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

Previous Article
Necessary optimality conditions for infinite horizon variational problems on time scales
NACO Home
This Issue
Next Article
Incremental quadratic stability

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

Abstract
Related Papers

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.

Keywords: royal road of calculus of variations, History of calculus of variations and optimal control, Carathéodory., maximum principle, Bellman equation.

Mathematics Subject Classification: Primary: 01A60, 49-03; Secondary: 90-03, 91-0.

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

show all references

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

[1]	Bernard Dacorogna, Giovanni Pisante, Ana Margarida Ribeiro. On non quasiconvex problems of the calculus of variations. Discrete & Continuous Dynamical Systems - A, 2005, 13 (4) : 961-983. doi: 10.3934/dcds.2005.13.961
[2]	Daniel Faraco, Jan Kristensen. Compactness versus regularity in the calculus of variations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (2) : 473-485. doi: 10.3934/dcdsb.2012.17.473
[3]	Felix Sadyrbaev. Nonlinear boundary value problems of the calculus of variations. Conference Publications, 2003, 2003 (Special) : 760-770. doi: 10.3934/proc.2003.2003.760
[4]	Ivar Ekeland. From Frank Ramsey to René Thom: A classical problem in the calculus of variations leading to an implicit differential equation. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 1101-1119. doi: 10.3934/dcds.2010.28.1101
[5]	Agnieszka B. Malinowska, Delfim F. M. Torres. Euler-Lagrange equations for composition functionals in calculus of variations on time scales. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 577-593. doi: 10.3934/dcds.2011.29.577
[6]	Delfim F. M. Torres. Proper extensions of Noether's symmetry theorem for nonsmooth extremals of the calculus of variations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 491-500. doi: 10.3934/cpaa.2004.3.491
[7]	Nuno R. O. Bastos, Rui A. C. Ferreira, Delfim F. M. Torres. Necessary optimality conditions for fractional difference problems of the calculus of variations. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 417-437. doi: 10.3934/dcds.2011.29.417
[8]	Nikos Katzourakis. Nonuniqueness in vector-valued calculus of variations in $L^\infty$ and some Linear elliptic systems. Communications on Pure & Applied Analysis, 2015, 14 (1) : 313-327. doi: 10.3934/cpaa.2015.14.313
[9]	Gisella Croce, Nikos Katzourakis, Giovanni Pisante. $\mathcal{D}$-solutions to the system of vectorial Calculus of Variations in $L^∞$ via the singular value problem. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6165-6181. doi: 10.3934/dcds.2017266
[10]	Ioan Bucataru, Matias F. Dahl. Semi-basic 1-forms and Helmholtz conditions for the inverse problem of the calculus of variations. Journal of Geometric Mechanics, 2009, 1 (2) : 159-180. doi: 10.3934/jgm.2009.1.159
[11]	Nikos Katzourakis. Corrigendum to the paper: Nonuniqueness in Vector-Valued Calculus of Variations in $ L^\infty $ and some Linear Elliptic Systems. Communications on Pure & Applied Analysis, 2019, 18 (4) : 2197-2198. doi: 10.3934/cpaa.2019098
[12]	Zaidong Zhan, Shuping Chen, Wei Wei. A unified theory of maximum principle for continuous and discrete time optimal control problems. Mathematical Control & Related Fields, 2012, 2 (2) : 195-215. doi: 10.3934/mcrf.2012.2.195
[13]	Md. Haider Ali Biswas, Maria do Rosário de Pinho. A nonsmooth maximum principle for optimal control problems with state and mixed constraints - convex case. Conference Publications, 2011, 2011 (Special) : 174-183. doi: 10.3934/proc.2011.2011.174
[14]	Steven Richardson, Song Wang. The viscosity approximation to the Hamilton-Jacobi-Bellman equation in optimal feedback control: Upper bounds for extended domains. Journal of Industrial & Management Optimization, 2010, 6 (1) : 161-175. doi: 10.3934/jimo.2010.6.161
[15]	Frank Sottile. The special Schubert calculus is real. Electronic Research Announcements, 1999, 5: 35-39.
[16]	Fabrizio Colombo, Graziano Gentili, Irene Sabadini and Daniele C. Struppa. A functional calculus in a noncommutative setting. Electronic Research Announcements, 2007, 14: 60-68. doi: 10.3934/era.2007.14.60
[17]	H. O. Fattorini. The maximum principle for linear infinite dimensional control systems with state constraints. Discrete & Continuous Dynamical Systems - A, 1995, 1 (1) : 77-101. doi: 10.3934/dcds.1995.1.77
[18]	Shaolin Ji, Xiaole Xue. A stochastic maximum principle for linear quadratic problem with nonconvex control domain. Mathematical Control & Related Fields, 2019, 9 (3) : 495-507. doi: 10.3934/mcrf.2019022
[19]	Yan Wang, Yanxiang Zhao, Lei Wang, Aimin Song, Yanping Ma. Stochastic maximum principle for partial information optimal investment and dividend problem of an insurer. Journal of Industrial & Management Optimization, 2018, 14 (2) : 653-671. doi: 10.3934/jimo.2017067
[20]	Shanjian Tang. A second-order maximum principle for singular optimal stochastic controls. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1581-1599. doi: 10.3934/dcdsb.2010.14.1581

Impact Factor:

American Institute of Mathematical Sciences