A very non-smooth maximum principle with state constraints

We present a version of the Pontryagin Maximum Principle with state-space constraints and very weak technical hypotheses. 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 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." 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.