Optimal control: nonlocal conditions, computational methods, and the variational principle of maximum

A. V. Arguchintsev ; V. A. Dykhta ; V. A. Srochko

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 1 (2009), pp. 3-43.

Résumé

This paper surveys theoretical results on the Pontryagin maximum principle (together with its conversion) and nonlocal optimality conditions based on the use of the Lyapunov-type functions (solutions to the Hamilton–Jacobi inequalities). We pay special attention to the conversion of the maximum principle to a sufficient condition for the global and strong minimum without assumptions of the linear convexity, normality, or controllability. We give the survey of computational methods for solving classical optimal control problems and describe nonstandard procedures of nonlocal improvement of admissible processes in linear and quadratic problems. Furthermore, we cite some recent results on the variational principle of maximum in hyperbolic control systems. This principle is the strongest first order necessary optimality condition; it implies the classical maximum principle as a consequence.

Détail

Keywords: maximum principle, Hamilton–Jacobi inequalities, nonlocal computational methods, variational maximum principle.

Bibliographie

[1] Pontryagin L. S., Boltyanskii V. G., Gamkrelidze R. V., Mischenko E. F., Matematicheskaya teoriya optimalnykh protsessov, Fizmatgiz, M., 1961, 391 pp. | MR

[2] Pontryagin L. S., Printsip maksimuma, Fond matem. obrazovaniya i prosvescheniya, M., 1998, 70 pp.

[3] Ioffe A. D., Tikhomirov V. M., Teoriya ekstremalnykh zadach, Nauka, M., 1974, 480 pp. | MR | Zbl

[4] Alekseev V. M., Tikhomirov V. M., Fomin S. V., Optimalnoe upravlenie, Nauka, M., 1979, 429 pp. | MR

[5] Vasilev F. P., Metody optimizatsii, Faktorial, M., 2002, 824 pp.

[6] Gabasov R., Kirillova F. M., Printsip maksimuma v teorii optimalnogo upravleniya, Nauka i tekhnika, Minsk, 1974, 272 pp. | MR

[7] Matveev A. S., Yakubovich V. A., Optimalnye sistemy upravleniya: Obyknovennye differentsialnye uravneniya. Spetsialnye zadachi, Ucheb. posobie, Izd-vo S.-Peterburgsk. un-ta, SPb., 2003, 540 pp.

[8] Arutyunov A. V., Magaril-Ilyaev G. G., Tikhomirov V. M., Printsip maksimuma Pontryagina. Dokazatelstvo i prilozheniya, Faktorial, M., 2006, 144 pp.

[9] Milyutin A. A., Dmitruk A. V., Osmolovskii N. P., Printsip maksimuma v optimalnom upravlenii, Izd-vo MGU, M., 2004, 167 pp.

[10] Aschepkov L. T., Lektsii po optimalnomu upravleniyu, Ucheb. posobie, Izd-vo Dalnevostochnogo un-ta, Vladivostok, 1996, 208 pp.

[11] Dykhta V. A., “Lyapunov–Krotov inequality and sufficient conditions in optimal control”, J. Math. Sci., 121:2 (2004), 2156–2177 | DOI | MR | Zbl

[12] Dykhta V. A., “Neravenstvo Lyapunova–Krotova i dostatochnye usloviya v optimalnom upravlenii”, Itogi nauki i tekhn. Sovr. matem. i ee prilozh., 110, VINITI, M., 2006, 76–108

[13] Antipina N. V., Dykhta V. A., “Lineinye funktsii Lyapunova–Krotova i dostatochnye usloviya optimalnosti v forme printsipa maksimuma”, Izv. vuzov. Matematika, 2002, no. 12, 11–22 | MR | Zbl

[14] Dubovitskii A. Ya., Milyutin A. A., “Teoriya printsipa maksimuma”, Metody teorii ekstremalnykh zadach v ekonomike, Nauka, M., 1981, 6–47 | MR

[15] Afanasev A. P., Dikusar V. V., Milyutin A. A., Chukanov S. A., Neobkhodimoe uslovie v optimalnom upravlenii, Nauka, M., 1990, 319 pp. | MR

[16] Milyutin A. A., Printsip maksimuma v obschei zadache optimalnogo upravleniya, Fizmatlit, M., 2001, 304 pp.

