On the approximation of state constrained control problems

The author discusses some basic properties and applications of the variational inequality approach related to state constrained control problems. He first examines the simplest possible case of distributed control for linear elliptic partial differential equations, in the presence of constraints on the state. Next, the difficult example of controlling melting/solidification processes under temperature constraints is analyzed. This is expressed as a nonlinear evolution equation entering the class of two-phase Stephan problems. A fundamental ideal related to this approach is to handle the state constraints by means of the state system, not only by appropriate modifications of the cost functional