytical approach for approximating the time derivative. In order to get a better accuracy in the numerical computation, we use cubic spline function for approximating the time derivative in the least squares sense.</p><p>The inverse problem we want to solve, by using Cauchy data, is a nonlinear heat conduction problem in one space dimension. Since the temperature data u = g(t) is recorded, e.g. by a thermocouple, it usually contains some perturbation in the data. Thus the solution can be severely ill-posed if the Cauchy data become very noisy. Two experiments are presented to test the proposed approach.</p>

The inverse heat conduction problem also frequently referred as the sideways heat equation, in short SHE, is considered as a mathematical model for a real application, where it is desirable for someone to determine the temperature on the surface of a body. Since the surface itself is inaccessible for measurements, one is restricted to use temperature data from the interior measurements. From a  mathematical point of view, the entire situation leads to a non-characteristic Cauchy problem, where by using recorded temperature one can solve a well-posed nonlinear problem in the finite region for computing heat flux, and consequently obtain the Cauchy data [u, ux]. Further by using these data and by performing an appropriate method, e.g. a space marching method, one can eventually achieve the desired temperature at x = 0. The problem is severely ill-posed in the sense that the solution does not depend continuously on the data. The problem solved by two different methods, and for both cases we stabilize the computations by replacing the time derivative in the heat equation by a bounded operator. The first one, a spectral method based on finite Fourier space is illustrated to supply an analytical approach for approximating the time derivative. In order to get a better accuracy in the numerical computation, we use cubic spline function for approximating the time derivative in the least squares sense. The inverse problem we want to solve, by using Cauchy data, is a nonlinear heat conduction problem in one space dimension. Since the temperature data u = g(t) is recorded, e.g. by a thermocouple, it usually contains some perturbation in the data. Thus the solution can be severely ill-posed if the Cauchy data become very noisy. Two experiments are presented to test the proposed approach.

Cette thèse est une contribution à l’étude de problèmes de contrôle optimal dont le caractère non linéaire se traduit par la présence, dans les équations d’état, d’un terme bilinéaire relativement à l’état et au contrôle. Malgré les difficultés liées à la non linéarité, nous obtenons des propriétés spécifiques au cas bilinéaire. L’introduction générale constitue la première partie. La seconde partie est consacrée à l’étude des équations d’état ; ce sont des équations aux dérivées partielles d’évolution. Nous établissons des estimations a priori sur les solutions à partir des inégalités de Willett et Wong et nous démontrons que les équations d’états sont bien posées. Dans le cas où les contrôles subissent une contrainte liée aux états, ces estimations permettent de déduire l’existence de solutions dans le cadre des inclusions différentielles. Les troisième et quatrième parties de ce mémoire sont dévolues à la démonstration de l’existence de contrôles optimaux, puis à l’analyse de la sensibilité relative à une perturbation qui intervient de façon additive dans l’équation d’état. Le caractère bilinéaire permet de vérifier des conditions suffisantes d’optimalité du second ordre. Nous fournissons sur des exemples, une formule explicite des dérivées directionnelles de la fonction valeur optimale / This thesis is devoted to the analysis of nonlinear optimal control problems governed by an evolution state equation involving a term which is bilinear in state and control. The difficulties due to nonlinearity remain, but bilinearity adds a lot of structure to the control problem under consideration. In Section 2, by using Willet and Wong inequalities we establish a priori estimates for the solutions of the state equation. These estimates allow us to prove that the state equation is well posed in the sense of Hadamard. In the case of a feedback constraint on the control, the state equation becomes a differential inclusion. Under mild assumptions, such a differential inclusion is solvable. In Section 3, we prove the existence of solutions to the optimal control problem. Section 4 is devoted to the sensitivity analysis of the optimal control problem. We obtain a formula for the directional derivative of the optimal value function. This general formula is worked out in detail for particular examples

In this work we study, in details, the Cauchy problem of the nonlinear Schrödinger equation, with initial datas in periodic Sobolev spaces. Specifically, we prove that this problem is locally well posed for datas in Hsper, with s &#8805; 0. Particularly, for initial datas in L2 the problem is globally well posed, due to the conservation law of the equation in this space. Moreover, we prove the this result is the best one, seeing we expose examples that show that the equation flow is not locally uniformly continuous for initial datas with regularity less than L2. / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Neste trabalho, fazemos um estudo detalhado do problema de Cauchy para a equação não-linear cúbica de Schrödinger, com dados iniciais em espaços de Sobolev no toro. Especificamente, provaremos que este modelo é localmente bem posto para dados em Hsper, com s &#8805; 0. Em particular, para dados iniciais em L2 o modelo é globalmente bem posto, devido à lei de conservação da equação neste espaço. Além disso, provaremos que os resultados obtidos são os melhores possíveis, visto que exibiremos exemplos que mostram que o fluxo da equação não é localmente uniformemente contínuo para dados iniciais com regularidade menor que L2.

