# American Institute of Mathematical Sciences

2013, 3(1): 161-173. doi: 10.3934/naco.2013.3.161

## Carathéodory's royal road of the calculus of variations: Missed exits to the maximum principle of optimal control theory

 1 University of Bayreuth, Chair of Mathematics in Engineering Sciences, Bayreuth, D 95440, Germany

Received  January 2012 Revised  November 2012 Published  January 2013

The purpose of the present paper is to show that the most prominent results in optimal control theory, the distinction between state and control variables, the maximum principle, and the principle of optimality, resp. Bellman's equation are immediate consequences of Carathéodory's achievements published about two decades before optimal control theory saw the light of day.
Citation: Hans Josef Pesch. Carathéodory's royal road of the calculus of variations: Missed exits to the maximum principle of optimal control theory. Numerical Algebra, Control & Optimization, 2013, 3 (1) : 161-173. doi: 10.3934/naco.2013.3.161
##### References:
 [1] R. E. Bellman, The theory of dynamic programming,, Bull. Amer. Math. Soc., 60 (1954), 503. doi: 10.1090/S0002-9904-1954-09848-8. Google Scholar [2] R. E. Bellman, "Eye of a Hurricane, an Autobiography,", World Scientific Publishing Co Pte Ltd., (1984). Google Scholar [3] H. Boerner, Carathéodorys Eingang zur Variationsrechnung,, Jahresbericht der Deutschen Mathematiker Vereinigung, 56 (1953), 31. Google Scholar [4] V. G. Boltyanski, R. V. Gamkrelidze and L. S. Pontryagin, On the theory of optimal processes (in Russian),, Doklady Akademii Nauk SSSR, 110 (1956), 7. Google Scholar [5] M. H. Breitner, The genesis of differential games in light of Isaacs' contributions,, J. of Optimization Theory and Applications, 124 (2005), 523. doi: 10.1007/s10957-004-1173-0. Google Scholar [6] C. Carathéodory, Die Methode der geodätischen Äquidistanten und das Problem von Lagrange,, Acta Mathematica, 47 (1926), 199. Google Scholar [7] C. Carathéodory, "Variationsrechnung und Partielle Differentialgleichungen Erster Ordnung,", Teubner, (1935). Google Scholar [8] C. Carathéodory, The beginning of research in the calculus of variations,, Osiris, 3 (1937), 224. Google Scholar [9] C. Carathéodory, "Calculus of Variations and Partial Differential Equations of the First Order, Part 1, Part 2,", Holden-Day, (2001), 1965. Google Scholar [10] C. Carathéodory, "Variationsrechnung und partielle Differentialgleichungen erster Ordnung,", With Contributions of H. Boerner and E. Hölder (edited, (). Google Scholar [11] D. Carlson, An observation on two methods of obtaining solutions to Variational problems,, Journal of Optimization Theory and Applications, 114 (2002), 345. doi: 10.1023/A:1016035718160. Google Scholar [12] D. Carlson and G. Leitmann, Fields of extremals and sufficient conditions for the simplest problem of the calculus of variations,, Journal of Global Optimization, 40 (2008), 41. doi: 10.1007/s10898-007-9171-z. Google Scholar [13] D. Carlson and G. Leitmann, Fields of extremals and sufficient conditions for the simplest problem of the calculus of variations in $n$ variables,, in:, 33 (2009), 75. Google Scholar [14] D. Carlson and G. Leitmann, An equivalent problem approach to absolute extrema for calculus of variations problems with differential constraints,, Dynamics of Continuous, 18 (2011), 1. Google Scholar [15] M. R. Hestenes, "A General Problem in the Calculus of Variations with Applications to the Paths of Least Time,", Research Memorandum No. 100, (1123). Google Scholar [16] R. P. Isaacs, "Games of Pursuit,", Paper No. P-257, (1951). Google Scholar [17] R. P. Isaacs, Some fundamentals of differential games,, in, (1973), 25. Google Scholar [18] G. Leitmann, A note on absolute extrema of certain integrals,, International Journal of Nonlinear Mechanics, 2 (1967), 55. doi: 10.1016/0020-7462(67)90018-2. Google Scholar [19] G. Leitmann, On a class of direct optimization problems,, Journal of Optimization Theory and Appplications, 108 (2001), 467. doi: 10.1023/A:1017507006157. Google Scholar [20] S. MacLane, The Applied Mathematics Group at Columbia in World War II,, in, (1988), 495. Google Scholar [21] H. J. Pesch and R. Bulirsch, The maximum principle, Bellman's equation and Carathéodory's work,, J. of Optimization Theory and Applications, 80 (1994), 203. doi: 10.1007/BF02192933. Google Scholar [22] H. J. Pesch and M. Plail, The maximum principle of optimal control: a history of ingenious ideas and missed opportunities,, Control & Cybernetics, 38 (2009), 973. Google Scholar [23] M. Plail, "Die Entwicklung der optimalen Steuerungen,", Vandenhoeck & Ruprecht, (1998). Google Scholar [24] H. J. Sussmann J. C. and Willems:, 300 years of optimal control: from the brachystrochrone to the maximum principle,, IEEE Control Systems Magazine, 17 (1997), 32. doi: 10.1109/37.588098. Google Scholar [25] F. O. O. Wagener, On the Leitmann equivalent problem approach,, Journal of Optimization Theory and Applications, 142 (2009), 229. doi: 10.1007/s10957-009-9513-8. Google Scholar

