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Tytuł artykułu

On the development of Pontryagin's Maximum Principle in the works of A.Ya. Dubovitskii and A.A. Milyutin

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EN
Abstrakty
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We give a short review of the development and generalizations of the Pontryagin Maximum Principle, provided in the studies of Dubovitskii and Milyutin in the 1960s and later years.
Rocznik
Strony
923--957
Opis fizyczny
Bibliogr. 75 poz.
Twórcy
Bibliografia
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  • ARUTYUNOV, A.V. and ASEEV, S.M. (1997) Investigation of the degeneracy phenomenon of the maximum principle for optimal control problems with state constraints. SIAM J. Control Optimization, 35 (3), 930-952.
  • ARUTYUNOV, A.V., MAGARIL-ILYAEV, G.G. and TIKHOMIROV, V.M. (2006) Pontryagin Maximum Principle: Proofs and Applications. Factorial Press, Moscow.
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  • CHUKANOV, S.V. (1990) Maximum principle for optimal control problems governed by integral equations. Necessary condition in optimal control. Nauka, Moscow (in Russian), Ch. 6.
  • CHUKANOV, S.V. (1993a) A problem with a strictly positive functional and a control of full dimension. Optimal control in linear systems. Nauka, Moscow (in Russian), Ch. 4.
  • CHUKANOV, S.V. (1993b) Construction of integral funnels at interior points of the singular manifold, ibid., Ch. 7.
  • CHUKANOV, S.V. and MILYUTIN, A.A. (1994) Qualitative study of singularities for extremals of quadratic optimal control problem. Russian Journal of Math. Physics, 2 (1), 31-48.
  • DIKUSAR, V.V. (1990) Numerical determination of optimal trajectories in optimal control problems with mixed constraints. Optimization of the flight distance for an apparatus in the atmosphere. Necessary condition in optimal control. Nauka, Moscow (in Russian), Ch. 8-10.
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  • DIKUSAR, V.V. and SHILOV, A.A. (1970) Nonregular optimal trajectories of the apparatus flying in the atmosphere. Uchenye zapiski TsAGI (Scientific notes of Central Aero-Hydrodynam. Inst.), 4, 73-83.
  • DMITRUK, A.V. (1987) Quadratic conditions for a Pontryagin minimum in an optimal control problem, linear in the control. Mathematics of the USSR, Izvestiya, 28 (2), 275-303.
  • DMITRUK, A.V. (1990) Maximum principle for a general optimal control problem with state and regular mixed constraints. In: Optimal’nost’ upravlyaemyh dinamicheskih sistem. Nauka, Moscow, 14, 26-42; English translation in Computat. Math, and Modeling, 1993, 4 (4), 364-377.
  • DMITRUK, A.V. (1999) Quadratic order conditions of a local minimum for singular extremals in a general optimal control problem. In: G. Ferreyra et al., eds., Proc. of Symposia in Pure Mathematics, 64, “Diff. Geometry and Control”. American Math. Society, 163-198.
  • DMITRUK, A.V. (2007) Approximation theorem for a nonlinear control system with sliding modes. Proc. of Steklov Math. Institute, 256, 102-114.
  • DUBOVITSKII, A.YA. (1975) Integral Maximum principle in the general optimal control problem. Doctoral Dissertation, Computing Center of the USSR Academy of Sciences, Moscow (in Russian).
  • DUBOVICKII, A.J. (DUBOVITSKII, A.YA.) (1976) Solution of a problem of S. Ulam on optimal matching of segments. Mathematics of USSR-Izvestiya, 10 (3), 639-651.
  • DUBOVITSKII, A.YA. (1978) The separation, and translations of the Euler equation in linear topological spaces. Mathematics of the USSR-Izvestiya, 12 (1), 194-204.
  • DUBOVITSKIJ, A.YA. and DUBOVITSKIJ, V.A. (1987) The maximum principle in regular optimal control problems with phase trajectory endpoints lying at the boundary of the phase constraint. Autom. Remote Control, 48 (12), 1578-1585.
  • DUBOVITSKIJ, A.YA. and DUBOVITSKIJ, V.A. (1988) On pointwise nontriviality of the maximum principle in problems with phase constraints. Zeitschrift für Analysis und ihre Anwendungen, 7 (5), 387-404.
