DOCUMENTA MATHEMATICA, Vol. Extra Volume: Optimization Stories (2012), 317-329

Hans Josef Pesch

Carathéodory on the Road to the Maximum Principle

\begingroup \renewcommand{\thefootnote}{\fnsymbol{footnote}} \footnotetext{This article contains material from the author's paper: \emph{Carathéodory's Royal Road of the Calculus of Variations: Missed Exits to the Maximum Principle of Optimal Control Theory}, to appear in Numerical Algebra, Control and Optimization (NACO).} \endgroup On his Royal Road of the Calculus of Variations\footnote{Hermann Boerner coined the term «Königsweg der Variationsrechnung» in 1953; see H. Boerner: \emph{Carathéodorys Eingang zur Variationsrechnung}, Jahresbericht der Deutschen Mathematiker Vereinigung, 56 (1953), 31--58. He habilitated 1934 under Carathéodory.} the genious Constantin Carathéodory found several exits -- and missed at least one -- from the classical calculus of variations to modern optimal control theory, at this time, not really knowing what this term means and how important it later became for a wide range of applications. How far Carathéodory drove into these exits will be highlighted in this article. These exits are concerned with some of the most prominent results in optimal control theory, the distinction between state and control variables, the principle of optimality known as Bellman's equation, and the maximum principle. These acchievements either can be found in Carathéodory's work or are immediate consequences of it and were published about two decades before optimal control theory saw the light of day with the invention of the maximum principle by the group around the famous Russian mathematician Pontryagin.

2010 Mathematics Subject Classification: 01A60, 49-03, 49K15

Keywords and Phrases: History of calculus of variations, history of optimal control, maximum principle of optimal control, calculus of variations, optimal control

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