Higher order necessary conditions for optimal controls not ranging in the interior

TL;DR

This paper extends classical optimal control necessary conditions to cases where the optimal control may lie on the boundary of the control set, removing the usual interiority assumption.

Contribution

It develops Goh's and Legendre-Clebsch necessary conditions applicable even when the optimal control is on the boundary of the control set.

Findings

Necessary conditions valid without interior control assumption

Applicable to control sets with empty interiors

Includes controls touching the boundary of the control set

Abstract

Goh's and Legendre-Clebsch necessary conditions for optimal controls of affine-control systems are usually established under the hypothesis that the minimizing control lies in the interior of the control set $U$. In this paper we investigate the possibility of establishing Goh's and Legendre-Clebsch necessary conditions without this assumption, so that even control sets with empty interiors or optimal controls touching the boundary of $U$ can be taken into consideration.