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##### References:
 [1] R. E. Bellman, The theory of dynamic programming,, Bull. Amer. Math. Soc., 60 (1954), 503. doi: 10.1090/S0002-9904-1954-09848-8. Google Scholar [2] R. E. Bellman, "Eye of a Hurricane, an Autobiography,", World Scientific Publishing Co Pte Ltd., (1984). Google Scholar [3] H. Boerner, Carathéodorys Eingang zur Variationsrechnung,, Jahresbericht der Deutschen Mathematiker Vereinigung, 56 (1953), 31. Google Scholar [4] V. G. Boltyanski, R. V. Gamkrelidze and L. S. Pontryagin, On the theory of optimal processes (in Russian),, Doklady Akademii Nauk SSSR, 110 (1956), 7. Google Scholar [5] M. H. Breitner, The genesis of differential games in light of Isaacs' contributions,, J. of Optimization Theory and Applications, 124 (2005), 523. doi: 10.1007/s10957-004-1173-0. Google Scholar [6] C. Carathéodory, Die Methode der geodätischen Äquidistanten und das Problem von Lagrange,, Acta Mathematica, 47 (1926), 199. Google Scholar [7] C. Carathéodory, "Variationsrechnung und Partielle Differentialgleichungen Erster Ordnung,", Teubner, (1935). Google Scholar [8] C. Carathéodory, The beginning of research in the calculus of variations,, Osiris, 3 (1937), 224. Google Scholar [9] C. Carathéodory, "Calculus of Variations and Partial Differential Equations of the First Order, Part 1, Part 2,", Holden-Day, (2001), 1965. Google Scholar [10] C. Carathéodory, "Variationsrechnung und partielle Differentialgleichungen erster Ordnung,", With Contributions of H. Boerner and E. Hölder (edited, (). Google Scholar [11] D. Carlson, An observation on two methods of obtaining solutions to Variational problems,, Journal of Optimization Theory and Applications, 114 (2002), 345. doi: 10.1023/A:1016035718160. Google Scholar [12] D. Carlson and G. Leitmann, Fields of extremals and sufficient conditions for the simplest problem of the calculus of variations,, Journal of Global Optimization, 40 (2008), 41. doi: 10.1007/s10898-007-9171-z. Google Scholar [13] D. Carlson and G. Leitmann, Fields of extremals and sufficient conditions for the simplest problem of the calculus of variations in $n$ variables,, in:, 33 (2009), 75. Google Scholar [14] D. Carlson and G. Leitmann, An equivalent problem approach to absolute extrema for calculus of variations problems with differential constraints,, Dynamics of Continuous, 18 (2011), 1. Google Scholar [15] M. R. Hestenes, "A General Problem in the Calculus of Variations with Applications to the Paths of Least Time,", Research Memorandum No. 100, (1123). Google Scholar [16] R. P. Isaacs, "Games of Pursuit,", Paper No. P-257, (1951). Google Scholar [17] R. P. Isaacs, Some fundamentals of differential games,, in, (1973), 25. Google Scholar [18] G. Leitmann, A note on absolute extrema of certain integrals,, International Journal of Nonlinear Mechanics, 2 (1967), 55. doi: 10.1016/0020-7462(67)90018-2. Google Scholar [19] G. Leitmann, On a class of direct optimization problems,, Journal of Optimization Theory and Appplications, 108 (2001), 467. doi: 10.1023/A:1017507006157. Google Scholar [20] S. MacLane, The Applied Mathematics Group at Columbia in World War II,, in, (1988), 495. Google Scholar [21] H. J. Pesch and R. Bulirsch, The maximum principle, Bellman's equation and Carathéodory's work,, J. of Optimization Theory and Applications, 80 (1994), 203. doi: 10.1007/BF02192933. Google Scholar [22] H. J. Pesch and M. Plail, The maximum principle of optimal control: a history of ingenious ideas and missed opportunities,, Control & Cybernetics, 38 (2009), 973. Google Scholar [23] M. Plail, "Die Entwicklung der optimalen Steuerungen,", Vandenhoeck & Ruprecht, (1998). Google Scholar [24] H. J. Sussmann J. C. and Willems:, 300 years of optimal control: from the brachystrochrone to the maximum principle,, IEEE Control Systems Magazine, 17 (1997), 32. doi: 10.1109/37.588098. Google Scholar [25] F. O. O. Wagener, On the Leitmann equivalent problem approach,, Journal of Optimization Theory and Applications, 142 (2009), 229. doi: 10.1007/s10957-009-9513-8. Google Scholar
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