[17] Arutyunov A. V., Usloviya ekstremuma. Anormalnye i vyrozhdennye zadachi, Faktorial, M., 1997, 256 pp. | MR | Zbl

[18] Mordukhovich B. Sh., Metody approksimatsii v zadachakh optimizatsii i upravleniya, Nauka, M., 1988, 360 pp. | MR | Zbl

[19] Klark F., Optimizatsiya i negladkii analiz, Nauka, M., 1988, 280 pp. | MR | Zbl

[20] Vinter R. B., Optimal control, Burkhäuser, Boston–Basel–Berlin, 2000, 504 pp. | MR

[21] Mordukhovich B. Sh., “Optimalnoe upravlenie raznostnymi, differentsialnymi i differentsialno-raznostnymi vklyucheniyami”, Itogi nauki i tekhn. Sovr. matem. i ee prilozh., 61, VINITI, M., 1999, 33–65 | MR

[22] Aseev S. M., “Zadacha optimalnogo upravleniya dlya differentsialnogo vklyucheniya s fazovym ogranicheniem. Gladkie approksimatsii i neobkhodimye usloviya optimalnosti”, Itogi nauki i tekhn. Sovr. matem. i ee prilozh., 64, VINITI, M., 1999, 57–81 | MR | Zbl

[23] Clarke F., “The maximum principle in optimal control: then and now”, J. Contr. and Cybern., 34:3 (2005), 709–722 | MR

[24] Clarke F., Necessary conditions in dynamic optimization, Mem. Amer. Math. Soc., 173, No 816, 2005, 113 pp. | MR

[25] Arutunov A., Karamzin D., Pereira F., “A nondegenerate maximum principle for optimal control problem with state constraints”, SIAM J. Contr. Optim., 43:5 (2005), 1812–1843 | DOI | MR

[26] Dikusar V. V., Milyutin A. A., Kachestvennye i chislennye metody v printsipe maksimuma, Nauka, M., 1989, 144 pp. | MR | Zbl

[27] Milyutin A. A., Ilyutovich A. E., Osmolovskii N. P., Chukanov S. V., Optimalnoe upravlenie v lineinykh sistemakh, Nauka, M., 1993, 268 pp. | MR | Zbl

[28] V. A. Baturin, V. A. Dykhta, A. I. Moskalenko i dr., Metody resheniya zadach teorii upravleniya na osnove printsipa rasshireniya, Nauka, Novosibirsk, 1990, 190 pp. | MR | Zbl

[29] Milyutin A. A., “Calculus of variations and optimal control”, Proc. of the Int. Conf. on the calculus of variations and related topics, CRC Research Notes in Math. Series, 411, Chapman and Hall, Haifa, 1999, 159–172 | MR

[30] Milyutin A. A., Osmolovskii N. P., Calculus of variation and optimal control, Amer. Math. Soc., Providence, RI, 1998, 372 pp. | MR | Zbl

[31] Gurman V. I., Printsip rasshireniya v zadachakh upravleniya, Nauka, Fizmatlit, M., 1997, 288 pp. | MR | Zbl

[32] Caratheodory C., Calculus of variations and partial differential equations of the first order, V. 2, Holden-Day, San Francisco, 1965, 224 pp. | MR | Zbl

[33] Yang L., Lektsii po variatsionnomu ischisleniyu i teorii optimalnogo upravleniya, Mir, M., 1974, 488 pp. | MR

[34] Krotov V. F., Global methods in optimal control theory, Marcel Dekker, New York, 1996, 408 pp. | MR | Zbl

[35] Krotov V. F., Gurman V. I., Metody i zadachi optimalnogo upravleniya, Nauka, M., 1973, 448 pp. | MR | Zbl

[36] Clarke F. H., Ledyaev Yu. S., Stern B. J., “Proximal analysis and feedback consruction”, Tr. in-ta matem. i mekhan., 6, no. 1-2, 2000, 91–109

