Regularity of the Hamiltonian along optimal trajectories

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 costate trajectory, is independent of time. If the differential inclusion is Lipschitz continuous, then the Hamiltonian, 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 w.r.t. 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 conditions under which minimizers in the calculus of variations have bounded slope. The analysis is based on a recently proposed, local concept of differential inclusions that have bounded variation w.r.t. the time variable, in which conditions are imposed on the multifunction involved, only in a neighborhood of a given state trajectory.