From the brachystochrone to the maximum principle

Optimal control was born in 1696 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 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 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 show how a modern look at the BP, from the perspective of optimal control theory, can still yield new insights into this problem.