[37] Clarke F. H., Ledyaev Yu. S., Stern B. J., “Invariance, monotonicity and applications”, Nonlinear analysis, differential equations and control (NATO ASI, Montreal, 1998), eds. F. H. Clarke, B. J. Stern, Kluwer Acad. Publ., Dordreht, 1999, 207–305 | MR | Zbl

[38] Khrustalev M. M., “Tochnoe opisanie mnozhestv dostizhimosti i uslovie globalnoi optimalnosti dinamicheskikh sistem. I. Otsenki i tochnoe opisanie mnozhestv dostizhimosti i upravlyaemosti”, Avtomatika i telemekhan., 1988, no. 5, 62–71 | MR

[39] Khrustalev M. M., “Tochnoe opisanie mnozhestv dostizhimosti i uslovie globalnoi optimalnosti dinamicheskikh sistem. II. Usloviya globalnoi optimalnosti”, Avtomatika i telemekhan., 1988, no. 7, 70–80 | MR

[40] Clarke F. H., “Nonsmooth analysis in control theory: a survey”, European J. Control. Fundamental issues in control, 7:2–3 (2001), 145–159

[41] Pereira F. L., “Control design for autonomous vehicles: a dynamic optimization perspective”, European J. Control. Fundamental issues in control, 7:2–3 (2001), 178–202

[42] Subbotin A. I., Minimaksnye neravenstva i uravneniya Gamiltona–Yakobi, Nauka, M., 1991, 216 pp. | MR | Zbl

[43] Subbotin A. I., “Minimaksnye resheniya uravnenii Gamiltona–Yakobi”, Itogi nauki i tekhn. Sovr. matem. i ee prilozh., 64, VINITI, M., 1999, 222–231 | MR | Zbl

[44] Cannarsa P., Sinestrari C., Semiconcave functions, Hamilton–Jacobi equations and optimal control, Progress in nonlinear differential equations and their appications, 58, Birkhäuser, Boston, 2004, 304 pp. | MR | Zbl

[45] Ushakov V. N., Chentsov A. G., “Andrei Izmailovich Subbotin”, Tr. in-ta matem. i mekhan., 6:1-2 (2000), 3–26 | MR

[46] Chernousko F. L., Otsenivanie fazovogo sostoyaniya dinamicheskikh sistem. Metod ellipsoidov, Nauka, M., 1988, 320 pp. | MR

[47] Chernousko F. L., Banichuk N. V., Variatsionnye zadachi mekhaniki i upravleniya (chislennye metody), Nauka, M., 1973, 238 pp.

[48] Gabasov R., Kirillova F. M., Kachestvennaya teoriya optimalnykh protsessov, Nauka, M., 1971, 508 pp. | MR | Zbl

[49] Kirin N. E., Vychislitelnye metody teorii optimalnogo upravleniya, Izd-vo LGU, L., 1968, 144 pp. | MR

[50] Vasilev O. V., Lektsii po metodam optimizatsii, Izd-vo Irkutsk. un-ta, Irkutsk, 1994, 344 pp.

[51] Vasilev O. V., Arguchintsev A. V., Terletskii V. A., “Metody optimizatsii sistem s sosredotochennymi i raspredelennymi parametrami, osnovannye na dopustimykh variatsiyakh”, Tr. 12-i Baikalskoi mezhd. konf. “Metody optimizatsii i ikh prilozh.”, Plenarn. dokl., Irkutsk, 2001, 52–68

[52] Tyatyushkin A. I., Chislennye metody i programmnye sredstva optimizatsii upravlyaemykh sistem, Nauka, Novosibirsk, 1992, 193 pp. | MR | Zbl

[53] Srochko V. A., Iteratsionnye metody resheniya zadach optimalnogo upravleniya, Fizmatlit, M., 2000, 160 pp.

[54] Vasilev O. V., Dykhta V. A., Srochko V. A., “Zadachi optimalnogo upravleniya: variatsionnyi printsip maksimuma i metody chislennogo resheniya”, Nelineinaya teoriya upravleniya i ee prilozh., eds. V. M. Matrosov, S. N. Vasilev, A. I. Moskalenko, Fizmatlit, M., 2000, 320 pp.

