El objetivo principal de esta memoria es estudiar algunos problemas inversos en ecuaciones en derivadas parciales mediante el uso de desigualdades de Carleman. Estas últimas son una herramienta muy útil para obtener estabilidad para el problema inverso en torno a una solución regular.

Made available in DSpace on 2015-05-15T11:46:03Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 876230 bytes, checksum: 3a204615891ef1a7232794e0c75afdc8 (MD5) Previous issue date: 2010-09-30 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / This dissertation is devoted to prove the local exact controllability to the trajectories for a coupled system, of the Boussinesq kind. In the state system, the unknowns are the velocity field and pressure of the uid (y; p), the temperature (-) and an additional variable c that can be viewed as the concentration of a contaminant solute. We prove several results, that essentially show that it is sufficient to act locally in space on the equations satisfied by (-) and c. The controllability property of this system will be obtained by means of a Carleman inequality for apropriate system and of a inverse function theorem. / Esta dissertação é dedicada a provar a controlabilidade exata local ás trajetórias para um sistema acoplado do tipo Boussinesq. No sistema estado, as variáveis desconhecidas são o campo velocidade e pressão do fluido (y; p), a temperatura - e uma variável adicional c que pode ser vista como uma concentração de um soluto contaminante. A propriedade de controlabilidade nula desse sistema será obtida por meio de uma desigualdade de Carleman para um sistema apropriado e de um teorema de função inversa.

Made available in DSpace on 2015-05-15T11:46:21Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 1037417 bytes, checksum: faece7a3bb8a0ded8cbbcc9b6dbdb1da (MD5) Previous issue date: 2012-04-04 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / This work presents one global Carleman inequality for wave linear equation with bounded potential. Furthermore, we do two applications of this result. The first one refers to the study of exact controlabillity on the boundary and the second one deals with an inverse problem, where we want to recover the potential. / Este trabalho apresenta uma desigualdade de Carleman global para a equação de onda linear com potencial limitado. Além disso, são feitas duas aplicações desse resultado. A primeira delas refere-se ao estudo da controlabilidade exata na fronteira e a segunda trata de um problema inverso, onde buscamos recuperar o potencial.

In this thesis we developed the Green's Function for a tapered equivalent cylinder model of dendritic electrical propagation. We then use the Green's Function to develop a Carleman linear embedding scheme which is used to estimate the effects of a nonlinear ion channel hot-spot on the tapered cylinder solution. Mathematica© was used to implement the Carleman embedding scheme.

The probit function is the inverse of the cumulative distribution function associated with the standard normal distribution. It is of great utility in statistical modelling. The Carleman embedding technique has been shown to be effective in solving first order and, less efficiently, second order nonlinear differential equations. In this thesis, we show that solutions to the second order nonlinear differential equation for the probit function can be approximated efficiently using a variant of the Carleman embedding technique.

The Carleman Embedding is a method that allows us to embed a finite dimensional system of nonlinear differential equations into a system of infinite dimensional linear differential equations. This technique works well when dealing with first-order nonlinear differential equations. However, for higher order nonlinear ordinary differential equations, it is difficult to use the Carleman Embedding method. This project will examine the Carleman Embedding and a variation of the method which is very convenient in applying to second order systems of nonlinear equations.

Dans la première partie de ce mémoire, on s’attache à l’obtention d’Inégalités de Carleman elliptiques pour des opérateurs d’ordre deux au bord pour des conditions dites de Ventcel. Dans une seconde partie, on démontre une Inégalité adaptée aux multi-interfaces, pour des opérateurs elliptiques d’ordre quelconque, sous la condition classique de sous-ellipticité de Hörmander, ainsi que sous une condition de compatibilité entre les opérateurs sur la multi-interface et l’intérieur, dite de recouvrement. Cette condition généralise la condition de Lopatinskii. Enfin, dans une troisième partie, on s’intéresse à la contrôlabilté de l’équation de la chaleur et la stabilisation faible de l’équation des ondes dans des domaines polygonaux. / In the first part of this thesis, we derive elliptic Carleman estimates for second-order operators with Ventcel boundary conditions. In the second part, we prove a proper estimate near multi-interfaces for elliptic operatorsof any order, under the classical sub-ellipticity condition of Hörmander and under a compatibility condition between the operators in the interior and at the multi-interface, called the covering condition. This condition is a generalization of the well-known Lopatinskii condition. Finally, in the third part, we focus on controllability properties of the heat equation, and stabilization properties of the wave equation for polygonal domains, with mixed boundary conditions.

