Normality and Gap Phenomena in Optimal Unbounded Control

TL;DR

This paper investigates conditions under which extended impulsive control problems avoid infimum gaps, focusing on normality of extremals and providing verifiable criteria to ensure the equivalence of original and extended problems.

Contribution

It establishes sufficient conditions based on normality for the absence of infimum gaps in unbounded control problems, improving previous criteria.

Findings

Normality of extremals guarantees no infimum gap.

Verifiable criteria simplify checking normality conditions.

Extended impulsive control problems can be analyzed for gap phenomena.

Abstract

Optimal unbounded control problems with affine control dependence may fail to have minimizers in the class of absolutely continuous state trajectories. For this reason, extended impulsive versions --which cannot be of measure-theoretical type-- have been investigated, in which the domain is enlarged to include discontinuous state trajectories of bounded variation, and for which existence of minimizers is guaranteed. It is of interest to know whether the passage from the original optimal control problem to its extension introduces an infimum gap. This paper provides sufficient conditions for the absence of an infimum gap based on normality of extremals. In certain cases, the normality conditions reduce to simple verifiable criteria, which improve on earlier, directly-derived sufficient conditions for no infimum gap.