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Perturbations of extremal problems with constraints and necessary optimality conditions

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Literature cited

  1. E. R. Avakov, “Conditions for an extremum for smooth problems with constraints of equality type,” Zh. Vychisl. Mat. Mat. Fiz.,25, No. 5, 690–693 (1985).

    Google Scholar 

  2. E. R. Avakov, “The maximum principle for abnormal optimal control problems,” Dokl. Akad. Nauk SSSR,298, No. 6, 1289–1292 (1988).

    Google Scholar 

  3. E. R. Avakov, “Necessary minimum conditions for irregular problems in Banach spaces,” Tr. Mat. Inst. Akad. Nauk SSSR,185, 3–29 (1988).

    Google Scholar 

  4. E. R. Avakov, “An estimate of the distance to the level surface of a mapping in the neighborhood of a singular point,” in: Tenth All-Union School: Theoretical and Applied Problems of Numerical Mathematics and Mathematical Physics (Riga, Nov. 19–28, 1985), Latv. Gos. Univ., Riga (1985), pp. 74–76.

    Google Scholar 

  5. A. A. Agrachev, “Quadratic mappings in geometric control theory,” Itogi Nauki Tekh., Ser. Probl. Geom.,20, 111–205 (1988).

    Google Scholar 

  6. A. A. Agrachev, “Homology of the intersections of real quadrics,” Dokl. Akad. Nauk SSSR,229, No. 5, 1033–1036 (1988).

    Google Scholar 

  7. A. A. Agrachev, “A necessary condition for second order optimality in the general nonlinear case,” Mat. Sb.,102, No. 4, 551–568 (1977).

    Google Scholar 

  8. A. A. Agrachev, “The topology of quadratic and smooth mappings,” Itogi Nauki Tekh., Ser. Algebra. Topologiya. Geometriya,26, 85–124 (1988).

    Google Scholar 

  9. A. A. Agrachev, “One more condition for conditional extremum” [in print].

  10. A. A. Agrachev and R. V. Gamkrelidze, “The principle of second order optimality for time-optimal problems,” Mat. Sb.,100, No. 4, 610–643 (1976).

    Google Scholar 

  11. A. A. Agrachev and R. V. Gamkrelidze, “The index of extremality and quasiextremal controls,” Dokl. Akad. Nauk SSSR,284, No. 4, 777–781 (1985).

    Google Scholar 

  12. A. A. Agrachev and R. V. Gamkrelidze, “Calculation of the Euler characteristic of the intersections of real quadrics,” Dokl. Akad. Nauk SSSR,299, No. 1, 11–14 (1988).

    Google Scholar 

  13. V. M. Alekseev, V. M. Tikhomirov, and S. V. Fomin, Optimal Control, Consultants Bureau, New York (1987).

    Google Scholar 

  14. R. Henrion, “La théorie de la variation seconde et ses applications en comande optimale,” Acad. Roy. Belg. Cl. Sci. Mém. Coll. in-8o (2),41, No. 7, (1975).

  15. A. V. Arutyunov, “On necessary conditions for optimality in a problem with phase constraints,” Dokl. Akad. Nauk SSSR,280, No. 5, 1033–1037 (1985).

    Google Scholar 

  16. A. V. Arutyunov, “On the maximum principle theory in optimal control problems with phase constraints,” Dokl. Akad. Nauk SSSR,304, No. 1, 11–14 (1989).

    Google Scholar 

  17. A. V. Arutyunov, “First-order necessary conditions in an optimal control problem with phase constraints,” Tr. Inst. Prikl. Mat., Tbiliss. Gos. Univ.,27, 46–59 (1988).

    Google Scholar 

  18. A. V. Arutyunov, “Tools of perturbation theory in optimization problems,” in: Abstracts of Reports of the All-Union Conference “The Method of Lyapunov Functions in Modern Mathematics” (Kharkov, May 27–29, 1986), Khar'kov Gos. Univ., Kharkov (1986), p. 82.

    Google Scholar 

  19. A. V. Arutyunov, “On the first condition in L. S. Pontryagin's article: ‘On the theory of differential games’,” Differents. Uravn.,19, No. 6, 923–938 (1983).

    Google Scholar 

  20. A. V. Arutyunov, “The maximum principle and second-order necessary conditions in an optimal control problem in the presence of delays,” Soobshch. Akad. Nauk Gruzin. SSR,122, No. 2, 265–268 (1986).

