Controllability for some equations from fluid mechanics

Abstract

In this thesis we present controllability results for some models of fluid mechanics. More precisely, we investigate the existence of controls that drive the solution of the system from an initial state to a prescribed final state in a given positive time. In the first Chapter, the controllability of the Stokes equation with memory is analyzed. This model is a variant of the well-known Stokes equation, with the addition of a non-local term in time building a memory effect in the equation. This model can also be seen as a linearization around zero of an Oldroyd kind viscoelastic fluid system. We prove that the result of null controllability for this equation is not true, even if the control acts over the whole boundary. To this purpose, it is verified that the corresponding observability inequality is not satisfied. We also build explicit initial data such that, for any control, the corresponding solution is different from zero at final time. The second Chapter is dedicated to the controllability of fluids in which thermal effects are important. We prove the exact controllability to the trajectories of a coupled system of the Boussinesq type, for a fluid satisfying boundary conditions of the Navier kind for the velocity and of the Robin kind for the temperature. The control acts on a part of the boundary. First, using a domain extension procedure, we transform the problem into to distributed controllability problem. Then, we prove an approximate global controllability result, following the strategy of Coron et al [J. EUR. Mathematics. Soc., 22 (2020), pp. 1625-1673]. Through linearization and using appropriate Carleman estimates, we conclude with a local control result.