Set-valued differentials and a nonsmooth version of Chow´s theorem

We present a simple application of a method developed by one of the authors to prove general smooth, nonsmooth, high-order, and hybrid versions of the maximum principle (MP) for finite-dimensional, deterministic optimal control problems without state space constraints. In the classical approach, one constructs "packets of needle variations", linearly approximates these packets at the base value of the variation parameter, and propagates the resulting linear approximations to the terminal point of the trajectory by means of the differentials of the reference flow maps. This technique must be modified by (a) replacing the classical differential by other objects, called generalized differentials ( GDs), (b) replacing the time-varying vector fields that occur in the classical MP by flows, and (c) replacing the needle variations by abstract variations. A notion of CD will yield a version of the MP provided it satisfies some natural properties such as the chain rule and an appropriate "directional open mapping property". Here we show how to apply the general theory to derive a "nonsmooth" version of Chow\´s theorem, in which some of the vector fields involved are just continuous, while the others are Lipschitz continuous but not necessarily of class C¹