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Singular controls in control problems for distributed-parameter systems

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The paper gives an overview of necessary optimality conditions for singular controls and second-order necessary optimality conditions for certain classes of optimal control problems described by equations of mathematical physics, two-dimensional integral equations, and discrete two-parameter systems.

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References

  1. A. A. Abdullaev, “On optimality of quasi-singular controls in processes described by Volterra-type integral equations,” Izv. Azerbaijan NAN, No. 6, 35–42 (1998).

  2. A. A. Abdullaev and K. B. Mansimov, “Quasi-singular controls in processes described by a system of nonlinear Volterra-type integral equations,” in: Proceedings of Scientific Conference Devoted to the 80th Anniversary of the Azerbaijan Democratic Republic [in Russian], Vol. I, Baku (1998), pp. 45–46.

    Google Scholar 

  3. A. A. Abdullaev and K. B. Mansimov, “Second-order necessary optimality conditions for processes described by a system of two-dimensional nonlinear Volterra-type integral equations,” Avtomat. Telemekh., No. 1 (2000).

  4. A. A. Abuladze, Optimal Control Problems for Systems Described by Integral Equations [in Russian], Tbilisi Univ., Tbilisi (1988).

    MATH  Google Scholar 

  5. S. S. Akhiev and K. T. Akhmedov, “On integral representation of solutions of certain systems of differential equations,” Izv. Akad. Nauk AzSSR, Ser. Fiz.-Tekh. Mat. Nauk, No. 2, 116–120 (1973).

  6. S. S. Akhiev, “A method for separating a Lagrange multiplier into summands,” Dokl. Akad Nauk AzSSR, 32, No. 5, 3–5 (1976).

    MATH  Google Scholar 

  7. K. T. Akhmedov and S. S. Akhiev, “Necessary optimality conditions for certain problems of optimal control theory,” Dokl. Akad. Nauk AzSSR, 28, No. 5, 12–16 (1972).

    MATH  Google Scholar 

  8. K. T. Akhmedov and S. S. Akhiev, “On a certain optimal control problem in distributed-parameter systems,” Uch. Zap. Azerb. Gos. Univ., Ser. Fiz.-Mat. Nauk, No. 3, 3–6 (1972).

  9. S. T. Alieva and K. B. Mansimov, “Necessary optimality condition in a certain two-parameter control problem,” in: Proceedings of Scientific Conference “Contemporary Problems in Mathematics” Devoted to the Memory of A. A. Novruzov [in Russian], Baku (2001), pp. 75–76.

  10. S. T. Alieva, “Quasi-singular controls in a certain discrete boundary control problem,” Izv. Azerbaijan NAN, Ser. Fiz.-Tekh. Mat. Nauk, No. 3, 71–76 (2004).

  11. S. T. Alieva and K. B. Mansimov, “On a certain discrete two-parameter control problem with boundary condition, Vestn. BGU, Ser. Fiz.-Mat. Nauk, No. 4, 25–31 (2004).

  12. S. T. Alieva, “On optimality of singular boundary controls in two-parameter systems,” Dokl. Azerbaijan NAN, Nos. 5–6, 21–24 (2004).

  13. T. S. Angell, “On the optimal control of systems governed by nonlinear Volterra equations,” J. Optimiz. Theory Appl., 19, No. 1, 29–45 (1976).

    Article  MATH  MathSciNet  Google Scholar 

  14. A. I. Arsenashvili, “Necessary optimality conditions for neutral-type variable structure systems with deviated argument,” in: Optimal Problems in Variable Structure Systems [in Russian], TGU, Tbilisi (1985), pp. 72–93

    Google Scholar 

  15. A. I. Arsenashvili and T. A. Tadumadze, “Necessary optimality conditions for neutral-type control systems and continuous succession,” Tr. I. N. Vekua Insts. Prikl. Mat., 27, 9–45 (1988).

    MATH  MathSciNet  Google Scholar 

  16. L. T. Ashchepkov, Optimal Control with Discontinuous Systems [in Russian], Nauka, Novosibirsk (1987).

    Google Scholar 

  17. L. T. Ashchepkov and O. V. Vasil’ev, “On optimality of singular controls in Goursat-Darboux systems,” Zh. Vychisl. Mat. Mat. Fiz., 15, No. 5, 1157–1167 (1975).