    Google Scholar 

  21. A. V. Arutyunov, G. Z. Ber, and N. T. Tynyanskii, “A numerical method for solving complete-dimensional optimal control problems,” Izv. Akad. Nauk SSSR, Tekh. Kibern., No. 3, 163–169 (1987).

    Google Scholar 

  22. A. V. Arutyunov, S. A. Eroshov, and N. T. Tynyanskii, “On the sign definiteness of a quadratic form on the intersection of quadrics,” in: Mathematical Questions of Nonlinear Analysis and Control of Dynamical Systems [in Russian], Moscow (1985), pp. 104–114.

  23. A. V. Arutyunov and M. Dzh. Mardanov, “On the theory of optimal processes with delays,” Differents. Uravn.,22, No. 8, 1291–1298 (1986).

    Google Scholar 

  24. A. V. Arutyunov and N. T. Tynyanskii, “On second-order conditions in the time-optimal problem,” Dokl. Akad. Nauk SSSR,261, No. 3, 523–526 (1981).

    Google Scholar 

  25. A. V. Arutyunov and N. T. Tynyanskii, “Conditions of the first and second order in a problem of time optimality,” Usp. Mat. Nauk,36, No. 6, 199–200 (1981).

    Google Scholar 

  26. A. V. Arutyunov and N. T. Tynyanskii, “On the system of Jacobi equations in a time-optimality problem,” Izv. Akad. Nauk SSSR, Ser. Mat.,46, No. 5, 1082–1105 (1982).

    Google Scholar 

  27. A. V. Arutyunov and N. T. Tynyanskii, “On necessary conditions for a local minimum in optimal control theory,” Dokl. Akad. Nauk SSSR,275, No. 2, 268–272 (1984).

    Google Scholar 

  28. A. V. Arutyunov and N. T. Tynyanskii, “The fixed sign property of a quadratic form on a cone,” Usp. Mat. Nauk,39, No. 2, 133–134 (1984).

    Google Scholar 

  29. A. V. Arutyunov and N. T. Tynyanskii, “On the maximum principle in a problem with phase constraints,” Izv. Akad. Nauk SSSR, Tekh. Kibern., No. 4, 60–68 (1984).

    Google Scholar 

  30. A. V. Arutyunov and N. T. Tynyanskii, “On second-order conditions in the time-optimality problem,” in: Applied Methods and Programming in Numerical Analysis [in Russian], Moskov. Gos. Univ., Moscow (1985), pp. 128–146.

    Google Scholar 

  31. A. V. Arutyunov and N. T. Tynyanskii, “On first- and second-order conditions in a time-optimal problem,” in: Materials of the Scientific Conference on Numerical Mathematics [in Russian], Scientific-Research Computational Center, Moscow State Univ. (1981), pp. 66–70.

  32. L. A. Beklaryan, “A variational problem with retarded argument and its relation to some semigroup of mappings of a segment into itself,” Dokl. Akad. Nauk SSSR,271, No. 5, 1036–1040 (1983).

    Google Scholar 

  33. G. A. Bliss, Lectures on the Calculus of Variations, Univ. Chicago Press, Chicago (1946).

    Google Scholar 

  34. A. E. Bryson, Jr. and Yu-Chi Ho, Applied Optimal Control, Ginn, Waltham, Mass. (1969).

    Google Scholar 

  35. N. Bourbaki, Topologie Générale, Chap. 1–2, Hermann, Paris (1961).

    Google Scholar 

  36. J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New York (1972).

    Google Scholar 

  37. F. P. Vasil'ev, Numerical Methods for Solving Extremal Problems [in Russian], Nauka, Moscow (1988).

    Google Scholar 

  38. A. B. Vasil'eva and M. G. Dmitriev, “Singular perturbations in optimal control problems,” Itogi Nauki Tekh., Ser. Mat. Anal.,20, 3–77 (1982).

    Google Scholar 

  39. R. V. Gamkrelidze, “Time optimal processes under bounded phase coordinates,” Dokl. Akad. Nauk SSSR,125, No. 3, 475–478 (1959).

