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, This thesis is organised in two relatively independent chapters Chapter 1 is devoted to the study of some mathematical problems arising in the theory of hydrodynamic turbulence Our results focus on questions related to the large deviations principle (LDP), Gallavotti?Cohen type symmetry, and ergodicity (existence and uniqueness of a stationary measure and its mixing properties) for a family of randomly forced PDE's. We establish the LDP for parabolic PDE's, such as the Navier?Stokes system or the complex Ginzburg?Landau equation, perturbed by a random kick force
, with strong nonlinear dissipation, such as the Burgers equation. Finally, we prove a mixing property for the complex Ginzburg?Landau equation with a white-noise perturbation in any space dimension we first consider the problem of controllability of a quantum particle by the amplitude of an electric field. The position of the particle is described by a wave function which obeys the bilinear Schrödinger equation. We are mainly interested in the global controllability problems of this equation. Using some variational methods, we establish approximate controllability, feedback stabilisation, and simultaneous controllability results. The second part of this chapter is concerned with the problem of controllability of Lagrangian trajectories of the 3D Navier?Stokes system by a finite-dimensional force. We provide some examples of saturating spaces which ensure the approximate controllability of the system, Keywords: Navier?Stokes system equation, nonlinear wave equation, large deviations principle, Gallavotti?Cohen symmetry, kick force, white noise, coupling method; Schrödinger equation, Lyapunov function, approximate controllability, stabilisation, simultaneous controllability
, Nos résultats portent principalement sur des questions liées au principe de grandes déviations (PGD), relation de Gallavotti?Cohen et ergodicité (existence et unicité d'une mesure stationnaire et ses propriétés de mélange) pour une classe d'EDP perturbées par une force aléatoire Nous établissons un PGD pour des EDP paraboliques, comme les équations de Navier?Stokes ou de Ginzburg?Landau complexe, perturbées par une force aléatoire discrète en temps. Nous étendons ce résultat au cas de l'équation d'onde non linéaire amortie soumise à une force aléatoire de type bruit blanc en temps et lisse par rapport à la variable spatiale, en prouvant un PGD local. Nous obtenons une relation de type Gallavotti?Cohen pour la fonction de taux d'une fonctionnelle de production d'entropie pour des EDP avec une dissipation non linéaire forte, comme l'équation de Burgers. Enfin, nous prouvons une propriété de mélange pour l'équation complexe de Ginzburg?Landau avec un bruit blanc dans un espace de dimension quelconque. Dans le chapitre 2, nous considérons d'abord le problème de la contrôlabilité d'une particule quantique par l'amplitude d'un champ électrique. L'état de la particule est décrit par une fonction d'onde qui obéit à l'équation de Schrödinger bilinéaire, Ce mémoire est composé de deux chapitres relativement indépendants. Le chapitre 1 est consacré à l'étude de quelques problèmes mathématiques issus de la théorie de la turbulence en hydrodynamique En utilisant des méthodes variationnelles, nous obtenons des résultats de contrôlabilité approchée, stabilisation et contrôlabilité simultanée. La deuxième partie de ce chapitre aborde le problème de la contrôlabilité lagrangienne de l'équation de Navier?Stokes 3D par une force de dimension finie. Nous donnons des exemples d'espaces qui assurent la contrôlabilité approchée du système
, équation de Burgers, équation d'onde non linéaire, principe de grandes déviations, relation de Gallavotti?Cohen, bruit blanc, méthode de couplage