    MATH  Google Scholar 

  18. L. T. Ashchepkov, O. V. Vasil’ev, and I. L. Kozlenok, “Strengthened optimality condition for singular controls in Goursat-Darboux systems,” Differents. Uravn., No. 6, 1054–1069 (1980).

    Google Scholar 

  19. N. M. Avalishvili, “Maximum principle for a variable-structure optimal problem with delay,” in: Optimal Problems in Variable-Structure Systems [in Russian], TGU, Tbilisi (1985), pp. 48–71.

    Google Scholar 

  20. S. A. Bagirova and K. B. Mansimov, “Necessary optimality conditions in a certain control problem of Goursat-Darboux systems,” in: Proceedings of the Republic Scientific Conference “Contemporary Problems in Informatics, Cybernetics, and Informational Technologies” [in Russian], Vol. 2, Baku (2003), pp. 56–58.

    Google Scholar 

  21. S. A. Bagirova and K. B. Mansimov, “Singular controls in a certain step-like control problem,” Izv. Azerbaijan NAN, Ser. Fiz.-Tekh. Math. Nauk, No. 3, 54–57 (2004).

  22. S. A. Bagirova, “Necessary optimality conditions in a variable-structure, distributed-parameter control problem,” Dokl. Azerbaijan NAN, LX, Nos. 1–2, 50–57 (2004).

    Google Scholar 

  23. E. E. Barbashina, Necessary optimality conditions with matrix impulses for the Goursat-Darboux system,” Differents. Uravn., 26, No. 2, 342–343 (1990).

    MATH  Google Scholar 

  24. D. Batkhuyag, “Strengthened optimality condition for a singular control for a certain class of distributed parameter systems,” Deposited at VINITI, No. 5548-81 (1981).

  25. S. A. Blumin and R. G. Faradzhev, “Linear cellular machines: State space approach,” Avtomat. Telemekh., No. 2, 125–163 (1982).

    Google Scholar 

  26. A. N. Burdukovskii, “Second-order optimality conditions for a certain class of distributed-parameter systems,” in: Numerical Methods of Analysis and Their Applications [in Russian], Irkutsk (1983), pp. 39–48.

  27. A. N. Burdukovskii, “Optimality conditions and successive approximation methods in control problems for distributed-parameter systems,” Theses [in Russian], Leningrad (1985).

  28. A. G. Butkovskii, Optimal Control Theory for Distributed Parameter Systems [in Russian], Nauka, Moscow (1965).

    Google Scholar 

  29. V. I. Bykov, G. S. Yablonskii, and M. G. Slin’ko, “Application of the maximum principle to optimization of quasi-stationary catalitic processes of variable activity,” in: IFIP Technical Conference on Optimization Techniques [in Russian], Novosibirsk (1974), pp. 11–16.

  30. A. I. Egorov, “On optimal control in certain distributed parameter systems,” Avtomat. Telemech., 25, No. 5, 613–623 (1964).

    Google Scholar 

  31. A. I. Egorov, “Optimal conrol of distributed-parameter systems and certain problems of the invariance theory,” in: Proceedings of IIIrd All-Union Conference on Automated Control, Optimal Systems, and Statistical Methods [in Russian], Nauka, Moscow (1967), pp. 79–92.

    Google Scholar 

  32. R. Gabasov and F. M. Kirillova, “On the theory of higher-order necessary optimality conditions,” Differents. Uravn., 6, No. 4, 665–676 (1970).

    MATH  MathSciNet  Google Scholar 

  33. R. Gabasov and F. M. Kirillova, “Modern state of art in optimal process theory,” Avtomat. Telemekh., 8, 31–62 (1972).

    Google Scholar 

  34. R. Gabasov and F. M. Kirillova, Singular Optimal Controls [in Russian], Nauka, Moscow (1973).

    Google Scholar 

  35. R. Gabasov and F. M. Kirillova, Maximum Principle in Optimal Control Theory [in Russian], Nauka i Tekhnika, Minsk (1974).

    Google Scholar 

  36. R. Gabasov and F. M. Kirillova, “Optimal control methods,” in: Progress in Science and Technology, Series on Contemporary Problems in Mathematics [in Russian], Vol. 6, All-Union Institute for Scientific and Technical Information, USSR Academy of Sciences, Moscow (1979), pp. 131–204.

