Optimality System POD for Time-Variant, Linear-Quadratic Control Problems

Abstract

This thesis focuses on time-variant, linear-quadratic control problems, which occur as subproblems in SQP methods, for instance. In particular it treats a quadratic minimization problem governed by a linear heat equation and inequality constraints concerning the control. After showing unique solvability of the problem, optimality conditions are derived. In order to solve the problem numerically, the FE-Galerkin method is applied for spacial discretization. A primal-dual active set strategy is presented to solve the discretized problem. Since this requires the solution of several high-dimensional equation systems, the POD method is applied for model reduction. Optimality system POD (OS-POD) is introduced to improve the approximation quality of the reduced order model. This approach results in a POD based algorithm with an OS-POD initialization step and a-posteriori error estimation to classify the quality of an approximate solution. Finally the methods are analyzed in several numerical test runs. Besides, the results are compared to findings of other theses.