No infimum gap and normality in optimal impulsive control under state constraints

TL;DR

This paper establishes conditions under which there is no infimum gap in impulsive optimal control problems with state constraints, linking normal extremals to the absence of such gaps and providing verifiable criteria.

Contribution

It extends the understanding of infimum gaps and normality in impulsive control problems to include state constraints and additional controls, with verifiable conditions.

Findings

Normal extremals ensure no infimum gap in constrained impulsive control.

Verifiable constraint and endpoint qualifications guarantee normality.

Links between infimum gap absence and normality are established with state constraints.

Abstract

In this paper we consider an impulsive extension of an optimal control problem with unbounded controls, subject to endpoint and state constraints. We show that the existence of an extended-sense minimizer that is a normal extremal for a constrained Maximum Principle ensures that there is no gap between the infima of the original problem and of its extension. Furthermore, we translate such relation into verifiable sufficient conditions for normality in the form of constraint and endpoint qualifications. Links between existence of an infimum gap and normality in impulsive control have previously been explored for problems without state constraints. This paper establishes such links in the presence of state constraints and of an additional ordinary control, for locally Lipschitz continuous data.