Singular controls in control problems for distributed-parameter systems. (English. Russian original) Zbl 1211.49033
J. Math. Sci., New York 148, No. 3, 331-381 (2008); translation from Sovrem. Mat. Prilozh. 42, 38-93 (2006).
The paper contains an overview of necessary optimality conditions for singular controls and second order necessary optimality conditions for certain classes of optimal control problems with distributed parameter systems. After the introduction, the authors present second-order necessary optimality conditions in control problems of Goursat-Darboux systems; singular controls in systems described by a system of first-order partial differential equations of hyperbolic type; second-order necessary optimality conditions in discrete two-parameter systems; necessary optimality conditions for singular controls in processes described by hyperbolic and parabolic equations with mixed boundary conditions; necessary optimality conditions for singular controls in processes described by a system of two-dimensional Volterra-Type integral equations.
Reviewer: Martin Gugat (Erlangen)
MSC:
| 49K20 | Optimality conditions for problems involving partial differential equations |
Keywords:
distributed parameter systems; singular controls; optimal control problems; necessary optimality conditionsReferences:
| [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. |
| [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). · Zbl 1096.49516 |
| [4] | A. A. Abuladze, Optimal Control Problems for Systems Described by Integral Equations [in Russian], Tbilisi Univ., Tbilisi (1988). |
| [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). · Zbl 0342.49013 |
| [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). |
| [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). · Zbl 0305.49021 · doi:10.1007/BF00934050 |
| [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 · Zbl 0594.49014 |
| [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). · Zbl 0731.93055 |
| [16] | L. T. Ashchepkov, Optimal Control with Discontinuous Systems [in Russian], Nauka, Novosibirsk (1987). · Zbl 0688.49016 |
| [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). |
| [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). |
| [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. |
| [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. |
| [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). |
| [23] | E. E. Barbashina, Necessary optimality conditions with matrix impulses for the Goursat-Darboux system,” Differents. Uravn., 26, No. 2, 342–343 (1990). · Zbl 0698.49017 |
| [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). |
| [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). |
| [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). |
| [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. |
| [32] | R. Gabasov and F. M. Kirillova, ”On the theory of higher-order necessary optimality conditions,” Differents. Uravn., 6, No. 4, 665–676 (1970). |
| [33] | R. Gabasov and F. M. Kirillova, ”Modern state of art in optimal process theory,” Avtomat. Telemekh., 8, 31–62 (1972). |
| [34] | R. Gabasov and F. M. Kirillova, Singular Optimal Controls [in Russian], Nauka, Moscow (1973). · Zbl 1177.93037 |
| [35] | R. Gabasov and F. M. Kirillova, Maximum Principle in Optimal Control Theory [in Russian], Nauka i Tekhnika, Minsk (1974). · Zbl 0291.49009 |
| [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. |
| [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. |
| [38] | R. Gabasov and F. M. Kirillova, Optimization Methods [in Russian], BGU, Minsk (1981). · Zbl 0514.49001 |
| [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). |
| [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). |
| [41] | I. V. Gaishun, Multiparameter Control Systems [in Russian], Nauka i Tekhnika, Minsk (1996). |
| [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). · Zbl 0524.49013 |
| [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). |
| [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. |
| [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. · Zbl 0799.49018 |
| [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). |
| [48] | M. A. Kazemi-Dehkordi, ”Necessary conditions for optimality of singular controls,” J. Optimiz Theory Appl., 43, No. 4, 629–638 (1984). · Zbl 0518.49014 · doi:10.1007/BF00935010 |
| [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). · Zbl 0518.49015 · doi:10.1007/BF00935011 |
| [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. |
| [52] | G. L. Kharatishvili, ”Polyatomic optimal systems,” in: Optimal Problems in Variable-Structure Systems [in Russian], Tbilisi (1985), pp. 3–47. · Zbl 0591.49012 |
| [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). |
| [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). · Zbl 0352.49009 |
| [57] | K. B. Mansimov, ”On the theory of necessary optimality conditions in certain systems with aftereffect,” Theses [in Russian], Baku (1979). · Zbl 0422.49020 |
