Observateurs en dimension infinie. Application à l'étude de quelques problèmes inverses

Dans un grand nombre d'applications modernes, on est amene a estimer l'etat initial (ou final) d'un systeme infini-dimensionnel (typiquement un systeme gouverne par une Equation aux Derivees Partielles (EDP) d'evolution) a partir de la connaissance partielle du systeme sur un intervalle de temps limite. Un champ d'applications dans lequel apparait frequemment ce type de probleme d'identification est celui de la medecine. Ainsi, la detection de tumeurs par tomographie thermo-acoustique peut se ramener a des problemes de reconstruction de donnees initiales. D'autres methodes necessitent l'identification d'un terme source, qui, sous certaines hypotheses, peut egalement se reecrire sous la forme d'un probleme de reconstruction de donnees initiales. On s'interesse dans cette these a la reconstruction de la donnee initiale d'un systeme d'evolution, en travaillant autant que possible sur le systeme infini-dimensionnel, a l'aide du nouvel algorithme developpe par Ramdani, Tucsnak et Weiss (Automatica 2010). Nous abordons en particulier l'analyse numerique de l'algorithme dans le cadre des equations de Schrodinger et des ondes avec observation interne. Nous etudions les espaces fonctionnels adequats pour son utilisation dans les equations de Maxwell, avec observations interne et frontiere. Enfin, nous tentons d'etendre le cadre d'application de cet algorithme lorsque le systeme initial est perturbe ou que le probleme inverse n'est plus bien pose, avec application a la tomographie thermo-acoustique.

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