Given a full-range simply-invariant shift-invariant subspace <i>M</i> of the vector-valued <i>L²</i> space on the unit circle, the classical Beurling-Lax-Halmos (BLH) theorem obtains a unitary operator-valued function <i>W</i> so that <i>M</i> may be represented as the image of of the Hardy space <i>H²</i> on the disc under multiplication by <i>W</i>. The work of Ball-Helton later extended this result to find a single function representing a so-called dual shift-invariant pair of subspaces <i>(M,M^Ã)</i> which together form a direct-sum decomposition of <i>L²</i>. In the case where the pair <i>(M,M^Ã)</i> are finite-dimensional perturbations of the Hardy space <i>H²</i> and its orthogonal complement, Ball-Gohberg-Rodman obtained a transfer function realization for the representing function <i>W</i>; this realization was parameterized in terms of zero-pole data computed from the pair <i>(M,M^Ã)</i>. Later work by Ball-Raney extended this analysis to the case of nonrational functions <i>W</i> where the zero-pole data is taken in an infinite-dimensional operator theoretic sense. The current work obtains analogues of these various results for arbitrary dual shift-invariant pairs <i>(M,M^Ã)</i> of the <i>L²</i> spaces on the real line; here, shift-invariance refers to invariance under the translation group. These new results rely on recent advances in the understanding of continuous-time infinite-dimensional input-state-output linear systems which have been codified in the book by Staffans. / Ph. D.

Le problème du « Shape From Shading » est aujourd'hui considéré comme un problème mal posé et difficile à résoudre. Afin de bien comprendre les difficultés de ce problème et d'apporter des solutions fiables et pertinentes, nous proposons une approche rigoureuse basée sur la notion de solution de viscosité.<br />Après avoir considéré et exploité au maximum les équations (aux dérivées partielles) obtenues à partir de la modélisation classique du problème du « Shape From Shading », nous proposons et étudions de nouvelles équations provenant de modélisations plus réalistes que celles qui avaient été traitées classiquement dans la littérature. Cette démarche nous permet alors de démontrer qu'avec de telles nouvelles modélisations, le problème du « Shape From Shading » est généralement un problème complètement bien posé. En d'autres termes, nous prouvons que la version classique du problème du « Shape from Shading » est devenu mal posée à cause d'une trop grande simplification de la modélisation.<br />Dans ce travail, nous proposons aussi une extension de la notion de solutions de viscosité singulières développée récemment par Camilli et Siconolfi. Cette extension nous permet de proposer une nouvelle caractérisation des solutions de viscosité discontinues. Ce nouveau cadre théorique nous permet aussi d'unifier les différents résultats théoriques proposés dans le domaine du « Shape From Shading ».

Ce mémoire porte sur l’étude des équations d’évolution sur des variétés à coubure non nulle, plus particulièrement l’équation des ondes sur les espaces symétriques riemanniens de type non compact.Des propriétés de dispersion des solutions du problème de Cauchy homogène sont démontrées. Ces propriétés sont ensuite utilisées pour établir des estimations dites estimations de Strichartz. L’examen de ces estimées permet de déduire que le problème de Cauchy non linéaire avec des non-linéarités de type puissance est globalement bien posé pour des données initiales petites et localement bien posé pour des données arbitraires.Après un chapitre introductif dédié aux définitions, propriétés algébriques et géométriques des espaces symétriques et à quelques aspects élémentaires d’analyse harmonique sphérique sur ces espaces, un article est présenté : Wave equation on Riemannian symmetric spaces. Cet article contient nos résultats principaux. Dans le dernier chapitre nous présentons en détail deux problèmes ouverts qui prolongent nos travaux. Il s’agit respectivement d’établir le lien entre le comportement asymptotique des estimées et les orbites nilpotentes, et l’étude de l’équation des ondes pour les formes différentielles sur les espaces symétriques. / In this memoir we study evolution equations on curved manifolds. In particular we are interested in the wave equation on Riemannian symmetric spaces of the noncompact type.Dispersive properties of solutions of homogeneous Cauchy problem are proved. These properties are then used to establish Strichartz-type estimates. A closer study of these estimates shows that the nonlinear Cauchy problem with power-like nonlinearities is globally well posed for small initial data and locally well posed for arbitrary initial data.The first chapter is devoted to definitions, algebraic and geometric properties of symmetric spaces and to few elementary aspects of spherical analysis on these spaces. Then our main results are represented in an article : Wave equation on Riemannian symmetric spaces. In the last chapter we present in detail two open problems for future work. One issue is to establish a link between the asymptotic behavior of the estimates and nilpotent orbits, while another issue is the study of wave equation for differential forms on symmetric spaces.