  This paper concerns state constrained optimal control problems, in which the
dynamic constraint takes the form of a differential inclusion. If the
differential inclusion does not depend on time, then the Hamiltonian, evaluated
along the optimal state trajectory and the co-state trajectory, is independent
of time. If the differential inclusion is Lipschitz continuous, then the
Hamitonian, evaluated along the optimal state trajectory and the co-state
trajectory, is Lipschitz continuous. These two well-known results are examples
of the following principle: the Hamiltonian, evaluated along the optimal state
trajectory and the co-state trajectory, inherits the regularity properties of
the differential inclusion, regarding its time dependence. We show that this
principle also applies to another kind of regularity: if the differential
inclusion has bounded variation with respect to time, then the Hamiltonian,
evaluated along the optimal state trajectory and the co-state trajectory, has
bounded variation. Two applications of these newly found properties are
demonstrated. One is to derive improved conditions which guarantee the
nondegeneracy of necessary conditions of optimality, in the form of a
Hamiltonian inclusion. The other application is to derive new, less
restrictive, conditions under which minimizers in the calculus of variations
have bounded slope. The analysis is based on a new, local, concept of
differential inclusions that have bounded variation with respect to the time
variable, in which conditions are imposed on the multifunction involved, only
in a neighborhood of a given state trajectory.
