Geometric Approach to Pontryagin's Maximum Principle

TL;DR

This paper offers a geometric perspective on Pontryagin's Maximum Principle, clarifying its proof and the role of abnormal extremals, which are control-dependent solutions previously overlooked, through a detailed, self-contained mathematical exposition.

Contribution

It introduces a geometric approach to understanding Pontryagin's Maximum Principle, emphasizing the significance of abnormal extremals and providing a comprehensive, accessible proof.

Findings

Clarifies the role of abnormal extremals in optimal control.

Provides a detailed geometric proof of the principle.

Highlights the importance of abnormal extremals in control problems.

Abstract

Since the second half of the 20th century, Pontryagin's Maximum Principle has been widely discussed and used as a method to solve optimal control problems in medicine, robotics, finance, engineering, astronomy. Here, we focus on the proof and on the understanding of this Principle, using as much geometric ideas and geometric tools as possible. This approach provides a better and clearer understanding of the Principle and, in particular, of the role of the abnormal extremals. These extremals are interesting because they do not depend on the cost function, but only on the control system. Moreover, they were discarded as solutions until the nineties, when examples of strict abnormal optimal curves were found. In order to give a detailed exposition of the proof, the paper is mostly self\textendash{}contained, which forces us to consider different areas in mathematics such as algebra, analysis, geometry.