# 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 and 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), 503-516. doi: 10.1090/S0002-9904-1954-09848-8. [2] R. E. Bellman, "Eye of a Hurricane, an Autobiography," World Scientific Publishing Co Pte Ltd., Singapore, 1984. [3] H. Boerner, Carathéodorys Eingang zur Variationsrechnung, Jahresbericht der Deutschen Mathematiker Vereinigung, 56 (1953), 31-58. [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), 7-10. [5] M. H. Breitner, The genesis of differential games in light of Isaacs' contributions, J. of Optimization Theory and Applications, 124 (2005), 523-559. doi: 10.1007/s10957-004-1173-0. [6] C. Carathéodory, Die Methode der geodätischen Äquidistanten und das Problem von Lagrange, Acta Mathematica, 47 (1926), 199-236. [7] C. Carathéodory, "Variationsrechnung und Partielle Differentialgleichungen Erster Ordnung," Teubner, Leipzig, Germany, 1935. [8] C. Carathéodory, The beginning of research in the calculus of variations, Osiris, 3 (1937), 224-240; also in "Gesammelte Mathematische Schriften 1 (Variationsrechnung)" (edited by the Bayerische Akademie der Wissenschaften), C. H. Beck'sche Verlagsbuchhandlung, München, Germany, (1954), 212-248. [9] C. Carathéodory, "Calculus of Variations and Partial Differential Equations of the First Order, Part 1, Part 2," Holden-Day, San Francisco, California, 1965-1967; Reprint: 2nd AMS printing, AMS Chelsea Publishing, Providence, RI, USA, 2001. [10] C. Carathéodory, "Variationsrechnung und partielle Differentialgleichungen erster Ordnung," With Contributions of H. Boerner and E. Hölder (edited, commented and extended by R. Klötzler), [11] D. Carlson, An observation on two methods of obtaining solutions to Variational problems, Journal of Optimization Theory and Applications, 114 (2002), 345-361. doi: 10.1023/A:1016035718160. [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), 41-50. doi: 10.1007/s10898-007-9171-z. [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: "Variational Analysis and Aerospace Engineering" (eds. G. Buttazzo and A. Frediani), Springer Optimization and Its Applications, 33, Springer, New York, New York, (2009), 75-90. [14] D. Carlson and G. Leitmann, An equivalent problem approach to absolute extrema for calculus of variations problems with differential constraints, Dynamics of Continuous, Discrete and Impulsive Systems, Series B: Applications & Algorithms, 18 (2011), 1-15. [15] M. R. Hestenes, "A General Problem in the Calculus of Variations with Applications to the Paths of Least Time," Research Memorandum No. 100, ASTIA Document No. AD 112382, RAND Corporation, Santa Monica, CA, 1950. [16] R. P. Isaacs, "Games of Pursuit," Paper No. P-257, RAND Corporation, Santa Monica, CA, 1951. [17] R. P. Isaacs, Some fundamentals of differential games, in "Topics in Differential Games" (ed. A. Blaquiére), North-Holland Publishing Company, Amsterdam, The Netherlands, (1973), 25-31. [18] G. Leitmann, A note on absolute extrema of certain integrals, International Journal of Nonlinear Mechanics, 2 (1967), 55-59. doi: 10.1016/0020-7462(67)90018-2. [19] G. Leitmann, On a class of direct optimization problems, Journal of Optimization Theory and Appplications, 108 (2001), 467-481. doi: 10.1023/A:1017507006157. [20] S. MacLane, The Applied Mathematics Group at Columbia in World War II, in "A Century of American Mathematics, Part III" (eds. P. L. Duren, R. Askey and U. C. Merzbach), Providence, RI, (1988), 495-515. [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), 203-229. doi: 10.1007/BF02192933. [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), 973-995. [23] M. Plail, "Die Entwicklung der optimalen Steuerungen," Vandenhoeck & Ruprecht, Göt-tingen, Germany, 1998. [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), 32-44. doi: 10.1109/37.588098. [25] F. O. O. Wagener, On the Leitmann equivalent problem approach, Journal of Optimization Theory and Applications, 142 (2009), 229-242. doi: 10.1007/s10957-009-9513-8.