    Google Scholar 

  37. R. Gabasov and F. M. Kirillova, “Mathematical optimal control theory,” in: Progress in Science and Technology, Series on Mathematical Analysis [in Russian], Vol. 16, All-Union Institute for Scientific and Technical Information, Russian Academy of Sciences, Moscow (1979), pp. 55–97.

    Google Scholar 

  38. R. Gabasov and F. M. Kirillova, Optimization Methods [in Russian], BGU, Minsk (1981).

    MATH  Google Scholar 

  39. R. Gabasov, F. M. Kirillova, and K. B. Mansimov, “Second-order necessary optimality conditions for a distributed parameter system,” Preprint, No. 31 (156), Institute of Mathematics, BSSR Academy of Sciences, Minsk (1982).

    Google Scholar 

  40. R. Gabasov, F. M. Kirillova, V. A. Srochko, and N. V. Tarasenko, “Higher-order optimality conditions, I–III,” Avtomat. Telemekh., 5, 5–22; 6, 5–24; 7, 5–34 (1971).

    Google Scholar 

  41. I. V. Gaishun, Multiparameter Control Systems [in Russian], Nauka i Tekhnika, Minsk (1996).

    MATH  Google Scholar 

  42. K. K. Gasanov, “Strengthened optimality conditions for singular controls in distributed-parameter systems with delayed argument,” Izv. Akad. Nauk AzSSR, Ser. Fiz.-Tekhn. Mat. Nauk, No. 6, 132–138 (1982).

  43. K. K. Gasanov and G. F. Kuliev, “On optimality of singular controls in a boundary-value problem for second-order hyperbolic equations,” in: Boundary-Value Problems for Partial Differential Equations [in Russian], Baku (1981), pp. 44–51.

  44. K. K. Gasanov and A. Ya. Sharifov, “Second-order necessary optimality conditions for processes described by a system of hyperbolic equations,” in: Studies in Differential Equations [in Russian], BGU, Baku (1984).

    Google Scholar 

  45. A. K. Gasanov and Sh. Sh. Yusubov, “On necessary optimality conditions for Goursat-Darboux systems with delayed argument,” in: Collection of Scientific Works [in Russian], S. M. Kirov Azerbaijan State University, Baku (1986), pp. 34–40.

    Google Scholar 

  46. K. K. Gasanov and Sh. Sh. Yusubov, “On optimality of singular controls in Goursat-Darboux-type systems, in: Boundary-Value Problems for Partial Differential Equations [in Russian], Baku (1990), pp. 44–47.

  47. V. V. Gorkhovik, “Higher-order necessary optimality conditions for a control problem with terminal constraints,” Preprint No. 1 (126), Institute of Mathematics, BSSR Academy of Sciences, Minsk (1982).

    Google Scholar 

  48. M. A. Kazemi-Dehkordi, “Necessary conditions for optimality of singular controls,” J. Optimiz Theory Appl., 43, No. 4, 629–638 (1984).

    Article  MATH  MathSciNet  Google Scholar 

  49. M. A. Kazemi-Dehkordi, “Necessary conditions for optimality of singular controls in systems governed by partial differential equations,” J. Optimiz. Theory Appl., 43, No. 4, 639–661 (1984).

    Article  MATH  MathSciNet  Google Scholar 

  50. L. N. Kerimov and T. K. Melikov, “Certain necessary optimality conditions for singular controls in Goursat-Darboux systems,” Trudy Azerb. S.-Kh. Inst., Ser. Mekhanizatsiya, No. 29, 197–203 (1979).

  51. G. L. Kharatishvili, “Maximum principle in switched optimal problems,” in: Proceedings of the Institute of Control Systems, GSSR Akad. Nauk, Vol. 19, No. 1, Metsniereba, Tbilisi (1980), pp. 5–17.

    Google Scholar 

  52. G. L. Kharatishvili, “Polyatomic optimal systems,” in: Optimal Problems in Variable-Structure Systems [in Russian], Tbilisi (1985), pp. 3–47.

  53. F. M. Kirillova and O. V. Vasil’ev, “On optimal processes in two-parameter discrete systems,” Dokl. Akad. Nauk SSSR, 175, No. 1, 17–18 (1967).

    MathSciNet  Google Scholar 

  54. E. S. Levitin, A. A. Milyutin, and N. P. Osmolovskii, “Higher-order condition theory in constrained smooth extremal problems,” in: Theoretical and Applied Optimal Control Problems [in Russian], Novosibirsk, pp. 4–40.

