American Institute of Mathematical Sciences

Advanced
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

Hans Josef Pesch 1,

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.
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), 503-516. 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., Singapore, 1984. Google Scholar

[3]

H. Boerner, Carathéodorys Eingang zur Variationsrechnung, Jahresbericht der Deutschen Mathematiker Vereinigung, 56 (1953), 31-58.  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), 7-10.  Google Scholar

[5]

M. H. Breitner, The genesis of differential games in light of Isaacs' contributions, J. of Optimization Theory and Applications, 124 (2005), 523-559. 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), 199-236.  Google Scholar

[7]

C. Carathéodory, "Variationsrechnung und Partielle Differentialgleichungen Erster Ordnung," Teubner, Leipzig, Germany, 1935.  Google Scholar

[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. Google Scholar

[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. 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), 345-361. 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), 41-50. 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: "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.  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, Discrete and Impulsive Systems, Series B: Applications & Algorithms, 18 (2011), 1-15.  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, ASTIA Document No. AD 112382, RAND Corporation, Santa Monica, CA, 1950. Google Scholar

[16]

R. P. Isaacs, "Games of Pursuit," Paper No. P-257, RAND Corporation, Santa Monica, CA, 1951. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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), 203-229. 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), 973-995.  Google Scholar

[23]

M. Plail, "Die Entwicklung der optimalen Steuerungen," Vandenhoeck & Ruprecht, Göt-tingen, Germany, 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), 32-44. 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), 229-242. 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), 503-516. 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., Singapore, 1984. Google Scholar

[3]

H. Boerner, Carathéodorys Eingang zur Variationsrechnung, Jahresbericht der Deutschen Mathematiker Vereinigung, 56 (1953), 31-58.  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), 7-10.  Google Scholar

[5]

M. H. Breitner, The genesis of differential games in light of Isaacs' contributions, J. of Optimization Theory and Applications, 124 (2005), 523-559. 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), 199-236.  Google Scholar

[7]

C. Carathéodory, "Variationsrechnung und Partielle Differentialgleichungen Erster Ordnung," Teubner, Leipzig, Germany, 1935.  Google Scholar

[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. Google Scholar

[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. 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), 345-361. 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), 41-50. 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: "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.  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, Discrete and Impulsive Systems, Series B: Applications & Algorithms, 18 (2011), 1-15.  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, ASTIA Document No. AD 112382, RAND Corporation, Santa Monica, CA, 1950. Google Scholar

[16]

R. P. Isaacs, "Games of Pursuit," Paper No. P-257, RAND Corporation, Santa Monica, CA, 1951. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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), 203-229. 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), 973-995.  Google Scholar

[23]

M. Plail, "Die Entwicklung der optimalen Steuerungen," Vandenhoeck & Ruprecht, Göt-tingen, Germany, 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), 32-44. 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), 229-242. 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, 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, 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, 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, 2011, 29 (2) : 417-437. doi: 10.3934/dcds.2011.29.417
[8]

Jacky Cresson, Fernando Jiménez, Sina Ober-Blöbaum. Continuous and discrete Noether's fractional conserved quantities for restricted calculus of variations. Journal of Geometric Mechanics, 2021  doi: 10.3934/jgm.2021012
[9]

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
[10]

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
[11]

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
[12]

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, 2017, 37 (12) : 6165-6181. doi: 10.3934/dcds.2017266
[13]

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
[14]

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
[15]

Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020110
[16]

Zhen Wu, Feng Zhang. Maximum principle for discrete-time stochastic optimal control problem and stochastic game. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021031
[17]

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
[18]

Zhen-Zhen Tao, Bing Sun. A feedback design for numerical solution to optimal control problems based on Hamilton-Jacobi-Bellman equation. Electronic Research Archive, , () : -. doi: 10.3934/era.2021046
[19]

Frank Sottile. The special Schubert calculus is real. Electronic Research Announcements, 1999, 5: 35-39.

[20]

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

 Impact Factor: 

Tools

Metrics

  • PDF downloads (56)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences