Generalized Differentials, Variational Generators, and the Maximum Principle with State Constraints

In a series of previous papers (cf. [20-23]), we have developed a “primal” approach to the non-smooth Pontryagin Maximum Principle, based on generalized differentials, flows, and general variations. The method used is essentially the one of classical proofs of the Maximum Principle such as that of Pontryagin and his coauthors (cf. Pontryagin et al. [15], Berkovitz [1]), based on the construction of packets of needle variations, but with a refinement of the “topological argument,” and with concepts of differential more general than the classical one, and usually set-valued.

In this article we apply this approach to optimal control problems with state space constraints, and at the same time we state the result in a more concrete form, dealing with a specific class of generalized derivatives (the “generalized differential quotients”), rather than in the abstract form used in some of the previous work.

The paper is organized as follows. In Sect. 2 we introduce some of our notations, and review some background material, especially the basic concepts about finitely additive vector-valued measures on an interval. In Sect. 3 we review the theory of “Cellina continuously approximable” (CCA) set-valued maps, and prove the CCA version – due to A. Cellina – of some classical fixed point theorems due to Leray-Schauder, Kakutani, Glicksberg and Fan. In Sect. 4 we define the notions of generalized differential quotient (GDQ), and approximate generalized differential quotient (AGDQ), and prove their basic properties, especially the chain rule, the directional open mapping theorem, and the transversal intersection property. In Sect. 5 we define the two types of variational generators that will occur in the maximum principle, and state and prove theorems asserting that various classical generalized derivatives – such as classical differentials, Clarke generalized Jacobians, subdifferentials in the sense of Michel–Penot, and (for functions defining state space constraints) the object often referred to as ∂ > _x gin the literature – are special cases of our variational generators. In Sect. 6 we discuss the classes of discontinuous vector fields studied in detail in [24]. In Sect. 7 we state the main theorem. The rather lengthy proof will be given in a subsequent paper.