Contents

Pontryagin’s Maximum Principle has been widely discussed and used in Optimal Control Theory since the second half of the 20th century. Here, we focus on the proof and on the understanding of this Principle, using as much as possible geometric ideas and tools. This approach provides a better and clearer understanding of the Principle and, in particular, of the role of the abnormal extremals. In order to give a detailed exposition of the proof, the paper is mostly self-contained