  • DUBOVITSKII, A.YA. and DUBOVITSKIJ, V.A. (1995) Existence criterion for a significant maximum principle for a problem with phase restrictions. Differ. Equations, 31 (10), 1595-1602.
  • DUBOVITSKII, A.YA. and MILYUTIN, A.A. (1963) Extremum problems in the presence of restrictions. Soviet Math. Doklady, 4, 425-455.
  • DUBOVITSKII, A.YA. and MILYUTIN, A.A. (1965) Extremum problems in the presence of restrictions. USSR Comput. Math, and Math. Phys., 5 (3), 1-80.
  • DUBOVITSKII, A.YA. and MILYUTIN, A.A. (1968) Necessary conditions for a weak extremum in optimal control problems with mixed constraints of the inequality type, ibid., 8 (4), 24-98.
  • DUBOVITSKII, A.YA. and MILYUTIN, A.A. (1969) Translations of Euler’s equation, ibid., 9 (6), 37-64.
  • DUBOVITSKII, A.YA. and MILYUTIN, A.A. (1971) Necessary conditions of a weak minimum in the general optimal control problem. Nauka, Moscow (in Russian).
  • DUBOVITSKII, A.YA. and MILYUTIN, A.A. (1981) Theory of the Maximum principle. In: V.L. Levin, ed., Methods of the theory of extremal problems in economics. Nauka, Moscow, 6-47 (in Russian).
  • DUBOVITSKII, A.YA and MILYUTIN, A.A. (1985) Maximum Principle in linear control problems with convex mixed restrictions. Zeitschrift für Analysis und ihre Anwendungen, 4 (2), 133-191.
  • DUBOVITSKIJ, V.A. (1982) Necessary and sufficient conditions for a Pontryagin minimum in problems of optimal control with singular conditions and generalized controls. Russ. Math. Surveys, 37 (3), 202-203.
  • DUBOVITSKIJ, V.A. (1985) The Ulam problem of optimal motion of line segments. Translation Series in Mathematics and Engineering. Optimization Software, Inc., Springer-Verlag, New York.
  • DYKHTA, V.A. (1994) Variational maximum principle and quadratic optimality conditions for impulse and singular processes. Siberian Math. Journal, 35 (1), 65-76.
  • DYUKALOV, A.N. (1977) An optimally criterion in linear dynamic optimal control problems with mixed constraints. USSR Comput. Math. Math. Phys. 16 (4), 31-47.
  • FULLER, A.T. (1961) Relay control systems optimized for various performance criteria. Automatic and remote control (Proc. First IFAC Congress, Moscow, 1960), Butterworth, London (1961), 510-519.
  • GAMKRELIDZE, R.V. (1962) Optimal sliding states. Soviet Math. Doklady, 3 (2), 559-562.
  • GAMKRELIDZE, R.V. (1977) Foundations of Optimal Control. Metsniereba Tbilisi (in Russian).
  • GAMKRELIDZE, R.V. (1999) Discovery of the maximum principle. J. of Dynamical and Control Systems, 5 (4), 437-551.
  • GAPOSHKIN, V.F. (1972) Convergence and limit theorems for sequences of random variables. Theory of Probability and Appi, 17-3, 379-400.
  • GIRSANOV, I.V. (1970) Lectures on the theory of extremal problems. Moscow State University Press, Moscow (in Russian).
  • ILYUTOVICH, A.E. (1993) Numerical methods for optimal control problems, based on the Maximum principle. Optimal control in linear systems. Nauka, Moscow (in Russian), Ch. 8.
  • IOFFE, A.D. and TIKHOMlROV, V.M. (1974) Theory of extremal problems. Nauka, Moscow.
  • KADEC, M. and PELCZYNSKI, A. (1961/62) Bases, lacunary sequences and complemented subspaces in the spaces Lp. Studia Mathematica 21, 161—176.
  • KLUMOV, A.S. and MERKULOV, A.E. (1984) Speed-optimal control of an astatic second-order loop with nonregular mixed constraints on the control. Autom. Remote Control, 45 (12), 1544-1552.
