Stationary Optimal Control Problems with Pointwise State Constraints

First order optimality conditions for stationary pointwise state constrained optimal control problems are considered. It is shown that the Lagrange multiplier associated with the pointwise inequality state constraint is a regular Borel measure only, in general. In case of sufficiently smooth data and under a regularity assumption on the active set, the Lagrange multiplier can be decomposed into a regular L2-part concentrated on the active set and a singular part, which is concentrated on the interface between the active and the inactive set. In a second part of the paper numerical solution strategies are reviewed. These methods fall into two classes: the first class which includes interior-point methods as well as active set strategies is purely finite dimensional. The second class, however, admits an analysis in function space. The latter methods typically rely on regularization. In this respect, MoreauYosida-based and Lavrentiev-based techniques are discussed. The paper ends by a numerical comparison of the presented solution algorithms. Date: October 4, 2006. 1991 Mathematics Subject Classification. 49M15,49M37,65K05,90C33.