Abstract
The paper contains a survey of theoretical and practical results connected with sufficient conditions for global optimality of controlled dynamic processes. Both discrete and continuous time systems and systems with distributed parameters are discussed.
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References
Krotov V.F. Methods of solution of variational problems on the basis of sufficient conditions for absolute minimum. I. Avtomat. i Telemeh.†, 23 (1962) 12, 1571-1583.
Krotov V.F. Methods of solving variational problems. II. Sliding regimes. Avtomat. i Telemeh.†, 24 (1963) 5, 581-598.
Krotov V.F. Methods for variational problem solution based on absolute minimum sufficient conditions. III. Avtomat. i Telemeh.†, 25 (1964) 7, 1037-1046.
Krotov V.F. Approximate synthesis of optimal control. Avtomat. i Telemeh.†, 25 (1964) 11, 1521-1527.
Krotov V.F. Sufficient optimality conditions for discrete control systems. Dokl. AN SSSR, 172 (1967) 1, 18-21.
Krotov V.F., Bukreev V.Z., Gurman V.I. New Variational Methods in Flight Dynamics. Mashinostroyenie, Moskva, 1969. (English transl.: NASA, Transl. TTF-657, 1971).
Krotov V.F., Gurman V.I. Methods and Problems of Optimal Control. Nauka, Moskva, 1973 (in Russian).
Bellman R. Dynamic Programming. Princeton University Press, New Jersey, 1957.
Ioffe A.D., Tihomirov V.M. Theory of Optimal Problems. Nauka, Moskva, 1974 (English transl.: North Holland, Amsterdam, 1973).
Pontryagin L.S., Boltianski V.G., Gamkrelidze P.V., Mishchenko E.F. Mathematical Theory of Optimal Processes. Fizmagtiz, Moskva, 1961 (English transi.: Interscience Publishers Inc., New York, 1962).
Mereau P.M., Powers W.F. A direct sufficient condition for free final time optimal control problems. SIAM J. Control Optim., 14 (1976) 4, 613-622.
Rozenberg G.S. On the necessity and the sufficient conditions for minimum in variational problems. Differencialnye Uravnenija, (1968) 2 (transl, to English as Differential Equations).
Zeidan V. 1st and 2nd order sufficient conditions for optimal control and the calculus of variations. Appl. Math. Optim., 11 (1984) 3, 209-226.
Girsanov I.V. On a relation between Krotov and Bellman functions in the dynamic programming method. Vestnik Moskov. Univ., Ser. I, Mat. Meh., (1968), 56–59 (in Russian).
Gurman V.I. Singular Problems in Optimal Control. Nauka, Moskva, 1977 (in Russian).
Gurman V.I. Principle of Extensions in Control Problems. Nauka, Moskva, 1985 (in Russian).
Krotov V.F. A fundamental problem of variational calculus for a simple functional on the set of discontinuous functions. Dokl. AN SSSR, 137 (1961) 1, 31-34 (in Russian).
Gurman V.I. A method for investigating one class of optimal sliding regimes. Avtomat i Telemeh.†, 26 (1965) 7, 1169-1176.
Gurman V.I. On optimal trajectories between nonoriented orbits in the central field. Kosmicheskiye Issledovania, (1966) 3 (in Russian).
Ioslovich I.V. The optimal stabilization of a satellite in an inertial system of coordinates. Astronaut. Acta, 13 (1962), 37–47 (in Russian).
Borshchevski M.Z., Ioslovich I.V. Some problems in optimal stabilization of cosymmetric satellite. Kosmicheskiye Issledovania, (1966) 3 (in Russian).
Khrustalev M.M. On sufficient conditions of global maximum. Dokl. AN SSSR, Ser. Mat., 174 (1967) 5 (in Russian).
Ioffe A.D. Convex functions occuring in variational problems and the absolute minimum problem. Mat. Sb., 88 (1972), 194–210.
