Approximation of optimal control problems governed by PDEs

Many technical processes are described by systems of partial differential equations. Optimization of such processes or identification of process parameters lead to optimal control problems for partial differential equations. Usually, such problems are characterized by additional constraints, i.e. some quantities of the process have to fulfil certain equations and inequalities. This project is concerned with linear-quadratic optimal control problems: The optimization goal is a quadratic function of the process quantities. Moreover, these quantities occur linear in the equations and inequalities. This project is especially interested in investigating elliptic and parabolic differential equations. Although this class of problems has a simple structure, it is impossible to solve such problems exactly. Therefore, it is necessary to discretize such problems in a suitable manner. Consequently, approximation properties of the discretized problems with respect to the solution of the continuous problem represent a main focus of the project. There is a large progress in the theory of control constrained problems in the recent years. In contrast to this, approximation results for state constrained optimal control problems are nearly unknown. This project will lower the large gap between the well investigated control constrained case and the widely unknown field of state constrained problems. Moreover, the results should be used to construct stopping criteria for iterative methods. Stopping criteria based on error estimates can drastically reduce computational time for optimal control problems in a reliable way. Therefore, it is possible to solve larger and more complicate problems in future. So-called superconvergence effects appear by solving optimal control problems numerically. A better understanding of superconvergence effects should help to exploit them in numerical algoritms. Such algorithms deliver essential better numerical results for a given discretization.

Many technical processes are described by systems of partial differential equations. Optimization of such processes or identification of process parameter lead to optimal control problems for partial differential equations. Usually, such processes are characterized by additional constraints, i.e., some quantities of the process have to fulfil certain equations and inequalities. This project was concerned with linear-quadratic optimal control problems. The optimization goal was a quadratic functional. Moreover, the quantities occurred linear in the equations and inequalities. Main goal of the project was the investigation of discretization and regularization strategies in particular for state constrained optimal control problems. We achieved a lot of new and innovative results for this important class of optimal control problems. A first issue was the regularization of state constrained problems by problems with mixed control-state constraints. We were able to derive estimates for the regularization error for the Lavrentiev regularization and for the new virtual control concept. Moreover, we studied the discretization error for both approaches for finite element discretizations. By means of an improvement of the primal dual active set strategy we were able to construct an efficient method for solving such kind of problems. Since we derived also error estimates for the optimization algorithm itself, we developed a new concept for solving optimal control problems with state constraints efficiently with given accuracy.