  55. K. B. Mansimov, “Maximum principle for distributed parameter systems,” Uch. Zap. Minist. Vysh. Sred. Obraz. AzSSR, Ser. Fiz.-Mat Nauk, No. 6, 37–41 (1976).

  56. K. B. Mansimov, “Necessary optimality conditions in distributed-parameter systems with intermediate inequality type of constraints,” Izv. Akad. Nauk AzSSR, Ser. Fiz.-Tekh. Mat. Nauk, No. 3, 113–116 (1976).

  57. K. B. Mansimov, “On the theory of necessary optimality conditions in certain systems with aftereffect,” Theses [in Russian], Baku (1979).

  58. K. B. Mansimov and F. B. Mansimov, “Certain multipoint necessary optimality conditions for a certain distributed-parameter system,” Izv. Akad. Nauk AzSSR, Ser. Fiz.-Tekh. Mat Nauk, No. 3, 29 (1979)

  59. K. B. Mansimov, “On a certain scheme for studying the singular case in Goursat-Darboux systems,” Izv. Akad. Nauk AzSSR, Ser. Fiz.-Tekh. Mat. Nauk No. 2, 100–104 (1981).

  60. K. B. Mansimov, “Study of the singular case in nonlinear Goursat-Darboux systems,” in: Proceedings of IIIrd Conference of Young Scientists in Automated Control [in Russian], Tbilisi (1982), pp. 56–58.

  61. K. B. Mansimov, “Multipoint necessary optimality conditions for singular controls in quasililinear Goursat-Darboux systems,” Izv. Akad. Nauk AzSSR, Ser. Fiz.-Tekh. Mat Nauk, No. 5, 106–111 (1982).

  62. K. B. Mansimov, “On the theory of necessary optimality conditions in a certain control problem,” Deposited at VINITI, No. 60V3 (1985).

  63. K. B. Mansimov, “Krotov-type sufficient optimality conditions for discrete two-parameter systems,” Avtomat. Telemekh., No. 8, 15–20 (1985).

  64. K. B. Mansimov, “On the theory of necessary optimality conditions in a certain control problem, II,” Deposited at VINITI, No. 2157-V86 (1986).

  65. K. B. Mansimov, “On optimality of quasi-singular controls in Goursat-Darboux systems,” Differents. Uravn., 22 No. II, 1952–1960 (1986).

    MathSciNet  Google Scholar 

  66. K. B. Mansimov, “On the optimality of controls singular in the classical sense in the Goursat-Darboux systems,” Dokl. Akad. Nauk SSSR, 286, No. 4, 808–812 (1986).

    MathSciNet  Google Scholar 

  67. K. B. Mansimov, “On a certain control problem of a reaction vessel,” in: Problems of Automated Control [in Russian], Metsniereba, Tbilisi, (1986), pp. 63–65.

    Google Scholar 

  68. K. B. Mansimov, “Some necessary optimality conditions in a certain control problem for distributed-parameter systems,” Differents. Uravn., No. 12, 2164–2167 (1987).

  69. K. B. Mansimov, “On the optimality of singular boundary controls in a certain distributed parameter system,” Soobshch. Akad. Nauk GSSR, 128, No. 3, 489–492 (1987).

    MATH  MathSciNet  Google Scholar 

  70. K. B. Mansimov, “An analog of the Gabasov-Kirillova condition for a certain class of distributed parameter systems,” Izv. Akad. Nauk. AZSSR, Ser. Fiz.-Tekh. Mat Nauk, No. 2, 30–36 (1988).

  71. K. B. Mansimov, “On the theory of necessary optimality conditions in a certain control problem for distributed-parameter systems,” Dokl. Akad. Nauk SSSR, 301, No. 3, 546–550 (1988).

    Google Scholar 

  72. K. B. Mansimov, “On integral second-order necessary optimality conditions in Goursat-Darboux systems with functional constraints,” in: All-Union School “Optimal Control, Geometry and Analysis,” Abstracts of Reports [in Russian], Kemerovo, (1988), p. 85.

  73. K. B. Mansimov, “On the integral necessary second-order optimality conditions,” in: 7th IFAC Workshop on Contr. Appl. of Nonlinear Programm and Optimiz., Abstracts, Moscow (1988), p. 92.

