From the Brachystochrone to the Maximum Principle

x 1. Introduction Optimal control was born in 1696 |300 years ago this year| in the Netherlands, when Johann Bernoulli challenged his contemporaries with the \brachystochrone problem" (BP). The purpose of this paper is to give a brief outline of why this event truly deserves to be called the birth of optimal control, and how the research that began in 1696 has led to modern optimal control theory and, especially, to the maximum principle. In particular , we will argue that, as this path was followed, several opportunities were missed that would have led to much earlier discovery of the maximum principle. In at least one case |that of the formulation of Hamilton's equations| we will attempt to show that the discovery was missed for no reason other than the decision to rewrite an equation in terms of one formalism rather than another one that would have been equally suitable and was also available at the time. As a conclusion, we will show how a modern look at the BP, from the perspective of optimal control theory, can still yield new insights into this 300-year old problem. Johann Bernoulli's challenge attracted a lot of attention , and some of the greatest mathematicians of the time submitted solutions. The May 1697 issue of Acta Eruditorum contains Johann's own solution, as well as a rather diierent one by his elder brother Jakob, and contributions by Newton, Tschirnhaus, l'H^ opital and Leibniz. So there is no doubt that something important happened in 1696-7. For example, D. J. Struik, referring to the articles published in the May 1697 Acta Eru-ditorum, writes ((3], p. 392) that \these papers opened the history of a new eld, the calculus of variations." We want to go a bit farther, however, and make a case for a 1696 birth of optimal control theory. This, naturally, requires some explanation. The conventional wisdom holds that optimal control theory was born about 40 years ago with the work on the \Pontryagin maximum principle" by L. S. Pontrya-gin and his group, cf. 2], or perhaps a few years earlier with the work of McShane and Hestenes. On the other hand, if we take a careful look at those features that make optimal control diierent from the calculus of variations, we can already nd quite a few of them in the BP. The calculus of variations deals mainly with optimization problems of the \standard" …