    Google Scholar 

  40. R. V. Gamkrelidze, Foundations of Optimal Control [in Russian], Izd. Tbilisskogo Univ., Tbilisi (1977).

    Google Scholar 

  41. D. F. Davidenko, “On a certain method for the numerical solution of a system of nonlinear equations,” Dokl. Akad. Nauk SSSR,88, No. 4, 601–602 (1953).

    Google Scholar 

  42. A. V. Dmitruk, “A Jacobi type condition for the nonnegativity of a quadratic form on a finitely-faced cone,” Izv. Akad. Nauk SSSR, Ser. Mat.,45, No. 3, 608–619 (1981).

    Google Scholar 

  43. A. V. Dmitruk, “On the Euler-Jacobi equation in the calculus of variations,” Mat. Zametki,20, No. 6, 847–858 (1976).

    Google Scholar 

  44. A. Ya. Dubovitskii, Integral maximum principle in the general optimal control problem. Manuscript deposited at VINITI, 1974, No. 2639-74 Dep.

  45. A. Ya. Dubovitskii and V. A. Dubovitskii, The principle of maximum of trajectories, whose boundaries lie on the phase boundary. Preprint, Chernogolovka (1988).

  46. A. Ya. Dubovitskii and V. A. Dubovitskii, “Necessary conditions for a strong minimum in optimal control problems with degeneracy of the endpoint and phase constraints,” Usp. Mat. Nauk,40, No. 2, 175–176 (1985).

    Google Scholar 

  47. A. Ya. Dubovitskii and V. A. Dubovitskii, “The principle of the maximum in regular optimal control problems in which the endpoints of the phase trajectory lie on the boundary of the phase constraint,” Avtom. Telemekh., No. 12, 25–33 (1987).

    Google Scholar 

  48. A. Ya. Dubovitskii and V. A. Dubovitskii, “Conditions for the pointwise nontriviality of the maximum principle in a regular optimal control problem,” Tr. Inst. Prikl. Mat. Tbiliss. Gos. Univ.,27, 60–88 (1988).

    Google Scholar 

  49. A. Ya. Dubovitskii and A. A. Milyutin, “Extremal problems in the presence of constraints,” Zh. Vychisl. Mat. Mat. Fiz.,5, No. 3, 395–453 (1965).

    Google Scholar 

  50. V. A. Dubovitskii, “Necessary and sufficient conditions for the Pontryagin minimum in problems of optimal control with singular conditions and generalized controls,” Usp. Mat. Nauk,37, No. 3, 185–186 (1982).

    Google Scholar 

  51. Yu. N. Zhidkov, Necessary conditions for optimality in two-level hierarchical dynamical systems with vector criteria. Vychisl. Tsentr, Akad. Nauk SSSR, Moscow (1981).

    Google Scholar 

  52. S. A. Zhitnikov, “On the question of the sign definiteness and the sign constancy of quadratic forms in some cone,” in: Differential Equations and Their Applications. Collection of Scientific Works [in Russian], Alma-Ata (1979), pp. 50–55.

  53. M. G. Imanov, “Linear time-optimal problems with phase constraints,” Izv. Akad. Nauk Az. SSR, Ser. Fiz.-Tekh. Mat. Nauk,7, No. 5, 50–55 (1986 (1987)).

    Google Scholar 

  54. K. Yosida, Functional Analysis, Springer, Berlin (1965).

    Google Scholar 

  55. A. D. Ioffe and V. M. Tikhomirov, Theory of Extremal Problems, North-Holland, Amsterdam (1979).

    Google Scholar 

  56. L. V. Kantorovich and G. P. Akilov, Functional Analysis [in Russian], Nauka, Moscow (1977).

    Google Scholar 

  57. V. L. Kreps, “On quadratic forms that are nonnegative on the orthant,” Zh. Vychisl. Mat. Mat. Fiz.,24, No. 4, 497–503 (1984).

    Google Scholar 

  58. K. Kuratowski, Topology, Vol. I, Academic Press, New York (1966).

    Google Scholar 

  59. A. B. Kurzhanskii and Yu. S. Osipov, “On a control problem with bounded phase coordinates,” Prikl. Mat. Mekh.,32, No. 2, 194–202 (1968).

    Google Scholar 

  60. E. S. Levitin, A. A. Milyutin, and N. P. Osmolovskii, “Higher order conditions for a local minimum in problems with constraints,” Usp. Mat. Nauk,33, No. 6, 85–148 (1978).