Dans cette thèse, nous prouvons des résultats concernant le contrôle et la stabilisation d'équations dispersives étudiées sur un intervalle borné. Pour commencer, nous étudions la stabilisation interne du système de Gear-Grimshaw, qui est un système de deux équations de Korteweg-de-Vries (KdV) couplées. Nous obtenons une décroissance exponentielle de l'énergie totale associée au modèle en introduisant une fonction de Lyapunov convenable. Nous prouvons aussi des résultats de contrôlabilité à zéro et exacte pour l'équation de Korteweg-de Vries avec un contrôle distribué à support dans un sous-intervalle du domaine. Pour la contrôlabilité à zéro du système linéarisé, nous utilisons l'approche classique basée sur la dualité qui ramène le problème à l'étude d'une inégalité d'observabilité qui, dans ce travail, est établie à l'aide d'une inégalité de Carleman. Ensuite, utilisant des fonctions plateau, nous prouvons un résultat de contrôlabilité exacte. Dans les deux cas, le résultat concernant le système non linéaire est obtenu à l'aide d'un argument de point fixe. Enfin, dans la lignée du résultat de contrôlabilité au bord obtenu par L. Rosier pour KdV, nous prouvons que le système linéaire de Boussinesq de type KdV-KdV est exactement contrôlable lorsque des contrôles sont appliqués au bord. Notre méthode repose sur l'utilisation de multiplicateurs et l'approche de la dualité mentionnée ci-dessus. Lorsqu'un mécanisme d'amortissement est introduit au bord, nous montrons que le système non linéaire est aussi exactement contrôlable et que l'énergie associée au modèle décroit exponentiellement / This work is devoted to prove a series of results concerning the control and stabilization properties of dispersive models posed on a bounded interval. Initially, we study the internal stabilization of a coupled system of two Korteweg-de Vries equations (KdV), the so-called Gear-Grimshaw system. Defining a convenient Lyapunov function we obtain the exponential decay of the total energy associated to the model. We also prove results of null and exact controllability for the Korteweg-de Vries equation with a control acting internally on a subset of the domain. In the case of the null controllability for the linear model, we use a classical duality approach which reduces the problem to the study of an observability inequality that, in this work, is proved by means of a Carleman inequality. Then, making use of cut-off functions, the exact controllability is also investigated. In both cases, the result for the nonlinear system is obtained by means of fixed-point argument. Finally, in view of the result of the boundary controllability obtained by L. Rosier for the KdV equation, we prove that the linear Boussinesq system of KdV-KdV type is exactly controllable when the controls act in the boundary conditions. Our analysis is performed using multipliers and the duality approach mentioned above. Adding a damping mechanism in the boundary, it is proved that the nonlinear system is also exactly controllable and that the energy associated to the model decays exponentially

Submitted by Maike Costa (maiksebas@gmail.com) on 2016-03-28T14:37:42Z No. of bitstreams: 1 arquivototal.pdf: 2200397 bytes, checksum: fa2b77afd6348b68a616a33acb7c7cb2 (MD5) / Made available in DSpace on 2016-03-28T14:37:42Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 2200397 bytes, checksum: fa2b77afd6348b68a616a33acb7c7cb2 (MD5) Previous issue date: 2014-03-20 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / The aim of this thesis is to present some controllability results for some fluid mechanic models. More precisely, we will prove the existence of controls that steer the solution of our system from a prescribed initial state to a desired final state at a given positive time. The two first Chapters deal with the controllability of the Burgers-α and Leray-α models. The Leray-α model is a regularized variant of the Navier-Stokes system (α is a small positive parameter), that can also be viewed as a model for turbulent flows; the Burgers-α model can be viewed as a related toy model of Leray-α. We prove that the Leray-α and Burgers-α models are locally null controllable, with controls uniformly bounded in α. We also prove that, if the initial data are sufficiently small, the pair state-control (that steers the solution to zero) for the Leray-α system (resp. the Burgers-α system) converges as α → 0+ to a pair state-control(that steers the solution to zero) for the Navier-Stokes equations (resp. the Burgers equation). The third Chapter is devoted to the boundary controllability of inviscid incompressible fluids for which thermal effects are important. They will be modeled through the so called Boussinesq approximation. In the zero heat diffusion case, by adapting and extending some ideas from J.-M. Coron [14] and O. Glass [45], we establish the simultaneous global exact controllability of the velocity field and the temperature for 2D and 3D flows. When the heat diffusion coefficient is positive, we present some additional results concerning exact controllability for the velocity field and local null controllability of the temperature. In the last Chapter, we prove the local exact controllability to the trajectories for a coupled system of the Boussinesq kind, with a reduced number of controls. In the state system, the unknowns are: the velocity field and pressure of the fluid (y, p), the temperature θ and an additional variable c that can be viewed as the concentration of a contaminant solute. We prove several results, that essentially show that it is sufficient to act locally in space on the equations satisfied by θ and c. / O objetivo desta tese é apresentar alguns resultados controlabilidade para alguns modelos da mecânica dos fluidos. Mais precisamente, provaremos a existência de controles que conduzem a solução do nosso sistema de um estado inicial prescrito à um estado final desejado em um tempo positivo dado. Os dois primeiros Capítulos preocupam-se com a controlabilidade dos modelos de Burgers-α e Leray-α. O modelo de Leray-α é uma variante regularizada do sistema de Navier-Stokes (α é umparâmetro positivo pequeno), que pode também ser visto como um modelo de fluxos turbulentos; já o modelo Burgers-α pode ser visto como um modelo simplificado de Leray-α. Provamos que os modelos de Leray-α e Burgers-α são localmente controláveis a zero, com controles limitados uniformemente em α. Também provamos que, se os dados iniciais são suficientemente pequenos, o par estado-controle (que conduz a solução a zero) para o sistema de Leray-α (resp. para o sistema de Burgers-α) converge quando α → 0+ a um par estado-controle (que conduz a solução a zero) para as equações de Navier-Stokes (resp. para a equação de Burgers). O terceiro Capítulo é dedicado à controlabilidade de fluidos incompressíveis invíscidos nos quais os efeitos térmicos são importantes. Estes fluidos são modelados através da então chamada Aproximação de Boussinesq. No caso emque não há difusão de calor, adaptando e estendendo algumas idéias de J.-M. Coron [14] e O. Glass [45], estabelecemos a controlabilidade exata global simultaneamente do campo velocidade e da temperatura para fluxos em 2D e 3D. Quando o coeficiente de difusão do calor é positivo, apresentamos alguns resultados sobre a controlabilidade exata global para o campo velocidade e controlabilidade nula local para a temperatura. No último Capítulo, provamos a controlabilidade exata local à trajetórias de um sistema acoplado do tipo Boussinesq, com um número reduzido de controles. Nesse sistema, as incógnitas são: o campo velocidade e a pressão do fluido (y, p), a temperatura θ e uma variável adicional c que pode ser vista como a concentração de um soluto contaminante. Provamos vários resultados, que essencialmente mostram que é suficiente atuar localmente no espaço sobre as equações satisfeitas por θ e c.

