Regularity of the Hamiltonian along Optimal Trajectories

TL;DR

This paper explores how the regularity of the differential inclusion in optimal control problems influences the regularity of the Hamiltonian along optimal trajectories, extending known results to cases with bounded variation.

Contribution

It introduces a new concept of differential inclusions with bounded variation and demonstrates how this affects the Hamiltonian's regularity, leading to improved optimality conditions.

Findings

Hamiltonian inherits bounded variation from the differential inclusion.

New conditions for nondegeneracy of optimality conditions.

Less restrictive criteria for bounded slope in calculus of variations.

Abstract

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.