[55] Fedorenko R. P., Priblizhennoe reshenie zadach optimalnogo upravleniya, Nauka, M., 1978, 488 pp. | MR | Zbl

[56] Vasilev F. P., Metody resheniya ekstremalnykh zadach, Nauka, M., 1981, 400 pp. | MR

[57] Demyanov V. F., Rubinov A. M., Priblizhennye metody resheniya ekstremalnykh zadach, Izd-vo LGU, L., 1968, 180 pp. | MR

[58] Tyatyushkin A. I., Mnogometodnaya tekhnologiya optimizatsii upravlyaemykh sistem, Nauka, Novosibirsk, 2006, 343 pp. | Zbl

[59] Baturin V. A., Urbanovich D. E., Priblizhennye metody optimalnogo upravleniya, osnovannye na printsipe rasshireniya, Nauka, Novosibirsk, 1997, 175 pp. | MR | Zbl

[60] Gabasov R., Kirillova F. M., Konstruktivnye metody optimizatsii. Ch. 2. Zadachi upravleniya, Izd-vo “Universitetskoe”, Minsk, 1984, 207 pp. | MR | Zbl

[61] Gabasov R., Kirillova F. M., “Optimalnoe upravlenie v rezhime realnogo vremeni”, Vtoraya mezhd. konf. po problemam upravleniya, Plenarn. dokl., Institut problem upravleniya, M., 2003, 20–47

[62] Moiseev N. N., Elementy teorii optimalnykh sistem, Nauka, M., 1975, 528 pp. | MR | Zbl

[63] Evtushenko Yu. G., Metody resheniya ekstremalnykh zadach i ikh primenenie v sistemakh optimizatsii, Nauka, M., 1982, 432 pp. | MR | Zbl

[64] Ermolev Yu. M., Gulenko V. P., Tsarenko T. I., Konechno-raznostnyi metod v zadachakh optimalnogo upravleniya, Nauk. dumka, Kiev, 1978, 164 pp. | MR

[65] Sirazetdinov T. K., Optimizatsiya sistem s raspredelennymi parametrami, Nauka, M., 1977, 480 pp. | MR

[66] Egorov A. I., Osnovy teorii upravleniya, Fizmatlit, M., 2004, 504 pp. | Zbl

[67] Ostrovskii G. M., Volin Yu. M., Metody optimizatsii khimicheskikh reaktorov, Khimiya, M., 1967, 248 pp.

[68] Vasilev O. V., Srochko V. A., Terletskii V. A., Metody optimizatsii i ikh prilozheniya. Ch. 2. Optimalnoe upravlenie, Nauka, Novosibirsk, 1990, 151 pp. | MR

[69] Srochko V. A., “Printsip maksimuma dlya odnogo klassa sistem s raspredelennymi parametrami”, Voprosy ustoichivosti i optimizatsii dinamicheskikh sistem, Irkutsk, 1983, 170–182 | Zbl

[70] Srochko V. A., “Usloviya optimalnosti tipa printsipa maksimuma v sistemakh Gursa–Darbu”, Sib. matem. zhurn., 25:1 (1984), 126–133 | MR

[71] Srochko V. A., Variatsionnyi printsip maksimuma i metody linearizatsii v zadachakh optimalnogo upravleniya, Izd-vo Irkut. un-ta, Irkutsk, 1989, 160 pp. | MR

[72] Terletskii V. A., “Variatsionnyi printsip maksimuma v upravlyaemykh sistemakh odnomernykh giperbolicheskikh uravnenii”, Izv. vuzov. Matematika, 1999, no. 12, 82–90 | MR | Zbl

[73] Bokmelder E. P., Dykhta V. A., “K teorii printsipa maksimuma dlya upravlyaemykh sistem giperbolicheskogo tipa”, Teor. i prikl. voprosy optimalnogo upravleniya, Novosibirsk, 1985, 41–58 | MR