Submitted by ANA KARLA PEREIRA RODRIGUES (anakarla_@hotmail.com) on 2017-08-23T16:00:02Z No. of bitstreams: 1 arquivototal.pdf: 9090532 bytes, checksum: d4fefb1d97e9c6d585d5d18a33abf752 (MD5) / Made available in DSpace on 2017-08-23T16:00:02Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 9090532 bytes, checksum: d4fefb1d97e9c6d585d5d18a33abf752 (MD5) Previous issue date: 2016-11-30 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this thesis we study controllability results, asymptotic behavior and inverse problem related to some problems of the theory of partial di erential equations. Two particular systems are the focus of the study: the Mindin-Timoshenko system, describing the vibrational motion of a plate or a beam, and the phase eld system describing the temperature and phase of a medium having two distinct physical states. The rst chapter is devoted to the study of the 1-D Mindlin-Timoshenko system with discontinuous coe cient. A Carleman inequality is obtained under the assumption of monotonicity on the beam speed. Subsequently, two applications are provided: the controllability of the control system acting on the boundary and Lipschitzian stability of the inverse problem of recovering a potential from a single measurement of the solution. In the second chapter we consider a contact problem characterized by the behavior of a two-dimensional plate whose board makes contact with a rigid obstacle. The formulation of this problem is presented by the 2-D Mindlin-Timoshenko system with boundary conditions and suitable damping terms. Concerning such system, is proved via penalty techniques, the existence of solution and that the system energy has exponential decay when the time approaches in nity. In the third chapter, the study is aimed at a nonlinear phase- eld system de ned in a real open interval. Here we present some controllability results when a single control acts, by means of Dirichlet conditions, on the temperature equation of the system on one of the endpoints of the interval. To prove the results is used the method of moments, plus a spectral study of operators associated to the system and xed point theory to deal with the nonlinearity. / Nesta tese estudamos resultados de controlabilidade, comportamento assintótico e problema inverso relacionados a alguns problemas da teoria de equações diferenciais parciais. Dois sistemas particulares são foco do estudo: o sistema de Mindin-Timoshenko, que descreve o movimento vibratório de uma placa ou viga, e o sistema de campo de fases que descreve a temperatura e a fase de um meio onde ocorrem dois estados físicos distintos. O primeiro capítulo é dedicado ao estudo do sistema de Mindlin-Timoshenko 1-D com coe ciente descontínuos. Uma desigualdade de Carleman é obtida sob a hipótese de monotonicidade sobre velocidade da viga. Posteriormente, são fornecidas duas aplicações: a controlabilidade do sistema com controles agindo na fronteira e a estabilidade Lipschitziana do problema inverso de recuperar um potencial através de uma única informação obtida sobre a solução. No segundo capítulo consideramos um problema de contato caracterizado pelo comportamento de uma placa bidimensional cujo bordo faz contato com um obstáculo rígido. A formulação deste problema é apresentada pelo sistema de Mindlin-Timoshenko 2-D com condi ções de fronteira e termos de amortecimento (damping) adequados. Sobre tal sistema, é provada, através de técnicas de penalização, a existência de solução e, posteriormente, que sua energia possui decaimento exponencial quando o tempo tende ao in nito. No terceiro capítulo o estudo é voltado a um sistema de campo de fases não-linear de nido em um intervalo aberto real. Neste espaço apresentamos alguns resultados de controlabilidade quando um único controle age, sob condições de Dirichlet, na equação da temperatura em um dos bordos do intervalo. Para provar os resultados é utilizado o método dos momentos, além de uma estudo espectral de operadores associados ao sistema e teoria de ponto xo para lidar com a não-linearidade.