Vinter R.B. Weakest conditions for existence of Lipshitz continuous Krotov functions in optimal control theory. SIAM J. Control Optim., 21 (1983) 2, 215-234.
Vinter R.B. New global optimality conditions in optimal control theory. SIAM J. Control Optim., 21 (1983) 2, 235-245.
Vinter R.B., Lewis R.M. A verification theorem which provides a necessary and sufficient condition for optimality. IEEE Trans. Autom. Control, AC-25 (1980) 1, 84-89.
Klötzler R. Starke Dualität in der Steurungstheorie. Math. Nachr., 95 (1980), 253–263.
Pickenhain S. Starke Dualität bei verallgemeinerten Steuerungsproblemen. Zeit. f. Analysis u. ihre Anwendungen, 1 (1982) 4, 15-24.
Khrustalev M.M. Exact Description of Reachable Sets and Global Optimality Conditions for Dynamic Systems. Avtomat. i Telemeh.†, N5, pp.62–70, May, 1988.
Khrustalev M.M. Exact Description of Reachable Sets and Conditions of Global Optimality for Dynamic Systems. Avtomat. i Telemeh.†, N7, pp.874–881, December, 1988.
Khrustalev M.M. On sufficient optimality conditions in the problem with phase coordinates constraints. Avtomat. i Telemeh.†, (1967) 4, 18-29 (in Russian).
Dubovitzky A.Ya., Milyutin A.A. Extremum problems under constraints. Zh. Vychisl. Mat. i Mat. Fiz., 5 (1965) 3, 395-453 (in Russian).
Klötzler R. Dualität in der Steuerungstheorie. Sonderdruck Heft, (1981) 2/4.
Klötzler R. Globale Optimierung in der Steurungstheorie. Hauptvortrag zur GAMM-Tagung, 1982.
Klötzler R. Models and applications of duality in optimal control. IFIP Conf., Hanoi, Viet-Nam, 10–14 January 1983.
Klötzler R. Geometrical applications of duality in optimal control. Int. Conf. Math. Methods in OR, Sofia, 1983, 52–69.
Andreeva I.A., Klötzler R. Zur analytische Lösung geometrischer Optimierugsaufgaben mittels Dualität bei Steuerungsproblem. Teil I. Z. Angew. Math. Mech., 64 (1984), 35–44.
Konstantinov G.N. Normalization of Controls in Dynamic Systems. Izd. Irkutskogo Univ., Irkutsk, 1983 (in Russian).
Moskalenko A.I. Methods of Linear Transformations in Optimal Control. Nauka, Moskva, 1983 (in Russian).
Subbotin A.I., Subbotina N.N. On proving the dynamical programming method in an optimal control problem. Izv. AN SSSR, Tehnich. Kibern., (1983) 2, 24 (in Russian).
Kelley H.J. Gradient theory of optimal flight path. ARS J., 30 (1960) 10.
Eneev T.M. On application of gradient method in optimal control theory problems. Kosmicheskiye Issledovania, 4 (1966) 5 (in Russian).
Krylov I.A., Chernousko F.L. On a method of consecutive approximations for solving optimal control problems. Zh. Vychisl. Mat. i Mat. Fiz., 2 (1962) 6, 1132-1139 (in Russian).
Bryson A.E., Ho Y.Ch. Applied Optimal Control. Hemisphere Publ. Corp., Washington, 1975.
Krotov V.F., Feldman I.N. An iterative method for solving problems of optimal control. Izv. AN SSSR, Tehnich. Kibern., (1983) 2, 162–167 (in Russian).
Propoi A.I. Elements of the Theory of Optimal Discrete Systems. Nauka, Moskva, 1981 (in Russian).
Kazakov V.A., Krotov B.F. Optimal control of resonance interaction of the light and a substance. Avtomat i Telemeh.†, (1987) 4, 9–15.
Kazakov V.A., Somoloi J., Tannor D.J. Optimal Control of Quantum Systems. I. A Shortsighted Optimization Procedure. J. Chem. Phys., 1993 (in appear).