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  • MATVEEV, A.S. (1987) Necessary conditions for an extremum in an optimal-control problem with phase restrictions. Diff. Equations, 23, 427-436.
  • MAURER, H. (1979) On the minimum principle for optimal control problems with state constraints. Preprint no. 41, University of Münster.
  • MILYUTIN, A.A. (1966) Extremum problems in the presence of constraints. Doctoral Dissertation, Institute of Applied Mathematics, Moscow.
  • MILYUTIN, A.A. (1970) General schemes of necessary conditions for extrema and problems of optimal control. Russ. Math. Surveys, 25 (5), 109-115.
  • MILYUTIN, A.A. (1990a) Extremals and their properties. Necessary condition in optimal control Nauka, Moscow (in Russian), Ch. 2.
  • MILYUTIN, A. A. (1990b) Properties of the measure - the Lagrange multiplier at the state constraint, ibid., Ch. 3.
  • MILYUTIN, A. A. (1990c) A theory of invariance of extremals, ibid., Ch. 4.
  • MILYUTIN, A.A. (1990d) Maximum principle for the regular systems, ibid., Ch. 5.
  • MILYUTIN, A.A. (1990e) On the equation ψ(4) = signi ψ. In: Optimization of controllable dynamical systems. VNIISI, Moscow, 1, 76-84 (in Russian).
  • MILYUTIN, A.A. (1993) Quadratic optimization. Optimal control in linear systems. Nauka, Moscow (in Russian), Chapters 3 and 5.
  • MILYUTIN, A. A. (1994) An example of an optimal control problem whose extremals possess a continual set of discontinuities of the control function. Russian Journal of Math. Physics, 1 (3), 397-402.
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  • MILYUTIN, A. A. (2000) On a certain family of optimal control problems with a phase constraint. J. of Math. Sciences, 100 (5), 2564-2571.
  • MILYUTIN, A.A. (2001) Maximum principle in the general optimal control problem. Fizmatlit, Moscow (in Russian).
  • MILYUTIN, A.A. and CHUKANOV, S.V. (1993) The problem ∫ x2 dt → min, x = u, u ‹ U and similar problems. Optimal control in linear systems, Maximum Faculty of Moscow State Nauka, Moscow (in Russian), Ch. 6.
  • MILYUTIN, A.A., DMITRUK, A.V., OSMOLOVSKII, N.P. (2004) Principle in Optimal Control. Mech-Math. University, Moscow (in Russian).
  • MILYUTIN, A.A. and OSMOLOVSKII, N.P. (1998) Calculus of Variations and Optimal Control. American Math. Society, Providence, RI.
  • OSMOLOVSKII, N.P. (1988) Necessary and sufficient conditions of a high order for Pontryagin and bounded-strong minima in an optimal control problem. Soviet Phys. Doklady, 33 (12), 883-885.
  • OSMOLOVSKII, N.P. (1993) Quadratic order conditions of extremum in a canonical optimal control problem. Optimal control in linear systems. Nauka, Moscow (in Russian), Ch. 1.
  • OSMOLOVSKII, N.P. (1995) Quadratic conditions for nonsingular extremals in optimal control (a theoretical treatment). Russian Journal of Math. Physics, 2 (4), 487-516.
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  • ROBBINS, H.M. (1980) Junction phenomena for optimal control problems with state variable inequality constraints of third order. JOTA, 31 (1), 85-99.
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  • SUSSMANN, H.J. AND WILLEMS, J.C. (1997) 300 years of optimal control: From the brachystochrone to the Maximum principle. IEEE Control Systems Magazine, 17, 32-44.
  • TER-KRIKOROV, A.M. (1976) Some linear problems of optimal control theory with phase constraints. USSR Comput. Math. Math. Phys. 15 (1), 51-63.
  • YAKUBOVICH, V.A. and MATVEEV, A.S. (1994) Abstract theory of optimal control. St.-Petersburg University Press, St.-Petersburg (in Russian).
  • ZELIKIN, M.I. and BORISOV, V.F. (1994) Theory of Chattering Controls. Birkhäuser, Boston-Basel-Berlin.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-article-BAT5-0046-0001
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