The Maximum Principle of optimal control: A history of ingenious ideas and missed opportunities

• Autorzy: Pesch H. J. , Plail M.
• Języki publikacji: EN

Abstrakty

EN

On the occasion of (more than) 50 years of optimal control theory, this paper deals with the development of the maximum principle of optimal control during the early times of the Cold War, when mathematicians in the US and the USSR made every endeavor to solve minimum time interception problems, which later on became prototypes of the first optimal control problems. The paper is a short survey on the history of optimal control, a sequence of ingenious ideas and regrets of missed opportunities. The conclusions of the paper are extracted from the second author's monograph on the development of optimal control theory from its commencements until it became an independent discipline in mathematics. The present paper also includes secondary literature published thereafter.

Słowa kluczowe

EN

history of optimal control; maximum principle

• Wydawca: Systems Research Institute, Polish Academy of Sciences
• Czasopismo: Control and Cybernetics
• Rocznik: 2009
• Tom: Vol. 38, no 4A
• Strony: 973--995
• Opis fizyczny: Bibliogr. 67 poz., fot.

Twórcy

autor

Pesch H. J.

autor

Plail M.

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

Bibliografia

• AIZERMAN, M.A. (1958) Lektsii po teorii avtomaticheskogo regulirovaniya. Moscow, Fizmatgiz. Translated into English: Theory of Automatic Control. Oxford, London, Pergamon 1963.
• ANDRONOV, A.A., VITT, A.A. and KHAIKIN, C.E. (1949) Teoriya kolebanii. Gosudarstvennoye Izdatel’stvo Fiziko-Matematicheskoi Literatury, Moscow. Translated into English: Theory of Oscillators. Pergamon Press, Oxford, 1966. Dover Publications, New York, 1987.
• BELLMAN, R.E. (1954a) The Theory of Dynamic Programming. Bulletin of the American Mathematical Society 60, 503-516.
• BELLMAN, R.E. (1954b) Dynamic Programming and a New Formalism in the Calculus of Variations. Proceedings of the National Acadademy of Sciences U.S.A. 40, 231-235.
• BELLMAN, R. (1984) Eye of the Hurricane, an Autobiography. World Scientific Publishing, Singapore.
• BERDYSHEV, V.I., KURZHANSKI, A.B., MISHCHENKO, E.F., OSIPOV, Yu.S. and RUMYANTSEV, V.V. (2004) Nikolai Nikolaevich Krasovskii (On the occasion of his 80th birthday). Proceedings of the Steklov Institute of Mathematics, Suppl. 2, S 1-S 19. Translated from Trudy Instituta Matematiki i Mekhaniki UrO RAN, 10 (2), 2004.
• BERKOVITZ, L. (1961) Variational Methods in Problems of Control and Programming. Journal of Mathematical Analysis and Applications 3, 145-169.
• BLISS, G.A. (1946) Lectures on the Calculus of Variations. The University of Chicago Press, Chicago. Seventh edition, 1963. Translated into Russian, Moscow, 1950.
• BLISS, G.A. and UNDERHILL, A.L. (1914) The Minimum of a Definite Integral for Unilateral Variations in Space. American Mathematical Society Transactions 15, 291-310.
• BOLTYANSKII, V.G. (1958) The Maximum Principle in the Theory of Optimal Processes (in Russian). Doklady Akademii Nauk SSSR 119, 1070-1073.
• BOLTYANSKII, V.G. (1969) The Mathematical Methods of Optimal Control (in Russian). Nauka, Moscow. Translated into German. Mathematische Methoden der optimalen Steuerungen. Geest und Portig, Leipzig 1971. Carl Manser Verlag, München, 1972.
