Set transversality, approximating multicones, Warga derivate containers and Mordukhovich cones

Smooth and nonsmooth versions of the Pontryagin Maximum Principle can be proved using necessary conditions for set separation in terms of approximating multicones arising from generalized differentiation theories. We propose a notion of approximating multicone derived from J. Warga’s theory of derivate containers and the notion of Mordukhovich normal cone, and state and prove the corresponding set separation theorem.