Set-valued differentials and the hybrid maximum principle

In recent years, it has become clear that most (smooth, nonsmooth, high-order, and hybrid) versions of the maximum principle (abbr. MP) for finite-dimensional, deterministic optimal control problems without state space constraints can be derived in a unified way, by using a modified version of the approach of the classical book [4] by Pontryagin et al.—that is, by constructing “packets of needle variations,” linearly approximating (that is, “differentiating”) these packets at the base value of the variation parameter, and propagating the resulting linear approximations to the terminal point of the trajectory by means of the differentials of the reference flow maps, in order to construct an “approximating cone” to a suitable reachable set. The classical approach must be modified in three basic ways. First, the classical differential (which is used twice, first to obtain linear approximations to the variations and then to propagate these approximations to the terminal point) must be replaced by other objects, called “generalized differentials” (abbr. GDs). Examples of GDs are: J. Warga’s “derivate containers” (cf. [9]), H. Halkin’s “screens” (cf. [3]), the “semidifferentials” and “multidifferentials” proposed by us in previous work (cf. [6]), and our more recent “generalized differential quotients” (abbr. GQDs) and “path-integral generalized differentials” (abbr. PIGDs), presented in [7] and [8]. Second, the time-varying vector fields that occur in the classical MP must be replaced by flows. Third, the needle variations must be replaced by abstract variations. A notion of GD 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” (abbr. DOMP). In the GD setting the “differentials” of a map at a point are sets of linear maps rather than single linear maps. It follows that these differentials cannot quite be used to propagate cones in a natural way, since the image of a cone under a set of linear maps is a set of cones rather than a single cone. Hence, if we agree to call a set of cones a “multicone,” the natural class of objects that must be used as linear approximations to sets is that of multicones. In particular, the “Pontryagin cone” of the classical theory is now replaced by a “Pontryagin multicone.” In this note, we propose an axiomatic definition of the concept of a “generalized differentiation theory” (abbr. GDT) and a precise statement of the DOMP, and we outline the definitions of our two most recent GDTs, namely, the GDQs and PIGDs. In addition, we give a complete statement of a “hybrid MP for general GDTs,”, which now amounts to saying that “to every GDT that has the DOMP is associated a version of the hybrid MP.” (We find it convenient not to include the DOMP among the GDT axioms, thus allowing the