[74] Bokmelder E. P., Dykhta V. A., “Printsip maksimuma dlya polulineinykh giperbolicheskikh sistem pri funktsionalnykh ogranicheniyakh”, Differents. uravneniya i chislennye metody, Nauka, Novosibirsk, 1986, 200–207 | MR

[75] Bokmelder E. P., Dykhta V. A., Moskalenko A. I., Ovsyannikova N. A., Usloviya ekstremuma i konstruktivnye metody resheniya v zadachakh optimizatsii giperbolicheskikh sistem, Nauka, Novosibirsk, 1993, 197 pp. | MR

[76] Arguchintsev A. V., “Neklassicheskoe uslovie optimalnosti v zadache upravleniya granichnymi usloviyami polulineinoi giperbolicheskoi sistemy”, Izv. vuzov. Matematika, 1994, no. 1, 3–11 | MR | Zbl

[77] Arguchintsev A. V., “Reshenie zadachi optimalnogo upravleniya nachalno-kraevymi usloviyami giperbolicheskoi sistemy na osnove tochnykh formul prirascheniya”, Izv. vuzov. Matematika, 2002, no. 12, 23–29 | MR | Zbl

[78] Arguchintsev A. V., “Optimizatsiya giperbolicheskikh sistem s upravlyaemymi nachalno-kraevymi usloviyami v vide differentsialnykh svyazei”, Zhurn. vychisl. matem. i matem. fiz., 44:2 (2004), 287–296 | MR

[79] Arguchintsev A. V., Optimalnoe upravlenie giperbolicheskimi sistemami, Fizmatlit, M., 2007, 168 pp.

[80] Dykhta V. A., “Variatsionnyi printsip maksimuma i kvadratichnye usloviya optimalnosti impulsnykh i osobykh protsessov”, Sib. matem. zhurn., 35:1 (1994), 70–82 | MR | Zbl

[81] Dykhta V. A., “Variatsionnyi printsip maksimuma dlya klassicheskikh zadach optimalnogo upravleniya”, Avtomatika i telemekhan., 2002, no. 4, 47–54 | MR | Zbl

[82] Dykhta V. A., Samsonyuk O. N., Optimalnoe impulsnoe upravlenie s prilozheniyami, Fizmatlit, M., 2003, 256 pp. | MR | Zbl

[83] Aschepkov L. T., Optimalnoe upravlenie razryvnymi sistemami, Nauka, Novosibirsk, 1987, 226 pp. | Zbl

[84] Clarke F. H., Vinter R. B., “Applications of optimal multiprocesses”, SIAM J. Contr. Optim., 27:5 (1989), 1048–1071 | DOI | MR | Zbl

[85] Clarke F. H., Vinter R. B., “Optimal multiprocesses”, SIAM J. Contr. Optim., 27:5 (1989), 1072–1091 | DOI | MR | Zbl

[86] Miller B. M., Rabinovich E. Ya., Optimizatsiya dinamicheskikh sistem s impulsnymi upravleniyami, Nauka, M., 2005, 429 pp.

[87] Dmitruk A. V., Kaganovich A. M., Printsip maksimuma dlya zadach optimalnogo upravleniya s promezhutochnymi ogranicheniyami, Nelineinye dinamicheskie sistemy i upravlenie, 6, Nauka, M., 2008

[88] Levitin E. S., Milyutin A. A., Osmolovskii N. P., “Usloviya vysshikh poryadkov v zadachakh s ogranicheniyami”, UMN, 33:6 (1978), 85–148 | MR | Zbl

[89] Arutyunov A. V., Vinter R. B., “Metod konechnoi approksimatsii v teorii optimalnogo upravleniya”, Differents. uravneniya, 39:11 (2003), 1443–1451 | MR | Zbl

[90] Arutunov A. V., Vinter R. B., “A simple “finite approximation” proof of the Pontryagin maximum principle under reduced differentiability hypotheses”, Set-valued analysis, 12:1–2 (2004), 5–24 | DOI | MR

[91] Sussman H. J., “Geometry and optimal control”, Mathematical control theory, eds. J. Baillieul, J. C. Willems, Springer, New York, 1998, 140–198 | MR | Zbl

