Optimal control of elliptic stationary problems with state constraints

An optimal control problem associated with a stationary elliptic partial differential equation system is considered, and optimality conditions when the state is constrained are obtained. The method used makes it possible to obtain first-order conditions more or less `strong´ (the `weak´ conditions being obtained without assuming the well-known Slater condition). The method is applied to a very simple example of a distributed control problem