Years Of Optimal Control: From The Brachystochrone To The Maximum Principle

This paper was presented in the history session of the 35th Conference on Decision and Control in Kobe, Japan, on December 11, 1996. 300 years of optimal control 1 Optimal control was born in 1697 --- 300 years ago --- in Groningen, a university town in the north of The Netherlands, when Johann Bernoulli, professor of mathematics at the local university from 1695 to 1705, published his solution of the brachystochrone problem. The year before he had challenged his contempories to solve this problem. We will tell the story of some of the events of 1696 and 1697 ---when solutions were submitted by Johann Bernoulli and such other giants as Newton, Leibniz, Tschirnhaus, l'Hopital, and Johann's brother, Jakob Bernoulli --- and then sketch the evolution of this field until it reached maturity in our century. Since the birth of optimal control, like all births, did not take place in a vacuum, the historical context will first be described, by outlining briefly some of the main ideas and discoveries on curve minimization problems from classical Greece up to Bernoulli's time. We will then state the brachystochrone problem, present Bernoulli's solution, and also provide a short nontechnical interlude, dealing with Bernoulli's personality and with his exceptionally gifted family. Subsequently we will follow the intricate path that has led to the modern versions of the necessary conditions for a minimum, from the Euler-Lagrange equations to the work of Legendre and Weierstrass and, eventually, the maximum principle of optimal control theory. Finally, we will "close the loop" by returning to the brachystochrone from the perspective of modern optimal control. Our thesis, that the brachystochrone marks the birth of optimal control, is undoubtedly somewhat controversial, and some re..