  74. K. B. Mansimov, “Optimality conditions of singular controls for a certain class of discrete two-parameter systems,” in: International Soviet-Poland Workshop “Mathematical Methods of Optimal Control and Their Applications,” Abstracts of Reports [in Russian], Minsk (1989), pp. 80–81.

  75. K. B. Mansimov, “Second-order optimality conditions in Goursat-Darboux systems with constraints,” Differents. Uravn., 26, No. 6, 954–965 (1990).

    MATH  MathSciNet  Google Scholar 

  76. K. B. Mansimov, “Necessary optimality condition for a certain control problem of Goursat-Darboux systems,” Deposited at VINITI, No. 1709-B91 (1991).

  77. K. B. Mansimov, “Optimization of a certain class of discrete two-parameter systems,” Differents. Uravn., No. 2, 359–361 (1991).

  78. K. B. Mansimov, “Integral necessary optimality conditions for quasi-singular controls in Goursat-Darboux systems,” Avtomat. Telemekh., No. 5, 36–43 (1993).

    Google Scholar 

  79. K. B. Mansimov, “Integral necessary optimality conditions for quasisingular controls in the Goursat-Darboux systems,” Avtomat. Telemekh., No. 5 (1993).

  80. K. B. Mansimov, “Necessary optimality conditions for singular processes in optimal control problems,” Theses [in Russian], Baku (1994).

  81. K. B. Mansimov, “Study of quasi-singular processes in a certain control problem of a reactor vessel,” Differents. Uravn., 33, No. 4, 433–439 (1997).

    MathSciNet  Google Scholar 

  82. K. B. Mansimov, “Second-order necessary optimality conditions in discrete two-parameter systems,” Izv. Azerbaijan NAN, Ser. Fiz.-Tekh. Mat Nauk, No. 2, 56–60 (1998).

  83. K. B. Mansimov and S. A. Bagirova, “On a certain control problem of composite distributed-parameter systems,” in: Proceedings of Ist International Scientific Conference “Inverse Problems of Theoretical and Mathematical Physics” [in Russian], Sumgait Gos. Univ, Sumgait (2003), pp. 36–37.

    Google Scholar 

  84. M. Dzh. Mardanov, “Second-order necessary optimality conditions in distributed-parameter systems,” Izv. Akad. Nauk. AzSSR, Ser. Fiz.-Tekh. Mat. Nauk, No. 4, 181–187 (1987).

  85. M. Dzh. Mardanov, “On a proof of the maximum principle in Goursat-Darboux systems with delays,” Partial Differential Equations and Their Applications [in Russian], Baku (1989), pp. 43–55.

  86. T. K. Melikov, “Study of singular processes in certain optimal systems,” Theses [in Russian], Baku (1976).

  87. T. K. Melikov, “On necessary optimality conditions for distributed parameter systems,” Deposited at VINITI, No. 2637-79 (1979).

  88. T. K. Melikov, “Recursive optimality conditions for singular controls in Goursat-Darboux systems,” Dokl. Akad. Nauk AzSSR, No. 8, 6–9 (1989).

    Google Scholar 

  89. T. K. Melikov, “Necessary optimality conditions for quasi-singular controls in Goursat-Darboux systems,” Deposited at AZNIIHTI, No. 1845-Az.92 (1992).

  90. T. K. Melikov, Controls Singular in the Classical Sense in Goursat-Darboux Systems [in Russian], Elm, Baku (2003).

    Google Scholar 

  91. T. K. Melikov, “Investigation of optimality of singular controls in Goursat-Darboux systems,” Proc. Inst. Math. Mech., 18, 227–236 (2003).

    MATH  MathSciNet  Google Scholar 

  92. T. K. Melikov, “Necessary optimality conditions of quasi-singular controls in Goursat-Darboux systems,” Trans. NAS Azerbaijan, Ser. Phys.-Tech. Math. Sci., No. 1, 35–41 (2004).

  93. T. K. Melikov, “On conditions of Kelley type optimality in Goursat-Darboux systems, Proc. Inst. Math. Mech., XXI, 229–236 (2004).

    MathSciNet  Google Scholar 

  94. P. G. Mironov, “On a certain optimality criterion for systems described by two-dimensional Volterra-type integral equations,” in: Problems of the Qualitative Theory of Differential Equations [in Russian], Cheboks. Gos. Univ, Cheboksary (1982), pp. 66–72.

