Chattering variations, finitely additive measures, and the nonsmooth maximum principle with state space constraints

We discuss the proof of a version of the maximum principle with state space constraints for data with very weak regularity properties, using the classical method of packets of needle variations (PNVs), as in Pontryagin’s book, but coupling it with a nonclassical theory of multivalued differentials, the so-called “generalized differential quotients” (GDQs). The key technical point of our argument is the use of a different type of PNVs, that we call “chattering PNVs.” These variations make it possible to get a conclusion involving finitely additive vector-valued measures of finite total variation. The theory presented here applies to control dynamics without uniqueness of trajectories (so that the flow maps are set-valued) and to differential inclusions (so that the “differentials” of maps are also set-valued). Keywords— Maximum Principle, state constraints, additive measures