| [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) · Zbl 0422.49020 |
| [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). · Zbl 0479.49013 |
| [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). · Zbl 0536.49014 |
| [62] | K. B. Mansimov, ”On the theory of necessary optimality conditions in a certain control problem,” Deposited at VINITI, No. 60V3 (1985). · Zbl 0629.49010 |
| [63] | K. B. Mansimov, ”Krotov-type sufficient optimality conditions for discrete two-parameter systems,” Avtomat. Telemekh., No. 8, 15–20 (1985). · Zbl 0592.49013 |
| [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). · Zbl 0596.49020 |
| [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). · Zbl 0596.49020 |
| [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. |
| [68] | K. B. Mansimov, ”Some necessary optimality conditions in a certain control problem for distributed-parameter systems,” Differents. Uravn., No. 12, 2164–2167 (1987). · Zbl 0635.49015 |
| [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). · Zbl 0639.49019 |
| [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). · Zbl 0674.49024 |
| [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). |
| [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. · Zbl 0694.49019 |
| [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). · Zbl 0707.49008 |
| [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). · Zbl 0734.49016 |
| [78] | K. B. Mansimov, ”Integral necessary optimality conditions for quasi-singular controls in Goursat-Darboux systems,” Avtomat. Telemekh., No. 5, 36–43 (1993). · Zbl 0813.49025 |
| [79] | K. B. Mansimov, ”Integral necessary optimality conditions for quasisingular controls in the Goursat-Darboux systems,” Avtomat. Telemekh., No. 5 (1993). · Zbl 0813.49025 |
| [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). |
| [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. |
| [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). · Zbl 0695.49017 |
| [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). · Zbl 0756.49012 |
| [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). · Zbl 1068.49019 |
| [91] | T. K. Melikov, ”Investigation of optimality of singular controls in Goursat-Darboux systems,” Proc. Inst. Math. Mech., 18, 227–236 (2003). · Zbl 1068.49019 |
| [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). · Zbl 1243.49025 |
| [93] | T. K. Melikov, ”On conditions of Kelley type optimality in Goursat-Darboux systems, Proc. Inst. Math. Mech., XXI, 229–236 (2004). · Zbl 1176.49028 |
| [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. |
| [95] | M. A. Monastyrskii, ”Optimal control of systems described by Volterra integral equations,” Avtomat. Telemekh., No. 1, 29–36 (1975). |
| [96] | V. B. Muslumov, ”Second-order necessary optimality conditions in one boundary control problem,” Proc. Inst. Math. Mech., Baku, XXI, 237–244 (2004). · Zbl 1068.49020 |
| [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). · Zbl 0617.49009 |
| [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. · Zbl 1126.93035 |
| [103] | G. M. Ostrovskii and Yu. M. Volin, Modeling of Complicated Chemical-Thechnological Schemes [in Russian], Khimiya, Moscow (1975). |
| [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). |
| [105] | L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, Mathematical Theory of Optimal Processes [in Russian], Nauka, Moscow (1969). |
| [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). · Zbl 0255.49015 · doi:10.1137/0310002 |
| [109] | V. A. Srochko, ”Optimality conditions for a certain class of distributed-parameter systems,” Sib. Mat. Zh., 17, No. 5, 1108–1115 (1976). · Zbl 0364.49009 |
| [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). · Zbl 0798.49005 |
| [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). · Zbl 0770.49016 |
| [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. · Zbl 0591.49019 |
| [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). |
| [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. |
| [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). |
| [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). |
| [128] | V. B. Velichenko, ”Optimal control of composite systems,” Dokl. Akad. Nauk SSSR, 176, No. 4, 754–756 (1967). · Zbl 0162.14101 |
| [129] | V. R. Vinokurov, ”Optimal control of processes described by integral equations, I-III,” Izv. Vuzov, Ser. Mat., Nos. 7–9 (1967). · Zbl 0171.06003 |
| [130] | A. S. Wilskey, ”Interrelation between the digital signal processing theory and the control and estimation theory, TIIER, 66, No. 9, 5–33 (1978). |
| [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). · Zbl 1104.49023 |
| [134] | G. K. Zakharov, ”Optimization of step-like control systems,” Avtomat. Telemekh., No. 8, 5–9 (1981). · Zbl 0491.93035 |
| [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). · Zbl 0469.49015 |
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