Quadratures of Pontryagin Extremals for Optimal Control Problems

TL;DR

This paper introduces a method combining Noether's principle with Kozlov-Kolesnikov's theorem to compute first integrals, providing conditions for integrability of optimal control problems and illustrating with examples.

Contribution

It presents a novel approach to derive first integrals for optimal control problems using a combination of classical and modern integrability theorems.

Findings

Provided a sufficient condition for integrability by quadratures.

Applied the method to problems from the literature.

Offered an alternative proof for the integrability of a specific sub-Riemannian Lie group.

Abstract

We obtain a method to compute effective first integrals by combining Noether's principle with the Kozlov-Kolesnikov integrability theorem. A sufficient condition for the integrability by quadratures of optimal control problems with controls taking values on open sets is obtained. We illustrate our approach on some problems taken from the literature. An alternative proof of the integrability of the sub-Riemannian nilpotent Lie group of type (2,3,5) is also given.