First order necessary optimal conditions in Gursat-Darboux stochastic systems |
Mastaliyev R.O. |
2021, issue 1, P. 89–104 DOI: https://doi.org/10.47910/FEMJ202108 |
Abstract |
For optimal control problems, described by the Gursat-Darboux stochastic system, a number of first-order necessary optimality conditions are formulated and proved, which are the stochastic analogue - the Pontryagin maximum principle, the linearized maximum principle and the Euler equation. |
Keywords: nonlinear Gursat-Darboux stochastic system, optimal control, necessary optimality conditions, analogue of the Euler equation |
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References |
[1] A. I. Egorov, “Ob optimal'nom upravlenii protsessami v nekotorykh sistemakh raspredelennymi parametrami” , Avtomatika i telemekhanika, 25:5 (1964), 613–623. [2] A.I. Egorov, “Optimal'nye protsessy v sistemakh s raspredelennymi parametrami i nekotorye zadachi teorii invariantnosti”, Izv. AN SSSR, Ser. matem., 29:6 (1965), 1205–1260. [3] V.I. Plotnikov, V.I. Sumin, “Optimizatsiia ob"ektov s raspredelennymi parametrami, opisyvaemykh sistemami Gursa–Darbu” , Zh. vychisl. matem. i matem. fiz., 12:1 (1972), 61–77. [4] S.S. Akhiev, K.T. Akhmedov, “Neobkhodimye usloviia optimal'nosti dlia nekotorykh zadach teorii optimal'nogo upravleniia” , Dokl. AN Azerb. SSR, 28:5 (1972), 12–16. [5] L.T. Ashchepkov, O.V. Vasil'ev, “Ob optimal'nosti osobykh upravlenii v sistemakh Gursa–Darbu”, Zhurn. vychisl. mat. i mat. fiziki, 15:5 (1975), 1157–1167. [6] L.T. Ashchepkov, O.V. Vasil'ev, I.L. Kovalenok, “Usilennoe uslovie optimal'nosti osobykh upravlenii v sisteme Gursa–Darbu”, Differents. uravneniia, 6 (1980), 1054–1059. [7] V.A. Srochko, “Usloviia optimal'nosti dlia odnogo klassa sistem s raspredelennymi parametrami”, Sib. mat. zhurnal, 2 (1984), 56–65. [8] T.K. Melikov, Issledovanie osobykh protsessov v nekotorykh optimal'nykh sistemakh // Avtoref. dis. kand. fiz.-mat. nauk, Baku, 1976, 17 s. [9] T.K. Melikov, Osobye v klassicheskom smysle upravleniia v sistemakh Gursa–Darbu, Elm, Baku, 2003, 96 s. [10] K.B. Mansimov, Neobkhodimye usloviia optimal'nosti osobykh protsessov v zadachakh optimal'nogo upravleniia // Avtoref. diss. d-ra fiz.-mat. nauk, Baku, 1994, 43 s. [11] K.B. Mansimov, Kachestvennaia teoriia optimal'nogo upravleniia sistemami Gursa-Darbu, Elm, Baku, 2010, 360 s. [12] Iu.M. Ermol'ev, V.P. Gulenko, T.I. Tsarenko, Konechno-raznostnyi metod v zadachakh optimal'nogo upravleniia, Naukova Dumka, Kiev, 1978, 164 s. [13] A. N. Tikhonov, A. A. Samarskii, Uravneniia matematicheskoi fiziki, Nauka, M., 1977, 736 s. [14] V.V. Rachinskii, Vvedenie v obshchuiu teoriiu dinamiki sorbtsii i khromatografii, Nauka, M., 1964, 136 s. [15] M.Kh. Bikchentaev, “K optimal'nomu upravleniiu sistemami s raspredelennymi parametrami pri sluchainykh vozdeistviiakh” , Izv.AN SSSR. Tekhn. kibernetika, 3 (1969), 158–167. [16] L.E. Shaikhet, “Ob optimal'nom upravlenii odnim klassom stokhasticheskikh differentsial'nykh uravnenii v chastnykh proizvodnykh” , Matem. zametki, 31:6 (1982), 933–936. [17] L.E. Shaikhet, “O neobkhodimom uslovii optimal'nosti upravleniia stokhasticheskimi differentsial'nymi uravneniiami giperbolicheskogo tipa. Teoriia sluchainykh protsessov”, Teoriia sluchainykh protsessov. T. 12, Kiev, 1984, 96–101. [18] L.E. Shaikhet, “Optimal'noe upravlenie nekotorymi giperbolicheskimi i integral'nymi uravneniiami”, Teoriia sluchainykh protsessov. T. 15, Kiev, 1987, 110–116. [19] L.E. Shaikhet, “About an unsolved optimal control problem for stochastic partial differential equation”, XVI International Confrence: Dynamical systems modeliling and stability investigation, Kiev, Ukraine, 29-31 May, 2013, 2013, 332–334. [20] J. Yeh, “Wiener measure in a space of functions of two variables”, Trans. Amer. Math. Soc., 95 (1960), 433–450. [21] V.I. Plotnikov, V.I. Sumin, “Problema ustoichivosti nelineinykh sistem Gursa–Darbu”, Differents. uravneniia, 8:5 (1972), 845-856. [22] L.L. Ponomarenko, “Stokhasticheskaia beskonechnomernaia zadacha Gursa”, Matematicheskii analiz i teoriia veroiatnostei, Kiev, 1978, 140-143. [23] V. M. Alekseev, V. M. Tikhomirov, S. V. Fomin, Optimal'noe upravlenie, Nauka, M., 1979, 429 s. [24] R. Gabasov, F. M. Kirillova, Printsip maksimuma v teorii optimal'nogo upravleniia, URSS, M., 2011, 272 s. [25] L.S. Pontriagin, V.G. Boltianskii, R.V. Gamkrelidze, E.F. Mishchenko, Matematicheskaia teoriia optimal'nykh protsessov, Nauka, M., 1983, 392 s. [26] V.A. Srochko, Vychislitel'nye metody optimal'nogo upravleniia, Izd-vo IGU, Ir- kutsk, 1982, 110 s. [27] R. Gabasov, F. M. Kirillova, Osobye optimal'nye upravleniia, Knizhnyi dom “Librokom” , M., 2013, 256 s. |