On Normality in Optimal Control Problems with State Constraints

A general optimal control problem with endpoint, mixed and state constraints is considered. The question of normality of the known necessary optimality conditions is investigated. Normality stands for the positiveness of the Lagrange multiplier λ0 corresponding to the cost functional. In order to prove the normality condition, an appropriate derivative operator for the state constraints is constructed, which acts in a specific Hilbert space and has the properties of surjection and continuity.