    Google Scholar 

  61. J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, New York (1971).

    Google Scholar 

  62. L. A. Lyusternik, “On conditional extrema of functionals,” Mat. Sb.,41, No. 3, 390–401 (1934).

    Google Scholar 

  63. M. Dzh. Mardanov, “Necessary conditions for optimality in systems with delays and phase constraints,” Mat. Zametki,42, No. 5, 691–702 (1987).

    Google Scholar 

  64. M. Dzh. Mardanov, “Necessary second-order optimality conditions in problems with delays in the controls,” Usp. Mat. Nauk,43, No. 4, 213–214 (1988).

    Google Scholar 

  65. A. S. Matveev, “On the necessary conditions for an extremum in an optimal control problem with phase constraints,” Differents. Uravn.,23, No. 4, 629–640 (1987).

    Google Scholar 

  66. A. A. Milyutin, “On quadratic conditions for an extremum in smooth problems with a finite-dimensional image,” in: Methods of the Theory of Extremal Problems in Economics [in Russian], Nauka, Moscow (1981), pp. 137–177.

    Google Scholar 

  67. B. Sh. Mordukhovich, “Penalty functions and necessary conditions in nonsmooth and nonconvex optimization problems,” Usp. Mat. Nauk,36, No. 1, 215–216 (1981).

    Google Scholar 

  68. I. P. Natanson, Theory of Functions of a Real Variable [in Russian], Nauka, Moscow (1974).

    Google Scholar 

  69. L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, The Mathematical Theory of Optimal Processes [in Russian], Nauka, Moscow (1969).

    Google Scholar 

  70. W. Rudin, Functional Analysis, McGraw-Hill, New York (1973).

    Google Scholar 

  71. A. V. Sarychev, “The index of the second variation of a control system,” Mat. Sb.,113, No. 3, 464–486 (1980).

    Google Scholar 

  72. M. F. Sukhinin, “The Lagrange multipliers rule in locally convex spaces,” Sib. Mat. Zh.,23, No. 4, 153–165 (1982).

    Google Scholar 

  73. A. V. Tavadze, “Certain questions of the qualitative theory of optimal control of neutral type,” Tr. Inst. Prikl. Mat. Tbil. Univ.,27, 107–137 (1988).

    Google Scholar 

  74. T. A. Tadumadze, Some Problems in the Qualitative Theory of Optimal Control [in Russian], Tbilis. Gos. Univ., Tbilisi (1983).

    Google Scholar 

  75. T. A. Tadumadze and N. M. Avalishvili, Regular Perturbations in Optimal Problems with Variable Structure [in Russian], Izd. Tbil. Univ., Tbilisi (1983).

    Google Scholar 

  76. A. M. Ter-Krikorov, Optimal Control and Mathematical Economics [in Russian], Nauka, Moscow (1977).

    Google Scholar 

  77. V. V. Fedorov, Numerical Maximin Methods [in Russian], Nauka, Moscow (1979).

    Google Scholar 

  78. V. V. Fedorov, “The maximum principle in minimax control problem with phase constraints,” Vestn. Mosk. Univ., Ser. Mat. Mekh., No. 15, No. 4, 36–46 (1977).

    Google Scholar 

  79. A. V. Fiacco and G. P. McCormick, Nonlinear Programming: Sequential Uncosntrained Minimization Techniques, Wiley, New York (1968).

    Google Scholar 

  80. A. F. Filippov, “On certain questions of optimal control,” Vestn. Mosk. Univ., Ser. 1, No. 2, 25–38 (1959).

    Google Scholar 

  81. A. V. Finkel'shtein, “Application of Ekeland's theorem for derivation of necessary and sufficient conditions for optimality of sequences,” Optimizatsiya, No. 31 (48), 33–47 (1983).

    Google Scholar 

  82. G. L. Kharatishvili, “The maximum principle in the theory of optimal processes with delays,” Dokl. Akad. Nauk SSSR,136, No. 1, 39–42 (1961).

    Google Scholar 

  83. G. L. Kharatishvili and T. A. Tadumadze, “Nonlinear optimal control systems with variable delays,” Mat. Sb.,107, No. 4, 613–633 (1978).

    Google Scholar 

  84. G. L. Kharatishvili and T. A. Tadumadze, “On optimal problems with delays and non-fixed initial time,” Soobshch. Akad. Nauk Gruzin. SSR,129, No. 3, 473–476 (1988).

