High-order Point Variations and Generalized Differentials

In a series of nonsmooth versions of the Pontryagin Maximum Principle, we used generalized differentials of set-valued maps, flows, and abstract variations. Bianchini and Stefani have introduced a notion of possibly high-order variational vector that has the summability property. We consider a slightly more general class of variational vectors than that defined by Bianchini and Stefani, and prove that the convex combinations of these vectors arise as “differentials” of variations that are differentiable in the sense of one of our generalized differentiation theories, namely, that of “approximate generalized differential quotients” (AGDQs).