POD-Based Mixed-Integer Optimal Control of Convection-Diffusion Equations

Abstract

In this thesis we consider the numerical treatment of mixed-integer optimal control problems governed by linear convection-diffusion equations and binary control constraints. We use relaxation techniques for the optimal control problem which have been already used in mixed-integer optimal control problems for ordinary differential equations. The goal is to construct binary admissible controls such that the corresponding optimal state and the optimal value of the cost function of the relaxed problem can be approximated with arbitrary accuracy. To solve the optimal control problems of the relaxation we have to solve many state and adjoint equations. Using finite element methods to discretize the state and adjoint equations yields often to extensive systems which make the frequently calculations time-consuming. Therefore we apply a model-order reduction by the proper orthogonal decomposition (POD) method. This yields to a significant acceleration of the CPU time while the error stays small. Finally we present numerical experiments to verify the functionality of the presented algorithm and the quality of the solutions of the reduced problem.