A New Constraint Qualification for Mixed Constrained Optimal Control

TL;DR

This paper introduces a new constraint qualification for mixed constrained optimal control problems, linking it to an asymptotic weak maximum principle and enhancing the theoretical framework for optimality conditions.

Contribution

It develops a novel constraint qualification that ensures the asymptotic weak maximum principle implies the classical one, and establishes its minimality and sufficiency criteria.

Findings

The new constraint qualification guarantees the classical weak maximum principle under its verification.

In smooth settings, it is the weakest qualification with this property.

Provides sufficient conditions for the qualification's validity.

Abstract

In recent developments, a novel set of necessary optimality conditions for mixed constrained optimal control problems, termed the asymptotic weak maximum principle, has been formulated. These novel conditions deviate from the classical ones by virtue of their sequential nature and the fact that they are satisfied regardless of the regularity conditions imposed on the mixed constraints. Furthermore, due to their asymptotic behaviour, these conditions serve as a precise tool for use as stopping criteria in numerical methods of solution. However, it should be noted that, in certain instances, these conditions may not be sufficiently robust to fully characterize optimal solutions, as they can be satisfied by processes that are not extremals. The present study proposes a novel constraint qualification, meticulously developed to address these asymptotic optimality conditions. It is demonstrated that the asymptotic weak maximum principle implies the classical weak maximum principle when the newly proposed constraint qualification is verified. It is further demonstrated that, in the smooth setting, this constraint qualification is the weakest one that possesses such a property. Additionally, this study present sufficient criteria for the validity of the newly proposed constraint qualification.