A Short Proof of the Pontryagin Maximum Principle on Manifolds

TL;DR

This paper presents a concise and novel proof of the Pontryagin Maximum Principle for control systems on smooth manifolds by embedding the manifold into Euclidean space and leveraging the classical principle.

Contribution

It introduces a new proof technique using the Tubular Neighborhood Theorem to extend and project control systems between manifolds and Euclidean spaces.

Findings

Proof simplifies the derivation of the Pontryagin Maximum Principle on manifolds.

Method leverages embedding into Euclidean space for easier application.

Provides a new geometric perspective on optimal control on manifolds.

Abstract

Applying the Tubular Neighborhood Theorem, we give a short and new proof of the Pontryagin Maximum Principle on a smooth manifold. The idea is as follows. Given a control system on a manifold $M$, we embed it into an open subset of some $\mathbb R^n$, and extend the control system to the open set. Then, we apply the Pontryagin Maximum Principle on $\mathbb R^n$ to the extended system and project the consequence to $M$.