#### 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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