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The objective of the present paper is investigation of the optimal synthesis and normality of the maximum principle for the Mayer optimal control problem under pure state constraints. Such models do arise in many applied areas such as space industry, robotics, drug administration, economy, etc. We express the optimal synthesis using Dini derivatives of an associated cost-to-go function and derive the normal maximum principle from a new Neighboring Feasible Trajectories theorem (NFT). For a state constraint with smooth boundary, NFT theorems imply that under standard assumptions on control system and an inward pointing condition, feasible trajectories depend in a Lipschitz way on the initial states. Some recent counterexamples indicate that, if the state constraint is an intersection of two half spaces in ℝn, surprisingly conclusions of NFT theorems might be no longer valid. We propose here a new inward pointing condition implying a new NFT theorem.