• BOLTYANSKII, V.G. (1994) The Maximum Principle - How it came to be? Schwerpunktprogramm der Deutschen Forschungsgemeinschaft „Anwen-dungsbezogene Optimierung und Steuerung“, Report No. 526, University of Technology Munich, Munich.
• BOLTYANSKII, V.G., GAMKRELIDZE, R.V. and PONTRYAGIN, L.S. (1956) On the Theory of Optimal Processes (in Russian). Doklady Akademii Nauk SSSR 110, 7-10.
• BOLTYANSKII, V.G., GAMKRELIDZE, R.V. and PONTRYAGIN, L.S. (1960) The Theory of Optimal Processes I. The Maximum Principle (in Russian). Izvestija Akademii Nauk SSSR Ser. Mat. 24, 3-42.
• BOLZA, O. (1904/1909) Lectures on the Calculus of Variations. University of Chicago Press, Chicago, 1904. Republished by Dover Publications, New York, 1961. Vorlesungen über Variationsrechnung. Revised and considerably extended German edition, Teubner, Leipzig, Berlin, 1909.
• BREITNER, M.H. (2005) The Genesis of Differential Games in Light of Isaacs’ Contributions. Journal of Optimization Theory and Applications 124, No. 3, 523-559.
• BRYSON JR., A.E. (1996) Optimal Control - 1950 to 1985. IEEE Control Systems Magazine 16, No. 3, 26-33.
• BUSHAW, D.W. (1952) Differential Equations with a Discontinuous Forcing Term. PhD thesis, Princeton University.
• CARATHÉODORY, C. (1926) Die Methode der geodätischen Äquidistanten und das Problem von Lagrange. Acta Mathematica 47, 199-236; see also Gesammelte Mathematische Schriften von Constantin Carathéodory 1 (Variationsrechnung), 212-248, 1954. Edited by the Bayerische Akademie der Wissenschaften. C. H. Beck’sche Verlagsbuchhandlung, München.
• CARATHÉODORY, C. (1935) Variationsrechnung und partielle Differential-gleichungen erster Ordnung. Teubner, Leipzig. Calculus of Variations and Partial Differential Equations of the First Order, Part 1, Part 2. Holden-Day, San Francisco, California, 1965-1967. Reprint of the German edition with contributions by H. Boerner and E. Hölder (Eds.), commented and extended by R. Klötzler. Teubner-Archiv der Mathematik 18, Teubner, Stuttgart, Leipzig, 1994.
• DMITRUK, A.V. (1993) Maximum Principle for the General Optimal Control Problem with Phase and Regular Mixed Constraints. Computational Mathematics and Modeling 4, No. 4, 364-377.
• DUBOVITSKII, A.YA. and MILYUTIN, A.A. (1965) Extremum Problems in the Presence of Restrictions (in Russian). Journal of Computational Mathematics and Mathematical Physics 5, 395-453.
• FEL’DBAUM, A.A. (1949) Jump Controls in Control Systems (in Russian). Avtomatika i Telemekhanika 10, 249-266.
• FEL’DBAUM, A. A. (1955) On the synthesis of Optimal Systems of Automatic Control (in Russian). Transactions of the Second National Conference on the Theory of Automatic Control 2. Izdatel’stvo Akademii Nauk SSSR, Moscow, 325-360.
• GAMKRELIDZE, R.V. (1957) On the Theory of Optimal Processes in Linear Systems (in Russian). Doklady Akademii Nauk SSSR 116, 9-11.
• GAMKRELIDZE, R.V. (1958) On the General Theory of Optimal Processes (in Russian). Doklady Akademii Nauk SSSR 123, 223-226. Completely translated into English in: Automation Express 2, March 1959
• GAMKRELIDZE, R.V. (1959) Processes which have an Optimal Speed of Response for Bounded Phase Coordinates (in Russian). Doklady Akademii Nauk SSSR 125, 475-478. Extracts in: Automation Express 2, 1959.
• GAMKRELIDZE, R.V. (1960) Principles of Optimal Control Theory. Plenum Press.
• GAMKRELIDZE, R.V. (1999) Discovery of the Maximum Principle. Journal of Dynamical and Control Systems 5, No. 4, 437-451.