[92] Sussman H. J., “New theories of set-valued differentionals and new versions of the maximum principle of optimal control theory”, Nonlinear control in the year 2000, eds. A. Isidory, F. Lamanbhi-Lagarrique, W. Respondek, Springer, London, 2000, 487–526 | MR

[93] Dmitruk A. V., “Printsip maksimuma dlya obschei zadachi optimalnogo upravleniya s fazovymi i smeshannymi ogranicheniyami”, Optimalnost upravlyaemykh dinamicheskikh sistem, Vyp. 14, VNIISI, M., 1993, 26–42

[94] Milyutin A. A., “Vypukloznachnye lipshitsevy differentsialnye vklyucheniya i printsip maksimuma Pontryagina”, Itogi nauki i tekhn. Sovr. matem. i ee prilozh., 65, 1999, 175–187 | MR

[95] Sussmann H. J., “Needle variations and almost lower semicontinuous differential inclussions”, Set-valued analysis, 10:2–3 (2002), 233–285 | DOI | MR | Zbl

[96] Aseev S. M., Kryazhimskii A. V., Printsip maksimuma Pontryagina i zadachi optimalnogo ekonomicheskogo rosta, Tr. MIAN, 257, Nauka, M., 2007, 272 pp. | MR

[97] Arutyunov A. V., “Printsip maksimuma Pontryagina i dostatochnye usloviya dlya nelineinykh zadach”, Differents. uravneniya, 39:12 (2003), 1587–1595 | MR | Zbl

[98] Guseinov Kh. G., Ushakov V. N., “Silno i slabo invariantnye mnozhestva otnositelno differentsialnogo vklyucheniya, ikh proizvodnye i primenenie k zadacham upravleniya”, Differents. uravneniya, 26:11 (1990), 1888–1894 | MR | Zbl

[99] Donchev T., Rios V., Wolenski P., “Strong invariance and one-sided Lipschitz multifunctions”, Nonlinear anal., 60:5 (2005), 849–862 | DOI | MR | Zbl

[100] Aubin J.-P., Cellina A., Differential inclusions, Springer-Verlag, Berlin, 1984, 342 pp. | MR

[101] Kurzhanskii A. B., Filippova T. F., “On the theory of trajectory tubes – a mathematical formalism for uncertain dynamics, viability and control”, Advances in nonlinear dynamics and control: a report from Russia, ed. A. B. Kurzhanskii, Burkhäuser, Boston, 1993, 122–188 | MR

[102] Moskalenko A. I., Metody nelineinykh otobrazhenii v optimalnom upravlenii, Nauka, Novosibirsk, 1983, 222 pp. | MR | Zbl

[103] Dykhta V. A., “Obschaya skhema preobrazovanii ekstremalnykh zadach i ee prilozheniya v optimalnom upravlenii”, Integrodifferents. uravneniya i ikh prilozh., Irkutsk, 1987, 82–91

[104] Dykhta V. A., Antipina N. V., “Dostatochnoe uslovie optimalnosti dlya zadach impulsnogo upravleniya”, Izv. RAN. Teoriya i sistemy upravleniya, 2004, no. 4, 76–83 | MR

[105] Milyutin A. A., “An example of an optimal control problem whose extremals possess a continual set of discontinuities of the control function”, Russian J. Math. Physics, 1:3 (1994), 397–402 | MR

[106] Kurzhanskii A. B., Valyi I., Ellipsoidal calculus for estimation and control, Birkhäuser, Boston, 1997, 220 pp. | MR

[107] Clarke F. H., “A proximal characterization of the reachable set”, System and Control Letters, 27 (1996), 195–197 | DOI | MR | Zbl

[108] Vinter R. B., “A characterization of the reachable set for nonlinear control system”, SIAM J. Contr. Optim., 18:6 (1980), 599–610 | DOI | MR | Zbl

[109] Cannarsa P., Frankowska H., “Some characterizations of optimal trajectories in control theory”, SIAM J. Contr. Optim., 29:6 (1991), 1322–1347 | DOI | MR | Zbl

