Generalized differentials , variational generators , and the maximum principle with state constraints

We present the technical background material for a version of the Pontryagin Maximum Principle with state space constraints and very weak technical hypotheses, based on a primal approach that uses generalized differentials and packets of needle variations. In particular, we give a detailed account of two theories of generalized differentials, the " generalized differential quotients " (GDQs) and the " approximate generalized differential quotients " (AGDQs), and prove the corresponding open mapping and separation theorems. We state—but do not prove—the resulting version of the Maximum Principle. The result does not require the time-varying vector fields corresponding to the various control values to be continuously differentiable, Lipschitz, or even continuous with respect to the state, since all that is needed is that they be " co-integrably bounded integrally continuous. " This includes the case of vector fields that are continuous with respect to the state, as well as large classes of discontinuous vector fields, containing, for example, rich sets of single-valued selections for almost semicontinuous differential inclusions. Uniqueness of trajectories is not required, since our methods deal directly with multivalued maps. The dynamical reference vector field and reference Lagrangian are only required to be " differentiable " along the reference trajectory in a very weak sense, namely, that of possessing suitable " variational generators. " This includes— but is much more general than—the conditions of the classical cases when the reference vector field and Lagrangian are differentiable with respect to the state and the variational generator can be taken to be the singleton of the classical differential, as well as the case when they are Lipschitz and the variational generator can be chosen to be the Clarke generalized Jacobian. In addition, for the Lagrangian one can chose the variational generator to be the Michel-Penot subdifferential. For the functions defining the state space constraints, all that is needed is the existence of a variational generator in a slightly different technical sense, which includes as a special case the object often referred to as ∂ > x g in the literature, as well as many non-Lipschitz cases. The conclusion yields finitely additive measures, as in earlier work by other authors, and a Hamiltonian maximization inequality valid also at the jump times of the adjoint covector.