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Nonseparation of Sets and Optimality Conditions

Two new concepts of uniform approximating cones are introduced and discussed. The main result is a theorem for nonseparability of two closed sets. As an application of this result, an abstract Lagrange multiplier rule and a necessary optimality condition of Pontryagin maximum principle type for an optimal control problem in infinite-dimensional state space are obtained. These results allow one to treat infinite-dimensional optimal control problems with nonconvex target. The proposed approach reveals the importance of the uniformity of a tangent cone for obtaining necessary optimality conditions in an infinite-dimensional setting.