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Necessary optimality conditions for infinite horizon variational problems on time scales
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 |
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. |
[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.
|
[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.
|
[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. |
[6] |
C. Carathéodory, Die Methode der geodätischen Äquidistanten und das Problem von Lagrange,, Acta Mathematica, 47 (1926), 199.
|
[7] |
C. Carathéodory, "Variationsrechnung und Partielle Differentialgleichungen Erster Ordnung,", Teubner, (1935).
|
[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, ().
|
[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. |
[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. |
[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.
|
[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.
|
[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. |
[19] |
G. Leitmann, On a class of direct optimization problems,, Journal of Optimization Theory and Appplications, 108 (2001), 467.
doi: 10.1023/A:1017507006157. |
[20] |
S. MacLane, The Applied Mathematics Group at Columbia in World War II,, in, (1988), 495.
|
[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. |
[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.
|
[23] |
M. Plail, "Die Entwicklung der optimalen Steuerungen,", Vandenhoeck & Ruprecht, (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.
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.
doi: 10.1007/s10957-009-9513-8. |
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. |
[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.
|
[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.
|
[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. |
[6] |
C. Carathéodory, Die Methode der geodätischen Äquidistanten und das Problem von Lagrange,, Acta Mathematica, 47 (1926), 199.
|
[7] |
C. Carathéodory, "Variationsrechnung und Partielle Differentialgleichungen Erster Ordnung,", Teubner, (1935).
|
[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, ().
|
[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. |
[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. |
[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.
|
[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.
|
[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. |
[19] |
G. Leitmann, On a class of direct optimization problems,, Journal of Optimization Theory and Appplications, 108 (2001), 467.
doi: 10.1023/A:1017507006157. |
[20] |
S. MacLane, The Applied Mathematics Group at Columbia in World War II,, in, (1988), 495.
|
[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. |
[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.
|
[23] |
M. Plail, "Die Entwicklung der optimalen Steuerungen,", Vandenhoeck & Ruprecht, (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.
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.
doi: 10.1007/s10957-009-9513-8. |
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