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##### References:
 [1] R. E. Bellman, The theory of dynamic programming, Bull. Amer. Math. Soc., 60 (1954), 503-516. doi: 10.1090/S0002-9904-1954-09848-8. [2] R. E. Bellman, "Eye of a Hurricane, an Autobiography," World Scientific Publishing Co Pte Ltd., Singapore, 1984. [3] H. Boerner, Carathéodorys Eingang zur Variationsrechnung, Jahresbericht der Deutschen Mathematiker Vereinigung, 56 (1953), 31-58. [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), 7-10. [5] M. H. Breitner, The genesis of differential games in light of Isaacs' contributions, J. of Optimization Theory and Applications, 124 (2005), 523-559. doi: 10.1007/s10957-004-1173-0. [6] C. Carathéodory, Die Methode der geodätischen Äquidistanten und das Problem von Lagrange, Acta Mathematica, 47 (1926), 199-236. [7] C. Carathéodory, "Variationsrechnung und Partielle Differentialgleichungen Erster Ordnung," Teubner, Leipzig, Germany, 1935. [8] C. Carathéodory, The beginning of research in the calculus of variations, Osiris, 3 (1937), 224-240; also in "Gesammelte Mathematische Schriften 1 (Variationsrechnung)" (edited by the Bayerische Akademie der Wissenschaften), C. H. Beck'sche Verlagsbuchhandlung, München, Germany, (1954), 212-248. [9] C. Carathéodory, "Calculus of Variations and Partial Differential Equations of the First Order, Part 1, Part 2," Holden-Day, San Francisco, California, 1965-1967; Reprint: 2nd AMS printing, AMS Chelsea Publishing, Providence, RI, USA, 2001. [10] C. Carathéodory, "Variationsrechnung und partielle Differentialgleichungen erster Ordnung," With Contributions of H. Boerner and E. Hölder (edited, commented and extended by R. Klötzler), [11] D. Carlson, An observation on two methods of obtaining solutions to Variational problems, Journal of Optimization Theory and Applications, 114 (2002), 345-361. doi: 10.1023/A:1016035718160. [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), 41-50. doi: 10.1007/s10898-007-9171-z. [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: "Variational Analysis and Aerospace Engineering" (eds. G. Buttazzo and A. Frediani), Springer Optimization and Its Applications, 33, Springer, New York, New York, (2009), 75-90. [14] D. Carlson and G. Leitmann, An equivalent problem approach to absolute extrema for calculus of variations problems with differential constraints, Dynamics of Continuous, Discrete and Impulsive Systems, Series B: Applications & Algorithms, 18 (2011), 1-15. [15] M. R. Hestenes, "A General Problem in the Calculus of Variations with Applications to the Paths of Least Time," Research Memorandum No. 100, ASTIA Document No. AD 112382, RAND Corporation, Santa Monica, CA, 1950. [16] R. P. Isaacs, "Games of Pursuit," Paper No. P-257, RAND Corporation, Santa Monica, CA, 1951. [17] R. P. Isaacs, Some fundamentals of differential games, in "Topics in Differential Games" (ed. A. Blaquiére), North-Holland Publishing Company, Amsterdam, The Netherlands, (1973), 25-31. [18] G. Leitmann, A note on absolute extrema of certain integrals, International Journal of Nonlinear Mechanics, 2 (1967), 55-59. doi: 10.1016/0020-7462(67)90018-2. [19] G. Leitmann, On a class of direct optimization problems, Journal of Optimization Theory and Appplications, 108 (2001), 467-481. doi: 10.1023/A:1017507006157. [20] S. MacLane, The Applied Mathematics Group at Columbia in World War II, in "A Century of American Mathematics, Part III" (eds. P. L. Duren, R. Askey and U. C. Merzbach), Providence, RI, (1988), 495-515. [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), 203-229. doi: 10.1007/BF02192933. [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), 973-995. [23] M. Plail, "Die Entwicklung der optimalen Steuerungen," Vandenhoeck & Ruprecht, Göt-tingen, Germany, 1998. [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), 32-44. doi: 10.1109/37.588098. [25] F. O. O. Wagener, On the Leitmann equivalent problem approach, Journal of Optimization Theory and Applications, 142 (2009), 229-242. doi: 10.1007/s10957-009-9513-8.
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