Emelichev V.A., Komlik V.I. A Method for Constructing a Sequence of Plans for Solving Discrete Optimal Problems. Nauka, Moskva, 1981 (in Russian).
Wagner G. Principles of Operations Research with Application to Managerial Decisions. Prentice Hall, Englewood Cliffs, 1969.
Gurman V.I., Baturin V.A., Danilina E.V. New Methods of Improving Controlled Process. Nauka, Novosibirsk, 1987 (in Russian).
Krotov V.F. Computing algorithms for solution and optimization of a controlled system of equations. I. Izv. AN SSSR, Tehnich. Kibern., (1975) 5, 3–15 (in Russian).
Krotov V.F. Computing algorithms for solution and optimization of a controlled system of equations. II. Izv. AN SSSR, Tehnich. Kibern., (1975) 6, 3–13 (in Russian).
Krotov V.F., Sergeev S.I. Computing algorithms for solution of some problems in linear and linear-integer programming. Avtomat i Telemeh.†, Part I (1980) 12, 86–96, Part II (1981) 1, 86-96, Part III (1981) 3, 83-94, Part IV (1981) 4, 103-112.
Egorov V.A., Gusev L.I. The Dynamics of the Flight between the Earth and the Moon. Nauka, Moskva, 1980 (in Russian).
Gukasian M.H. On computing algorithms for optimization of linear controlled systems with distributed parameters. In: Investigations in Mechanics of a Solid Deforming Body. Erewan, 1981, 93–97.
Bukreev V.Z. Concerning certain method of approximate synthesis of optimal control. Avtomat i Telemeh.†, (1968) 11, 5–13.
Bukreev V.Z., Rozenblat G.M. Approximate design of optimal control for discrete-time systems. Avtomat i Telemeh.†, (1976) 1, 90–93.
Bakhito R.U., Krapchetkov N.P., Krotov V.F. The synthesis of an approximately optimal control for one class of controlled systems. Avtomat i Telemeh.†, (1972) 10, 33–43.
Trigub M.B. Approximately optimal stabilization of one class of nonlinear systems. Avtomat i Telemeh.†, (1987) 1, 34–47.
Lekkerke P.I., Derleman T.W. Calculation method for dynamic optimization using quadratic approximations to minimum surface. Automatica, 7 (1971) 6, 713.
Krotov V.F. Optimization methods of control processes with minimax criteria.I, Automat, i Telemeh.†, No. 12 (1973).
Krotov V.F. Optimization methods of control processes with minimax criteria.II, Avtomat. i Telemeh.†, No.1 (1974).
Krotov V.F. Optimization methods of control processes with minimax criteria.III. Avtomat. i Telemeh.†, No.2 (1974).
Khrustalev M.M. Necessary and sufficient conditions of invariance. Avtomat. i Telemeh.†, No.4 (1968).
Andreeva A.A., Kolmanovskii V.B., Shaychet L.E. Control of the systems with aftereffect. Moscow, Nauka, 1992, pp.64–98 (in Russian).
Kolmanovskii V.B., Myshkis A.D. Functional — differential equations. Kluwer Ac. Publ. N. Y. 1992, pp.162–173.
Khrustalev M.M. Global Optimality Conditons for Controlled Noninertial Systems. Control and Cybernetics, 17, N02-3, 1988.
Krotov V.F. A Technique of Global Bounds in Optimal Control Theory. Control and Cybernetics, 17, N02-3, 1988.
Stoddard A.W.J. Estimation of Optimality for Multidimensional Control Systems. J. of Opt. Theory and Appl., 1969, v.3, N03, pp.385–391.
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Krotov, V.F. (1993). Global Methods in Optimal Control Theory. In: Kurzhanski, A.B. (eds) Advances in Nonlinear Dynamics and Control: A Report from Russia . Progress in Systems and Control Theory, vol 17. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0349-0_3
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DOI: https://doi.org/10.1007/978-1-4612-0349-0_3
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