Carathéodory on the road to the maximum principle

  • Hans Josef Pesch

    Chair of Mathematics in Engineering Sciences University of Bayreuth 95440 Bayreuth Germany


On his Royal Road of the Calculus of Variations Hermann Boerner coined the term "Königsweg der Variationsrechnung" in 1953; see H. Boerner: 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.