Local Minimum Principle for an Optimal Control Problem with a Nonregular Mixed Constraint

TL;DR

This paper establishes a new necessary condition, called the local minimum principle, for optimal control problems with a nonregular mixed constraint, expanding theoretical understanding in control theory.

Contribution

It introduces a novel local minimum principle for problems with nonregular mixed constraints, avoiding the use of functionals on bounded measurable functions.

Findings

Proves a first-order necessary condition for weak minima.

Formulates the principle using functions of bounded variation and Lebesgue--Stieltjes measures.

Provides illustrative examples demonstrating the principle.

Abstract

We consider the simplest optimal control problem with one nonregular mixed inequality constraint, i.e. when its gradient in the control can vanish on the zero surface. Using the Dubovitskii--Milyutin theorem on the approximate separation of convex cones, we prove a first or der necessary condition for a weak minimum in the form of the so-called local minimum principle, which is formulated in terms of functions of bounded variation, integrable functions, and Lebesgue--Stieltjes measures, and does not use functionals on the space of measurable bounded functions. Two illustrative examples are given. The work is based on results by Milyutin.