AbstractsEngineering

The objective of this thesis is to study optimal control problems subject to equations arising in the field of fluid dynamics. This thesis is split into two essential parts. Each of them deals with an important partial differential equation, that are of interest in various applications and are widely considered in current research: The density dependent Navier – Stokes equation and the thin-film equation. These optimal control problems are motivated in many ways: First, the equations are mathematically interesting due to strong nonlinear effects occurring additionally as coupling effects in the context of optimization. Also, it is not immediate that properties (such as convergence of numerical approximations) are inherited by the optimal control problem. The literature on optimal control subject to nonlinear partial differential equation is rare, while the knowledge on those problems subject to the mentioned equations is even more rare: Only very few works are known, and the content of this thesis is a big contribution to this topic. Finally, for both control problems, there are industrial applications requiring the optimal control of fluid flows (which will also be addressed within the this thesis) such as the control of the interface in aluminum production, or the control of thin liquid layer on a silicon wafer. In both parts, the use of regularization parameters is vital in order to overcome analytical issues. The coupling of these parameters is specified, and (in the second part) a limiting problem is solved for these parameters tending to zero. In the first part of this thesis, we consider an optimal control problem for the interface in a two-dimensional two-phase fluid problem. The minimization functional consists of two parts: The $L^2$-distance to a given density profile and the interfacial length. We show existence of an optimal control and derive necessary first order optimality conditions for a corresponding phase field approximation. An unconditionally stable fully discrete scheme which is based on low order finite element discretization is proposed, and convergence of corresponding iterates to solutions of the continuous optimality conditions for vanishing discretization parameters is shown. The second part consists of an optimal control problem subject to the thin-film equation which is deduced from the Navier – Stokes equation. The thin-film equation lacks well-posedness for general controls due to possible degeneracies; state constraints are used to circumvent this problematic issue, and ensure well-posedness of the optimal control problem as well as the rigorous derivation of necessary first order optimality conditions for the optimal control problem. A multi-parameter regularization addressing both, the possibly degenerate term in the equation and the state constraint, is considered, and convergence is shown for vanishing regularization parameters by decoupling both effects. Both parts are concluded by corresponding numerical experiments, validating the models, comparing parameters…