Optimization with Partial Differential Equations

In the rst part of this article, we have shown how time-dependent optimal control for partial di erential equations can be realized in a modern high-level modeling and simulation package. In this second part we extend our approach to (state) constrained problems. Pure state constraints in a function space setting lead to non-regular Lagrange multipliers (if they exist), i.e. the Lagrange multipliers are in general Borel measures. This will be overcome by di erent regularization techniques. To implement inequality constraints, active set methods and interior point methods (or barrier methods) are widely in use. We show how these techniques can be realized in the modeling and simulation package Comsol Multiphysics. In contrast to the rst part, only the one-shot-approach based on space-time elements is considered. We implemented a projection method based on active sets as well as a barrier method and compare these methods by a specialized PDE optimization program, and a program that optimizes the discrete version of the given problem.