Quasi-Invariant Optimal Control Problems

TL;DR

This paper extends Noether's theorem in optimal control to include quasi-invariant problems and transformations depending on multiple parameters, broadening its applicability and providing new insights.

Contribution

It introduces a generalized version of Noether's theorem for optimal control that handles quasi-invariance and multi-parameter transformations, expanding theoretical understanding.

Findings

Extended Noether's theorem for quasi-invariant problems

Applicable to transformations with multiple parameters

Illustrated with new examples not covered before

Abstract

We study in optimal control the important relation between invariance of the problem under a family of transformations, and the existence of preserved quantities along the Pontryagin extremals. Several extensions of Noether theorem are provided, in the direction which enlarges the scope of its application. We formulate a more general version of Noether's theorem for optimal control problems, which incorporates the possibility to consider a family of transformations depending on several parameters and, what is more important, to deal with quasi-invariant and not necessarily invariant optimal control problems. We trust that this latter extension provides new possibilities and we illustrate it with several examples, not covered by the previous known optimal control versions of Noether's theorem.