PhD defence: OOC: Stabilization to trajectories and approximate controllability for the equations of fluid mechanics

In the work, we focus on three problems: Gevrey regularity, approximate controllability and stabilization to trajectories for the equations of fluid mechanics. The Gevrey regularity problem is addressed for Navier--Stokes equations under Lions boundary conditions in case the fluid is assumed to be viscous, incompressible and homogeneous. Sufficient conditions are given to guarantee that the solution of the systems is well-defined in a Gevrey regularity space. Some cases of boundaryless manifolds such as: the 2D Sphere; the 2D and 3D Torus which were mentioned in earlier studies are revisited. Four new results are obtained under Lions boundary conditions namely, the 2D Rectangle, the 2D Cylinder and the 2D Hemisphere; and the 3D Rectangle. The approximate controllability problem is considered for the Navier--Stokes equations and the 1D Burger equations. Following the Agrachev--Sarychev method, the approximate controllability by means of only few actuators is proven for Navier--Stokes system in a 2D infinite channel and in a 3D Rectangle.The actuators are the eigenfunctions of the Stokes operator. By ``few'' actuators, we mean that the number of them is independent of the viscosity coefficient. For the case of the 1D Burgers equations, a result is obtained for the case where the actuators are constrained in a small subset The exponential stabilization to trajectories problem is addressed for a general class of semilinear parabolic systems. The controls take values in a finite-dimensional space. Both internal and boundary actuators are supported in small region. We prove that under suitable conditions on the family of actuators, they allow us to stabilize the system. Moreover, some estimates on the number of actuators that we need to stabilize the system are established. Using the Dynamical Programming Principle, the stabilizing controls can be taken in feedback form and be computed by solving a suitable differential Riccati equation. Some simulations are presented by using Finite Element Method (FEM) and performed with MATLAB.