[110] Cannarsa P., Frankowska H., Sinestrari S., “Optimality conditions and synthesis for the minimum time problem”, Set-valued analysis, 8:1–2 (2000), 127–148 | DOI | MR | Zbl

[111] Wolenski P., Shuang Yu., “Proximal analysis and the minimal time function”, SIAM J. Contr. Optim., 36:3 (1998), 1048–1072 | DOI | MR | Zbl

[112] Khrustalev M. M., “Neobkhodimye i dostatochnye usloviya optimalnosti v forme uravneniya Bellmana”, DAN SSSR, 242:5 (1978), 1023–1026 | MR | Zbl

[113] Motta M., Rampazzo F., “Dynamic programming for nonlinear system driven by ordinary and impulsive controls”, SIAM J. Contr. Optim., 34:1 (1996), 199–225 | DOI | MR | Zbl

[114] Stefanova A. V., “Uravnenie Gamiltona–Yakobi–Bellmana v nelineinoi zadache impulsnogo upravleniya”, Tr. in-ta matem. i mekhan., 5 (1998), 301–318 | Zbl

[115] Pereira F. L., Matos A. S., “Hamilton–Jacobs conditions for a measure differential inclusion control problem”, Proc. of 12th Baikal Internat. Conf. “Optimization methods and their applications”, Plenary Lect., Irkutsk, 2001, 237–245

[116] Vinter R. B., “Weakest conditions for existence of Lipschitz continuous Krotov functions in optimal control theory”, SIAM J. Contr. Optim., 21:2 (1983), 215–234 | DOI | MR | Zbl

[117] Vinter R. B., “New global optimality conditions in optimal control theory”, SIAM J. Contr. Optim., 21:2 (1983), 235–245 | DOI | MR | Zbl

[118] Aubin J.-P., Frankowska H., Set-valued analysis, Birkhäuser, Boston–Basel–Berlin, 1990, 461 pp. | MR | Zbl

[119] Krasovskii N. N., Subbotin A. I., Pozitsionnye differentsialnye igry, Nauka, M., 1974, 455 pp. | MR | Zbl

[120] Krylov I. A., Chernousko F. L., “O metode posledovatelnykh priblizhenii dlya resheniya zadach optimalnogo upravleniya”, Zhurn. vychisl. matem. i matem. fiz., 2:6 (1962), 1132–1139 | MR | Zbl

[121] Krylov I. A., Chernousko F. L., “Algoritm metoda posledovatelnykh priblizhenii dlya zadach optimalnogo upravleniya”, Zhurn. vychisl. matem. i matem. fiz., 12:1 (1972), 14–34 | Zbl

[122] Kelly H. J., Kopp R. E., Moyer H. G., “Successive approximation techniques for trajectory optimization”, Proc. Symp. on vehicle system optimization, New York, 1961, 360–391

[123] Velichenko V. V., “Chislennyi metod resheniya zadach optimalnogo upravleniya”, Zhurn. vychisl. matem. i matem. fiz., 6:4 (1966), 635–647 | Zbl

[124] Vasilev O. V., Tyatyushkin A. I., “Ob odnom metode resheniya zadach optimalnogo upravleniya, osnovannom na printsipe maksimuma”, Zhurn. vychisl. matem. i matem. fiz., 21:6 (1981), 1376–1384 | MR

[125] Tarasenko N. V., “Odin metod resheniya zadachi optimalnogo upravleniya na osnove integralnogo printsipa maksimuma”, Diskretnye i raspredelennye sistemy, Irkutsk, 1981, 142–150 | MR | Zbl

[126] Lyubushin A. A., Chernousko F. L., “Metod posledovatelnykh priblizhenii dlya rascheta optimalnogo upravleniya”, Izv. AN SSSR. Tekhn. kibernetika, 1983, no. 2, 147–159 | Zbl

[127] Vasiliev O. V., Optimization methods, World Federation Publishers Company Inc., Atlanta, 1996, 276 pp. | MR | Zbl

