Minimizers that are not Impulsive Minimizers and Higher Order Abnormality

TL;DR

This paper explores the relationship between classical optimal control approaches, establishing conditions for their compatibility, and investigates higher-order abnormality phenomena related to infimum gaps and strict-sense minimizers.

Contribution

It introduces conditions under which set-separation and penalization methods are compatible and extends abnormality-gap results to strict-sense minimizers using higher-order conditions.

Findings

Compatibility conditions between set-separation and penalization approaches.

Extension of abnormality-gap correspondence to strict-sense minimizers.

Application to infimum-gap phenomena in control problems with unbounded controls.

Abstract

This paper addresses two related problems in optimal control. The first investigation consists of compatibility issues between two classical approaches to deriving necessary conditions for optimal control problems with a final target: the set-separation approach and penalization techniques. These methods generally lead to non-equivalent conditions, mainly due to their reliance on different notions of tangency at the target. We address this issue by considering Quasi Differential Quotient (QDQ) approximating cones (which are fit for the set-separation approach) and identifying conditions under which the Clarke tangent cone (which is a typical tool within penalization techniques) is also a QDQ approximating cone. In particular, we show that this property holds under suitable local invariance assumptions or when the target coincides locally with an $r$-prox regular set. In the second part of the paper we apply this compatibility result to the study of infimum-gap phenomena in optimal control problems with unbounded controls and impulsive extensions. In particular, we establish a connection between the occurrence of infimum gaps for strict-sense minimizers and abnormality in a higher-order Maximum Principle involving Lie brackets. While the abnormality-gap correspondence beyond first-order conditions has been already established for extended-sense --i.e. impulsive-- minimizers, a topological argument involving the former and the utilization of the above compatibility issues allow us to extend this correspondence to strict-sense minimizers.