• GAMKRELIDZE, R.V. (2000) The Mathematical Work of L. S. Pontryagin. Journal of Mathematical Sciences 100 (5), 2447-2457.
• GAMKRELIDZE, R.V. (2009) Hamiltonian form of the Maximum Principle. Control and Cybernetics, this issue.
• GODDARD, R.H. (1919) A Method of Reaching Extreme Altitudes. Smithsonian Miscellaneous Collections 71 (2). Smithsonian Institution, Washington.
• GRAVES, L.M. (1932) The Weierstrass Condition for the Problem of Bolza in the Calculus of Variations. Annals of Mathematics 33, 747-752.
• GRAVES, L.M. (1933) A Transformation of the Problem of Lagrange in the Calculus of Variations. Transactions of the American Mathematical Society 35, 675-682.
• HAMEL, G. (1927) Über eine mit dem Problem der Rakete zusammenhängende Aufgabe der Variationsrechnung. Zeitschrift für angewandte Mathematik und Mechanik 7, 451-452.
• HESTENES, M.R. (1949) Numerical Methods for Obtaining Solutions of Fixed End Point Problems in the Calculus of Variations. Research Memorandum No. 102, RAND Corporation, Santa Monica.
• HESTENES, M.R. (1950) 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.
• ISAACS, R.P. (1951) Games of Pursuit. Paper No. P-257, RAND Corporation, Santa Monica.
• ISAACS, R.P. (1965) Differential Games: A Mathematical Theory with Applications to Warfare and Pursuit, Control, and Optimization. John Wiley and Sons, New York. Revised second edition, John Wiley and Sons, New York, 1967. Translated into Russian. Mir, Moscow, 1967. Revised and extended third edition, Krieger, Huntington, 1975.
• ISAACS, R.P. (1973) Some Fundamentals of Differential Games. In: A. Blaquiere, ed., Topics in Differential Games. North-Holland Publishing Company, Amsterdam, The Netherlands.
• LERNER, A.YA. (1952) Improvement of the Dynamical Properties of Automatic Compensation by Nonlinear Feedback (in Russian). Avtomatika i Telemekhanika 13, 134-144, 429-444.
• KRASOVSKII, N.N. (1957a) On a Problem of Optimal Regulation (in Russian). Prikladnaya Matematika i Mekhanika 21 (5), 670-677.
• KRASOVSKII, N.N. (1957b) On the Theory of Optimal Regulation (in Russian). Avtomatika i Telemekhanika 18 (1), 960-970.
• KROTOV, V.F. and KURZHANSKI, A.B. (2005) National Achievements in Control Theory (The Aerospace Perspective). Annual Reviews in Control 29 (1), 13-31.
• MACLANE, S. (1988) The Applied Mathematics Group at Columbia in World War II. In: P.L. Duren, R. Askey and U.C. Merzbach, eds., A Century of American Mathematics, Part I-III. American Mathematical Society, Providence.
• McSHANE, E.J. (1939) On Multipliers for Lagrange Problems. American Journal of Mathematics 61, 809-819.
• McSHANE, E.J. (1978) The Calculus of Variations from the Beginning to Optimal Control. In: A.B. Schwarzkopf, W.G. Kelley and S.B. Eliason, eds., Optimal Control and Differential Equations. Academic Press, New York, 2-49.
• MIELE, A. (1950) Problemi di Minimo Tempo nel Volo Non-Stazionario degli Aeroplani (Problems of Minimum Time in the Nonsteady Flight of Aircraft). Atti della Accademia delle Scienze di Torino, Classe di Scienze Matematiche, Fisiche e Naturali 85, 41-52.
• MISHCHENKO, E.F. (2003) About Aleksandrov, Pontryagin and Their Scientific Schools. In: A.A. Bolibruch, Yu.S. Osipov, Ya.G. Sinai, V.I. Arnold, L.D. Faddeev, Ya.I. Manin, V.B. Philippov, V.M. Tikhomirov and A.M. Vershik, eds., Mathematical Events of the Twentieth Century. Original Russian edition published by PHASIS, Moscow. English edition published jointly with PHASIS, Moscow, 2006.