    Google Scholar 

  85. J. K. Hale, Theory of Functional Differential Equations, 2nd edition, Springer, New York (1977).

    Google Scholar 

  86. A. G. Chentsov, Applications of Measure Theory to Control Problems [in Russian], Sredn.-Ural. Izd., Sverdlovsk (1985).

    Google Scholar 

  87. R. E. Edwards, Functional Analysis. Theory and Applications, Holt, Rinehart and Winston, New York (1965).

    Google Scholar 

  88. I. Ekeland and R. Temam, Convex Analysis and Variational Problems, North-Holland, Amsterdam (1976).

    Google Scholar 

  89. L. C. Young, Lectures on the Calculus of Variations and Optimal Control Theory, Saunders, Philadelphia (1969).

    Google Scholar 

  90. J. V. Breakwell and Y.-C. Ho, “On the conjugate point condition for the control problem,” Int. J. Eng. Sci.,2, 565–579 (1965).

    Google Scholar 

  91. R. Courant, “Variational methods for the solution of problems of equilibrium and vibrations,” Bull. Am. Math. Soc.,49, No. 1, 1–23 (1943).

    Google Scholar 

  92. L. L. Dines, “On the mapping of n quadratic forms,” Bull. Am. Math. Soc.,48, 467–471 (1942).

    Google Scholar 

  93. J. W. Gaddum, “Linear inequalities and quadratic forms,” Pacific J. Math.,8, 411–414 (1958).

    Google Scholar 

  94. M. R. Hestenes, “Applications of the theory of quadratic forms in Hilbert space to the calculus of variations,” Pacific J. Math.,1, No. 4, 525–581 (1951).

    Google Scholar 

  95. S.-C. Huang, “Optimal control problems with retardations and restricted phase coordinates,” J. Optim. Theory Appl.,3, 316–360 (1969).

    Google Scholar 

  96. R. E. Kaiman, “Contributions to the theory of optimal control,” Bol. Soc. Mat. Mexicana,5, 102–119 (1960).

    Google Scholar 

  97. A. J. Krener, “The high order maximal principle and its applications to singular extremals,” SIAM J. Control Optim.,15, No. 2, 256–293 (1977).

    Google Scholar 

  98. J.-L. Lions, Control of Distributed Singular Systems, Gauthier-Villars, Paris (1985).

    Google Scholar 

  99. D. H. Martin and D. H. Jacobson, “Copositive matrices and definiteness of quadratic forms subject to homogeneous linear inequality constraints,” Linear Algebra Appl.,35, 227–258 (1981).

    Google Scholar 

  100. D. H. Martin, M. J. D. Powell, and D. H. Jacobson, “On a decomposition of conditionally positive-semidefinite matrices,” Linear Algebra Appl.,39, 51–59 (1981).

    Google Scholar 

  101. E. J. McShane, “On the second variation in certain anormal problems of the calculus of variations,” Am. J. Math.,63, 516–530 (1941).

    Google Scholar 

  102. M. Razzaghi, “Conjugate point conditions for linear quadratic control problems,” Automat. Control Theory Appl.,6, No. 3, 51–53 (1978).

    Google Scholar 

  103. I. B. Russak, “Second-order necessary conditions for general problems with state inequality constraints,” J. Optim. Theory Appl.,17, No. 1/2, 43–92 (1975).

    Google Scholar 

  104. J. Warga, “Second order controllability and optimization with ordinary controls,” SIAM J. Control Optim.,23, No. 1, 49–60 (1985).

    Google Scholar 

  105. V. Zeidan and P. Zezza, “Necessary conditions for optimal control problems: conjugate points,” SIAM J. Control Optim.,26, No. 3, 592–608 (1988).

    Google Scholar 

  106. V. Zeidan and P. Zezza, “The conjugate point condition for smooth control sets,” J. Math. Anal. Appl.,132, No. 2, 572–589 (1988).

    Google Scholar 

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Translated from Itogi Nauki i Tekhniki, Seriya Matematicheskii Analiz, Vol. 27, pp. 147–235, 1989.

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Arutyunov, A.V. Perturbations of extremal problems with constraints and necessary optimality conditions. J Math Sci 54, 1342–1400 (1991). https://doi.org/10.1007/BF01373649

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