On second order necessary conditions in infinite dimensional optimal control with state constraints

This paper is devoted to second order necessary optimality conditions for control problems in infinite dimensions. The main novelty of our work is the presence of pure state constraints together with end point constraints, quite useful in the applications. Second order analysis for control problems involving PDEs has been extensively discussed in the literature. The most usual approach to derive necessary optimality conditions is to rewrite the control problem as an abstract mathematical programming one. Our approach is different, we avoid the reformulation of the optimal control problem and use instead second order variational analysis. The necessary optimality conditions are in the form of a maximum principle and a second order variational inequality. They are first obtained in the form of nonintersection of convex sets. A suitable separation theorem allows to deduce their dual characterization.