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

The optimization of processes is an omnipresent task in industry and science. In many cases those processes are characterized by partial di erential equations (PDEs), which describe how the state of a considered system can be regulated by a control. The aim is to nd a control which induces a desired state or a close approximation of it. A task of this type is called an optimal control problem. In this thesis we study thermal processes. As an initial setting we assume a temperature distribution y0 on a domain Ω and a desired nal distribution y(T ). The temperature can be controlled by a heat source u acting on the boundary of the domain. In the considered cases of this work, the correlation between control and state is determined by a (non-)linear heat equation. In this way every control u gets assigned uniquely to a state y(u). The purpose is to nd a control, such that the associated state approximates the desired state as precise as possible at nal time T . Additional restrictions can be imposed on the control to meet technical limiting factors for example. We state this task mathematically by a quadratic minimization problem governed by a (non-)linear heat equation and inequality constraints concerning the control. We start this thesis with a short chapter providing fundamental facts and de nitions. To motivate the relevance of the considered problem we introduce a general optimal control problem governed by a nonlinear parabolic equation at the beginning of the second chapter. One possible approach to this problem is the application of a sequential quadratic programming (SQP) method. At this proceeding each iterative step yields a quadratic minimization problem with linearized constraints. This is the kind of problem we study in the course of the present thesis. In Chapter 3 we specify the parameters of the problem, show the existence of a unique solution, and state the corresponding optimality conditions. We perform a spatial discretization of the domain in Chapter 4 using the nite element (FE) Method and develop a reduction of the problem by the FE-Galerkin approach. This leads to high dimensional systems of ordinary di erential equations. The discretization is completed by the application of the implicit Euler method. Furthermore we present the primal-dual active set strategy (PDASS) as a possible way to solve the completely discretized problem with inequality constraints. To classify the quality of an approximate solution we make use of an a-posteriori error estimation, which we present in Chapter 5. In Chapter 6 we introduce the proper orthogonal decomposition (POD) method for model reduction and adjust the PDASS algorithm of the previous chapter to the POD-Galerkin reduced problem. By this method we can lower the dimension of the considered systems of ordinary di erential equations signi cantly. In order to improve the approximation quality of the reduced order model we introduce optimality system proper orthogonal decomposition (OS-POD) in Chapter 7. This approach results in a POD based algorithm with an OS-POD initialization step and a-posteriori error estimation. We apply the presented algorithms in several numerical test runs and analyze the results in the last chapter. Additionally, we compare our ndings with the ones of [Rog14] and [Grä14]. Finally, we draw a short conclusion in the last section.