POD-Based Bicriterial Optimal Control of Convection-Diffusion Equations

Abstract

In this thesis optimal control problems governed by linear convection-diffusion equations and bilateral control constraints are investigated. The optimal control problem is seen as a multiobjective optimization problem, with the objectives being the deviation of the state variable from a prescribed desired state on the one hand and the costs of the control function on the other hand. Therefore, techniques to handle multiobjective optimization problems are presented. As optimality notion the Pareto optimality is chosen and methods to provide Pareto optimal points are introduced. Analytical and geometrical properties are shown for these methods. The theoretical problem of how to get the set of all Pareto optimal points, the so-called Pareto front, is investigated by looking at two parameter-dependent method classes: the weighted sum method and reference point methods. A continuous dependency of the solution of Euclidean reference point problems on the reference points is proved. Based on that, a numerical algorithm to approximate the Pareto front using the Euclidean reference point method is proposed. The approximation quality is ensured by the algorithm which generates the reference points.It is shown how the above mentioned optimal control problems can be transformed, such that they fit into the framework of multiobjective optimization. The Euclidean reference point method is applied to the transformed problems and the adjoint equation is introduced to get a numerically evaluateable representation of the derivatives of the cost function.As the finite element discretization of the controlled partial differential equation (PDE) yields high dimensional equation systems, which have to be solved repeatedly, proper orthogonal decomposition (POD) is used to get a reduced-order approximation of the optimal control problem. A-priori convergence results of the solution of the reduced problem to the solution of the full problem and a-posteriori error estimates are shown.Lastly, numerical experiments are presented to show the successful functioning of the presented algorithm and to evaluate the quality of the solutions of the model order reduced problem.