    Google Scholar 

  95. M. A. Monastyrskii, “Optimal control of systems described by Volterra integral equations,” Avtomat. Telemekh., No. 1, 29–36 (1975).

    Google Scholar 

  96. V. B. Muslumov, “Second-order necessary optimality conditions in one boundary control problem,” Proc. Inst. Math. Mech., Baku, XXI, 237–244 (2004).

    MathSciNet  Google Scholar 

  97. V. B. Muslumov, “Study of quasi-singular controls in a certain boundary control problem of distributed-parameter systems,” Izv. Azerbaijan NAN, Ser. Fiz.-Tehh. Mat. Nauk, No. 2, 58–60 (2004).

  98. V. B. Muslumov, “Integral necessary optimality conditions for quasi-singular controls in a certain distributed-parameter boundary control problem with delay,” Izv. Azerbaijan NAN, Ser. Fiz.-Tekh. Mat. Nauk, No. 3, 77–81 (2004).

  99. V. B. Muslumov, “Necessary optimality conditions for boundary controls singular in the Pontryagin maximum principle sense,” Izv. Azerbaijan NAN, Ser. Fiz.-Tekh. Mat. Nauk, No. 2, 135–139 (2005).

  100. M. S. Nikol’skii, “Variable-structure variational problems,” Vestn. MGU, Ser. 15, Vychislitel’naya Matematika Kibernetika, No. 1, 36–41 (1987).

  101. Ch. Osmonova, “On optimality of a singular control in the optimization problem of heat-transfer processes,” in: Mathematical Methods of Control of Distributed-Parameter Systems [in Russian], Frunze (1978), pp. 29–42.

  102. G. M. Ostrovskii and Yu. M. Volin, Optimization Methods of Complicated Chemical-Technological Schemes [in Russian], Khimiya, Moscow.

  103. G. M. Ostrovskii and Yu. M. Volin, Modeling of Complicated Chemical-Thechnological Schemes [in Russian], Khimiya, Moscow (1975).

    Google Scholar 

  104. V. I. Plotnikov and V. I. Sumin, “Optimization of distributed parameter objects described by the Darboux-Goursat system,” Zh. Vychisl. Mat. Mat. Fiz., 12, No. 1, 61–77 (1972).

    MathSciNet  Google Scholar 

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

    Google Scholar 

  106. A. Ya. Sharifov, “Second-order necessary optimality conditions for processes described by a system of parabolic equations,” in: Proceedings of the Republic Scientific Conference of Post-Graduate Students [in Russian], Baku (1984).

  107. A. Ya. Sharifov, “On the existence of an optimal control and necessary optimality conditions for distributed parameter systems,” Theses [in Russian], Baku (1990).

  108. Shenq-Chao Huang, “Optimal control problems with a system of integral equations and restricted phase coordinates,” SIAM J. Control, 10, No. 1, 14–38 (1972).

    Article  MATH  MathSciNet  Google Scholar 

  109. V. A. Srochko, “Optimality conditions for a certain class of distributed-parameter systems,” Sib. Mat. Zh., 17, No. 5, 1108–1115 (1976).

    MATH  Google Scholar 

  110. V. A. Srochko, “Maximum principle for a certain class of distributed-parameter systems,” in: Stability and Optimization Problems of Dynamical Systems [in Russian], Irkutsk (1983), pp. 70–82.

  111. V. A. Srochko, Variational Maximum Principle and Linearization Methods in Optimal Control Problems [in Russian], IGU, Irkutsk (1989).

    Google Scholar 

  112. V. I. Sumin, “Optimization of generalized Volterra systems,” Theses [in Russian], Gor’kii (1975).

  113. V. I. Sumin, “Strong degeneration of singular controls in distributed optimiazation problems,” Dokl. Akad. Nauk SSSR, 320, No. 2, 295–299 (1991).

    Google Scholar 

  114. T. A. Tadumadze and N. M. Avalishvili, “Regular perturbations in variable-structure optimal control problems,” in: Optimal Problems in Variable-Structure Systems [in Russian], Tbilisi (1985), pp. 100–154.

  115. O. V. Vasil’ev, “On the optimality of processes in continuous and discrete two-parameter systems,” in: Informational Collection of Papers of the Irkutsk State University Computational Center [in Russian], No. 2, Irkutsk (1968), pp. 87–104.

