Generalized maximum principle in optimal control

TL;DR

This paper introduces a generalized maximum principle for optimal control problems, extending classical results by formulating necessary conditions for strong local minima, applicable to calculus of variations and beyond.

Contribution

It develops a new concept of strong local infimum and derives generalized maximum principles that strengthen classical optimal control conditions.

Findings

Contains a family of maximum principles including Pontryagin's as a special case.

Provides generalized necessary conditions for strong local minima.

Includes examples demonstrating the extension and strengthening of classical results.

Abstract

For an optimal control problem, the concept of a strong local infimum is introduce, for which necessary conditions consisting of some family of "maximum principles" are formulated. If a function delivers a strong local minimum in this problem (and therefore, a~strong local infimum), then this family contains the classical Pontryagin maximum principle. As a corollary, we derive generalized necessary conditions for a strong local minimum for a problem of the calculus of variations. Examples are given to show that the necessary conditions obtained in the present paper generalize and strengthen classical results.