[128] Vasilev O. V., “Odno zamechanie k algoritmu posledovatelnykh priblizhenii, osnovannomu na printsipe maksimuma”, Differents. i integralnye uravneniya, Irkutsk, 1980, 167–178

[129] Antonik V. G., Srochko V. A., “K resheniyu zadach optimalnogo upravleniya na osnove metodov linearizatsii”, Zhurn. vychisl. matem. i matem. fiz., 32:7 (1992), 979–991 | MR | Zbl

[130] Antonik V. G., Srochko V. A., “Metod proektsii v lineino-kvadratichnykh zadachakh optimalnogo upravleniya”, Zhurn. vychisl. matem. i matem. fiz., 38:4 (1998), 564–572 | MR | Zbl

[131] Zakharchenko V. G., Srochko V. A., “Metod priraschenii dlya resheniya kvadratichnykh zadach optimalnogo upravleniya”, Izv. RAN. Teoriya i sistemy upravleniya, 1995, no. 6, 145–154

[132] Mamonova N. V., Srochko V. A., “Iteratsionnye protsedury resheniya zadach optimalnogo upravleniya na osnove kvazigradientnykh approksimatsii”, Izv. vuzov. Matematika, 2001, no. 12, 55–67 | MR | Zbl

[133] Srochko V. A., “Modernizatsiya metodov gradientnogo tipa v zadachakh optimalnogo upravleniya”, Izv. vuzov. Matematika, 2002, no. 12, 66–78 | MR | Zbl

[134] Boldyrev V. I., “Chislennoe reshenie zadachi optimalnogo upravleniya”, Izv. RAN. Teoriya i sistemy upravleniya, 2000, no. 3, 85–92 | MR

[135] Baturin V. A., “Priblizhennye metody resheniya zadach optimalnogo upravleniya na osnove dostatochnykh uslovii optimalnosti V. F. Krotova”, Tr. IX mezhd.Chetaevskoi konf. “Analiticheskaya mekhan., ustoichivost i upravlenie dvizheniem”, posvyaschennoi 105-letiyu N. G. Chetaeva, T. 3. Upravlenie i optimizatsiya, Irkutsk, 2007, 30–47

[136] Avvakumov S. N., Kiselev Yu. N., Orlov M. V., “Metody resheniya zadach optimalnogo upravleniya na osnove printsipa maksimuma Pontryagina”, Tr. MIAN, 211, Nauka, M., 1995, 3–31 | MR

[137] Strekalovskii A. S., Elementy nevypukloi optimizatsii, Nauka, Novosibirsk, 2003, 356 pp.

[138] Balashevich N. V., Gabasov R., Kirillova F. M., “Chislennye metody programmnoi i pozitsionnoi optimizatsii kusochno-lineinykh sistem”, Zhurn. vychisl. matem. i matem. fiz., 41:11 (2001), 1658–1674 | MR | Zbl

[139] Gabasov R., Kirillova F. M., Khomitskaya T. G., “Programmnoe i pozitsionnoe resheniya terminalnoi lineino-vypukloi zadachi optimalnogo upravleniya”, Izv. vuzov. Matematika, 2004, no. 12, 3–16 | MR

[140] Kaganovich S. L., “Optimizatsiya lineinykh sistem s peremennoi strukturoi”, Zhurn. vychisl. matem. i matem. fiz., 21:2 (1981), 306–314 | MR | Zbl

[141] Krotov V. F., Feldman I. N., “Iteratsionnyi metod resheniya zadach optimalnogo upravleniya”, Izv. AN SSSR. Tekhn. kibernetika, 1983, no. 2, 160–168 | MR

[142] Gabasov R., Kirillova F. M., Osobye optimalnye upravleniya, Nauka, M., 1973, 256 pp. | MR

[143] Filippov A. F., Differentsialnye uravneniya s razryvnoi pravoi chastyu, Nauka, M., 1985, 224 pp. | MR

[144] Rozonoer L. I., “Printsip maksimuma L. S. Pontryagina v teorii optimalnykh sistem. I”, Avtomatika i telemekhan., 20:10 (1959), 1320–1334 | MR