# Optimal control of the unsteady Navier-Stokes equations

## Wachsmuth, Daniel

In dieser Arbeit werden Optimalsteuerungsprobleme bei der instationären Navier-Stokes-Gleichung behandelt. Zuerst werden die Zustandsgleichungen untersucht. Dabei beschränkten wir uns auf zweidimensionale Probleme, weil dort bekannt ist, dass die Lösung der Navier-Stokes-Gleichung eindeutig ist. Für das Optimalsteuerungsproblem werden notwendige Optimalitätsbedingungen erster und zweiter Ordnung bewiesen. Weiter werden hinreichene Optimalitätsbedingungen zweiter Ordnung behandelt. Erfüllt eine lokal optimale Steuerung diese Bedingung, dann ist sie stabil bezüglich kleinen Störungen des Ausgangsproblems. Weiterhin konvergiert die SQP-Methode quadratisch, wenn man nahe genung an solch einer Lösung startet. Weiterhin wurden diese Resultate für box-beschränkte Probleme auf Probleme mit konvexen Steuerungsbeschränkungen ausgedehnt. Die Arbeit wurde durch numerische Tests absgeschlossen.
My thesis is concerned with the study of optimal control problems for the non-stationary Navier-Stokes equations. These equations are a mathematical model to describe the behaviour of fluid flows. Simply speaking, we are considering the minimization of an objective function J(y, u) depending on control u and velocity field y subject to the state equation condensed to NS(y, u) = 0. This work is not the first on that particular subject. Many other authors have contributed their results. As a pioneering work stands the article of Abergel and Temam, where the first optimal control problem for fluid flows was considered, and where also the first existence and optimality conditions can be found. My thesis follows these classical attempt and will repeat known results, but also adds some new insight to this subject. At first an overview of the governing equations is given. It provides some functional analytic background material such as function spaces and weak solution concepts. Each control action requires a reaction of the flow. Though in real life this reaction should be unique, the mathematical theory does not yet provide such a result for three-dimensional problems for low regular data. In the two dimensional case this reaction is unique, and the mapping control-to-velocity field can be studied. It turns out that this mapping is even twice Fr´echet differentiable, which enables us to use well-known Banach space programming techniques later on. The contribution of the first chapter is the study of the linearized and adjoint state equations in Lp-spaces, a topic that was not recognized in optimal control of the instationary Navier-Stokes equations. These results are the basis for arguments in the following chapters, which would be not possible without them. The optimal control problem is then formulated exactly. Here also the question of existence of solutions is answered positively. The characterization of such a solution by necessary conditions follows. The first-order conditions imply a representation of optimal controls by projections, a fact which leads to new regularity results. In particular, an optimal control is a continuous function in space and time under some regularity assumptions. Sufficient second-order conditions are studied in detail. Such conditions are a-priori assumptions to prove local optimality of a reference control, as the name sufficient suggests. A refined analysis allows to prove local optimality of reference controls in weaker than L1-norms, which are usually encountered in optimal control problems for semilinear partial differential equations. This sufficient condition is an important assumption to prove stability of optimal controls under small perturbations. With the help of the Lp-estimates, we prove stability of optimal controls with respect to the L1-norm, a result which is stronger and was proven under weaker assumptions than other results of that type. This stability result directly gives the local quadratic convergence of the SQP-method for our optimal control problem. Furthermore, we investigated more general control constraints. The control in our problem is vector-valued, thus box-constraints are not the only possible choice. The material in this chapter especially regularity of optimal controls and sufficient optimality conditions is original. Infinite-dimensional optimization problems with such contraints are rarely investigated in the literature. Some questions remain open in this chapter and may initiate further research in this area. The thesis is completed by numerical tests. They confirm the convergence theory of the SQP-method. Furthermore, a new active-set method to solve problems with convex control constraints is presented.