  116. O. V. Vasil’ev, “On optimality of a singular control in distributed-parameter systems,” in: Abstracts of Report, IInd All-Union Conference on Problems of Theoretical Cybernetics [in Russian], Novosibirsk (1971), pp. 26–27.

  117. O. V. Vasil’ev, “On optimality of singular controls in distributed-parameter systems,” in: Control Systems [in Russian], No. 10, Novosibirsk (1972), pp. 27–34.

  118. O. V. Vasil’ev, “On the study of a singular control in a certain distributed parameter system,” in: Control Systems [in Russian], No. 15, Novosibirsk (1976), pp. 3–15.

  119. O. V. Vasil’ev, “On necessary optimality conditions in distributed parameter systems,” in: Optimization and Operation Research Methods (Applied Mathematics) [in Russian], Irkutsk (1976), pp. 21–28.

  120. O. V. Vasil’ev, “Optimality conditions for singular controls in a certain class of distributed parameter systems,” in: Proceedings of All-Union Symposium on Optimal Control and Differential Games [in Russian], Tbilisi (1977), 72–77.

  121. O. V. Vasil’ev and V. A. Srochko, “On optimization of a certain class of distributed parameter control processes,” Sib. Mat. Zh., 19, No. 2, 466–470 (1978).

    MATH  Google Scholar 

  122. O. V. Vasil’ev, “L. S. Pontryagin Maximum principle in optimal system theory,” in: Applied Mathematics [in Russian], No. 3, Novosibirsk (1978), pp. 109–138.

  123. O. V. Vasil’ev and V. A. Teletskii, “On optimization of a certain class of distributed-parameter systems,” in: Optimization of Dynamical Systems [in Russian], BGU, Minsk (1978), pp. 26–30.

    Google Scholar 

  124. O. V. Vasil’ev, “On optimality of singular boundary controls in distributed parameter systems,” in Control Systems [in Russian], No. 18, (1979), 4–13.

  125. F. P. Vasil’ev, Methods for Solving Extremal Problems [in Russian], Nauka, Moscow (1981).

    MATH  Google Scholar 

  126. O. V. Vasil’ev, “Qualitative and constructive methods for optimization of distributed parameter control processes,” Theses [in Russian], Leningrad (1984).

  127. O. V. Vasil’ev, V. A. Srochko, and V. A. Terletskii, Optimization Methods and their Applications, Part 2 [in Russian], Nauka, Novosibirsk (1990).

    MATH  Google Scholar 

  128. V. B. Velichenko, “Optimal control of composite systems,” Dokl. Akad. Nauk SSSR, 176, No. 4, 754–756 (1967).

    MathSciNet  Google Scholar 

  129. V. R. Vinokurov, “Optimal control of processes described by integral equations, I-III,” Izv. Vuzov, Ser. Mat., Nos. 7–9 (1967).

  130. A. S. Wilskey, “Interrelation between the digital signal processing theory and the control and estimation theory, TIIER, 66, No. 9, 5–33 (1978).

    Google Scholar 

  131. M. A. Yagubov, “Second-order necessary optimality conditions for processes described by the Mangeron equation,” in: Partial Differential Equations and Their Applications [in Russian], Baku (1983), pp. 108–116.

  132. M. Ya. Yagubov, “Regular Young representation and a necessary optimality condition for relaxed controls in the Goursat-Darboux system,” Partial Differential Equations and Their Applications [in Russian], Baku (1989), pp. 85–90.

  133. Sh. Sh. Yusubov, “On certain problems of optimal control theory for systems described by hyperbolic equations,” Theses [in Russian], Baku (1992).

  134. G. K. Zakharov, “Optimization of step-like control systems,” Avtomat. Telemekh., No. 8, 5–9 (1981).

  135. S. V. Zubarev, “On an analog of the Kelley condition for systems of hyperbolic equations,” Ukr. Mat. Zh., 31, No. 1, 3–8 (1981).

    MathSciNet  Google Scholar 

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  1. Institute of Cybernetics, Azerbaijan National Academy of Sciences, Azerbaijan

    K. B. Mansimov

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Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 42, Optimal Control, 2006.

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Mansimov, K.B. Singular controls in control problems for distributed-parameter systems. J Math Sci 148, 331–381 (2008). https://doi.org/10.1007/s10958-008-0009-0

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