INVERSE AND CONTROL PROBLEMS FOR SOME FLUID DYNAMIC EQUATIONS

Abstract

In this thesis, we study some inverse and control problems in the context of uid dynamic equations. We focus on the existence, uniqueness and stability as well as exact controllability to the trajectories issues related to those problems. Inverse problems are those problems in which the main concern is to nd causes for certain physical phenomena from the observations on e ects. It is a mathematical framework that is used to obtain information about a physical object or system from observed measurements. A common inverse problem is the recovery of coe cients of PDEs, such as conductivity, medium propagation characteristics, internal density distribution, external force, physical parameters, etc, from boundary or exterior measurements. On the other hand, the controllability problem may be formulated as follows: Consider an evolution PDE. We are allowed to act on the trajectories of the system using a suitable control (the right-hand side of the system (interior control), the boundary conditions (boundary control), etc). Fix T > 0, given a time interval t 2 (0; T) and initial and nal states, we aim to determine a control such that the solution coincides with the initial state t = 0 and the nal state at t = T. Applications of inverse and control problems appear in weather forecasting, stabilizing turbulent ows, the optimal design of aerodynamic pro les, hydrology, recovery of natural resources, electrical impedance tomography and control of uids, etc. The main aim of the thesis is to study the well-posedness of an inverse problem for convective Brinkman-Forchheimer (CBF) equations, a local in time existence and uniqueness result of an inverse problem for the Kelvin-Voigt uids, and local exact controllability to the trajectories of the CBF equations.