De type parabolique et soumis à l’action d’un couple de contrôles (h, k) où h et k jouent des rôles différents ; le contrôle k étant de type "contrôlabilité" et h de type "contrôle optimal".Il est alors naturel de considérer un problème d’optimisation multi-critères. Il existe plusieurs façons d’étudier de tels problèmes. Nous proposons, dans cette thèse, le contrôle de Stackelberg. Il s’agit d’une notion d’optimisation hiérarchique avec, ici, h qui est le "Leader" et k le "Follower". / In this thesis, we are interested in evolution problems governed by parabolic equations subjected to the action of a pair of controls (h, k) where h and k play different roles : the control k being of "controllability" type and h of "optimal control" type.It is then natural to consider a multi-criteria optimization problem. There are several ways to study such problems. We propose in this thesis, the Stackelberg control which is a notion of hierarchical optimization with here, h which is the "Leader" and k the "Follower".

Cette thèse est consacrée à l'étude de la contrôlabilité approchée et à zéro des systèmes paraboliques linéaires sur un domaine non vide borné Ω de (), contrôlés par moins de forces que d'équations. Les contrôles seront localisés sur un ouvert de Ω ou sur son bord. Nous étudierons deux problèmes différents. Le premier consiste à contrôler une des équations indirectement à l'aide d'un opérateur de couplage d'ordre un. Nous obtenons alors des résultats pour plusieurs classes d'opérateurs et de systèmes. La deuxième question que nous étudierons est de savoir s'il est possible de contrôler seulement certaines composantes de la solution du système. Nous donnons une condition nécessaire et suffisante lorsque les coefficients de couplage sont constants ou dépendent du temps et étudions un système simplifié quand ils dépendent de l'espace. Nous terminerons en détaillant un schéma numérique avec lequel nous fournirons des perspectives quant à quelques problèmes qui restent ouverts en contrôlabilité partielle des systèmes paraboliques linéaires. / This thesis is devoted to the study of the approximate and null controllability of linear parabolic systems on a nonempty bounded domain Ω of(), controlled by less controls than equations. The controls will be localized in an open set of Ω or on its boundary. We will study two different problems. The first of them involves controlling one of the equations indirectly with a coupling operator of order one. We obtain some results for different class of operators and systems. The second question we will study is to know if it is possible to control only some components of the solution of the system. We give a necessary and sufficient condition when the coupling coefficients are constant or time depending and study a simplified system when they are space dependent. We will finish by giving details on a numerical scheme with which we provide perspectives concerning some open problems in partial controllability of linear parabolic systems.

Made available in DSpace on 2015-05-15T11:46:19Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 2353317 bytes, checksum: d71ead9d4e0f785df35982fc9318c7da (MD5) Previous issue date: 2014-07-25 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this thesis, we study controllability results of some phenomena modeled by Partial Differential Equations (PDEs): Multi objective control problem, for parabolic equations, following the Stackelber-Nash strategy is considered: for each leader control which impose the null controllability for the state variable, we find a Nash equilibrium associated to some costs. The leader control is chosen to be the one of minimal cost. Null controllability for the linear Schrödinger equation: with a convenient space-time discretization, we numerically construct boundary controls which lead the solution of the Schrödinger equation to zero; using some arguments of Fursikov-Imanuvilov (see [Lecture Notes Series, Vol 34, 1996]) we construct controls with exponential decay at final time. Null controllability for a Schrödinger-KdV system: in this work, we combine global Carleman estimates with energy estimates to obtain an observability inequality. The controllability result holds by the Hilbert Uniqueness Method (HUM). Controllability results for a Euler type system, incompressible, inviscid, under the influence of a temperature are obtained: we mainly use the extension and return methods / Nesta tese, estudaremos resultados de controle para alguns problemas da teoria das equações diferenciais parciais (EDPs): Problema de controle multi objetivo para um problema parabólico, seguindo estratégias do tipo Stackelberg-Nash: para cada controle líder, que impõe a controlabilidade nula para o estado, encontramos seguidores, em equilíbrio de Nash, associados a funcionais custo. Em seguida, determinamos o líder de menor custo. Controlabilidade nula para a equação de Schrödinger linear: com uma discretização espaço-tempo adequada, construímos numericamente controles-fronteira que conduzem a solução de Schrödinger a zero; utilizando técnicas de Fursikov-Imanuvilov (veja [Lecture Notes Series, Vol 34, 1996]) contruímos controles que decaem exponencialmente no tempo final. Controlabilidade nula para um sistema acoplado Schrödinger-KdV: neste trabalho, combinando estimativas globais de Carleman com estimativas de energia, obtemos uma desigualdade de observabilidade. O resultado de controlabilidade segue pelo método de unicicade Hilbert (HUM). Controlabilidade para um sistema do tipo Euler, incompressível, invíscido, sob influência de uma temperatura: Utilizamos os métodos de extensão seguido do método do retorno para provar resultados de controlabilidade para este sistema