Faculty 1: University Award for the Best Master Thesis 2021

Integer optimal control problems have a variety of applications, for example in shape optimization, building energy management, or network transport problems. The goal is to control a system in such a way that a certain quantity is minimized or maximized, such as its energy consumption. However, it is often also of interest to obtain certain qualitative properties for the control function itself. In this sense, the paper looked more closely at total variation, which is a measure of how much a function varies, i.e., roughly speaking, it tells us something about its changing behavior. Penalizing strong variations in the control should lead to solutions that are easier to implement in practice. Such an approach also has other advantages and applications in mathematical modeling, for example, the existence of an optimal solution can be guaranteed for many problems if the total variation is used as a regularization term.

The master thesis investigates a general class of problems on such optimization problems. Besides proving the existence of a solution, the objective function was examined for differentiability and the findings were used to construct a solution algorithm based on the proximal gradient method as well as Bellman's optimality principle. After proving some convergence properties for sequences constructed by the algorithm, it was tested on two problem instances. Satisfactory solutions could be computed for only one problem. However, it is possible to transfer the ideas of the method to other promising algorithms.