• MISHCHENKO, E.F. and PONTRYAGIN, L.S. (1955) Periodic Solutions of Systems of Differential Equations near to Discontinuous Ones (in Russian). Doklady Akademii Nauk SSSR 110, 889-891.
• MORDUKHOVICH, B.S. (2006) Variational Analysis and Generalized Differentiation, I: Basic Theory, II: Applications. Series Fundamental Principles of Mathematical Sciences 330, 331. Springer, Berlin.
• MORSE, M. (1931) Sufficient Conditions in the Problem of Lagrange with Variable End Conditions. American Journal of Mathematics 53, 517-546.
• OKHOTSIMSKY, D.E. (1946) On the Theory of Rocket Propulsion (in Russian). Prikladnaya Matematika i Mekhanika 10 (2), 251-272.
• OKHOTSIMSKY, D.E. and ENEYEV, T.M. (1957) Some Variational Problems Related to the Launch of an Artificial Satellite of the Earth (in Russian). Uspekhi Fizicheskikh Nauk 63 (1).
• PATSKO, V.S. and TUROVA, V.L. (2008) Homicidal Chauffeur Game: History and Modern Studies. Scientific Report. Ural Branch of the Russian Academy of Sciences, Institute of Mathematics and Mechanics, Ekaterinburg. Annals of the International Society of Dynamic Games, to appear.
• PESCH, H.J., BULIRSCH, R. (1994) The Maximum Principle, Bellman’s Equation and Carathéodory’s Work. Journal of Optimization Theory and Applications 80 (2), 203-229.
• PLAIL, M. (1998) Die Entwicklung der optimalen Steuerungen. Vandenhoeck & Ruprecht, Göttingen.
• PONTRYAGIN, L.S. (1959) Optimal Control Processes II (in Russian). Uspekhi Matematicheskikh Nauk 14, 3-20. Translated into English in: Automation Express 2, July 1959. (Part I in Automation Express 1.)
• PONTRYAGIN, L.S. (1960) Optimal Processes of Regulation. Proceedings of the International Mathematical Congress, Edinburgh, August 14-21, 1958. Cambridge University Press.
• PONTRYAGIN, L.S. (1978) A Short Autobiography of L. S. Pontryagin (in Russian). Uspekhi Matematicheskikh Nauk 33, 7-21. Russian Mathematical Surveys 33, 7-24.
• PONTRYAGIN, L.S. (1998) The Life of Lev Semenovich Pontryagin, Mathematician (in Russian). Prima V, Private Publisher.
• PONTRYAGIN, L.S., BOLTYANSKII, V.G., GAMKRELIDZE, R.V. and MISHCHENKO, E.F. (1961) Matematicheskaya teoriya optimal’nykh prozessov. Fizmatgiz, Moscow. Translated into English. The Mathematical Theory of Optimal Processes. John Wiley and Sons (Interscience Publishers), New York, 1962. Translated into German. Mathematische Theorie optimaler Prozesse. Akademie-Verlag, Leipzig, 1964. Second revised German edition, Oldenbourg, München, 1967.
• REID, W.T. (1978) A Historical Note on the Maximum Principle. SIAM Review 20 (3), 580-582.
• ROZONOÉR, L.I. (1959) L. S. Pontryagin’s Maximum Principle in the Theory of Optimal Systems, Part I-III (in Russian). Avtomatika i Telemekhanika 20, 1320-1334, 1441-1458, 1561-1578. Translated into English: Automation and Remote Control 20, 1288-1302, 1405-1421, 1517-1532.
• SUSSMANN, H.J. and WILLEMS, J.C. (1997) 300 Years of Optimal Control: From the Brachistochrone to the Maximum Principle. IEEE Control Systems Magazine 17, 32-44.
• WEIERSTRASS, K.T.W. (1879) Vorlesungen über Variationsrechnung. In: R.E. Rothe, ed., Gesammelte Werke von Karl Weierstrass 7. Akademische Verlagsgesellschaft, Leipzig, 1927. Reprints: Georg Olms, Hildesheim, 1967; Johnson, New York, 1967.
• VALENTINE, F.A. (1937) The Problem of Lagrange with Differential Inequalities as Added Side Conditions. PhD Thesis, University of Chicago.