Volterra functional equations and optimization of distributed systems. Special optimal controls

V. I. Sumin

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh International Spring Mathematical School "Modern Methods of the Theory of Boundary-Value Problems. Pontryagin Readings – XXXII”, Voronezh, May 3–9, 2021, Part 2, Tome 209 (2022), pp. 53-76.

Résumé

The work is a brief review of some results of optimization theory obtained by using Volterra functional equations (VFE). We present the method proposed by the author for using the VFE-description of controlled initial-boundary-value problems for studying special controls on which necessary optimality conditions degenerate. Illustrative examples are given.

Détail

Keywords: Volterra functional equation, distributed control system, optimal control problem, necessary optimality conditions, maximum principle, special control.

Bibliographie

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

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

[3] Aschepkov L. T., Vasilev O. V., “Ob optimalnosti osobykh upravlenii v sistemakh Gursa—Darbu”, Zh. vychisl. mat. mat. fiz., 15:5 (1975), 1157–1167 | MR | Zbl

[4] Aschepkov L. T., Vasilev O. V., Kovalenok I. L., “Usilennoe uslovie optimalnosti osobykh upravlenii v sisteme Gursa—Darbu”, Differ. uravn., 16:6 (1980), 1054–1059 | MR | Zbl

[5] Burdukovskii A. N., “Usloviya optimalnosti osobykh upravlenii v zadache Gursa—Darbu”, Upravl. sist. (Novosibirsk)., 1986, no. 26, 16–24

[6] Bukhgeim A. L., Uravneniya Volterra i obratnye zadachi, Nauka, Novosibirsk, 1983

[7] Varga Dzh., Optimalnoe upravlenie differentsialnymi i funktsionalnymi uravneniyami, Nauka, M., 1977

[8] Vasilev O. V., “Ob optimalnosti osobykh upravlenii v sistemakh s raspredelennymi parametrami”, Upravl. sist. (Novosibirsk)., 1972, no. 10, 27–34

[9] Vasilev O. V., “Ob optimalnosti osobogo upravleniya v sistemakh s raspredelennymi parametrami”, Tez. dokl. II Vsesoyuz. konf. po problemam teoreticheskoi kibernetiki, Novosibirsk, 1971, 26–27 | MR

[10] Vasilev O. V., Kachestvennye i konstruktivnye metody optimizatsii upravlyaemykh protsessov s raspredelennymi parametrami, Diss. na soisk. uch. step. dokt. fiz.-mat. nauk, LGU, L., 1984 | MR

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

[12] Vasilev F. P., Metody optimizatsii. Ch. II, Izd-vo MTsNMO, M., 2011

[13] Gabasov R., Kirillova F. M., Osobye optimalnye upravleniya, Nauka, M., 1973

[14] Gabasov R., Kirillova F. M.. Mansimov K. B., Neobkhodimye usloviya optimalnosti vtorogo poryadka dlya sistem s raspredelennymi parametrami, In-t mat. AN BSSR, Minsk, 1982

[15] Gokhberg I. Ts., Krein M. G., Teoriya volterrovykh operatorov v gilbertovom prostranstve i ee prilozheniya, Nauka, M., 1967

[16] Gusarenko S. A., “Ob odnom obobschenii ponyatiya volterrova operatora”, Dokl. AN SSSR., 295:5 (1987), 1046–1049 | Zbl

[17] Danford N., Shvarts Dzh., Lineinye operatory. T. 1. Obschaya teoriya, IL, M., 1962

[18] Daletskii Yu. L., Krein M. G., Ustoichivost reshenii differentsialnykh uravnenii v banakhovom prostranstve, Nauka, M., 1970 | MR

[19] Zhukovskii E. S., “K teorii uravnenii Volterra”, Differ. uravn., 25:9 (1989), 1599–1605 | MR

[20] Zhukovskii E. S., Alvesh M. Zh., “Abstraktnye volterrovy operatory”, Izv. vuzov. Mat., 2008, no. 3, 3–17 | Zbl

[21] Zabreiko P. P., “Ob integralnykh operatorakh Volterra”, Usp. mat. nauk., 22:1 (1967), 167–168 | MR

[22] Zabreiko P. P. O spektralnom radiuse integralnykh operatorov Volterra, Lit. mat. sb., 7:2 (1967), 281–287 | MR | Zbl

[23] Zelikin M. I., Borisov V. F., “Osobye optimalnye rezhimy v zadachakh matematicheskoi ekonomiki”, Sovr. mat. prilozh., 11 (2003), 3–161 | Zbl

[24] Ioffe A. D., Tikhomirov V. M., Teoriya ekstremalnykh zadach, Nauka, M., 1974 | MR

[25] Kantorovich L. V., Akilov G. P., Funktsionalnyi analiz, Nauka, M., 1977 | MR

[26] Krasnoselskii M. A., Zabreiko P. P., Pustylnik E. I., Sobolevskii P. E., Integralnye operatory v prostranstvakh summiruemykh funktsii, Nauka, M., 1966 | MR

[27] Lions Zh.-L., Upravlenie singulyarnymi raspredelennymi sistemami, Nauka, M., 1987

[28] Lisachenko I. V., Sumin V. I., “Nelineinaya upravlyaemaya zadacha Gursa—Darbu: usloviya sokhraneniya globalnoi razreshimosti”, Differ. uravn., 47:6 (2011), 858–870 | MR | Zbl

[29] Lisachenko I. V., Sumin V. I., “Ob osobykh upravleniyakh printsipa maksimuma dlya zadachi optimizatsii sistemy Gursa—Darbu”, Vestn. Udmurt. un-ta. Ser. Mat. Mekh. Komp. naukiyu, 2015 25, no. 4, 483–491 | Zbl

[30] Lisachenko I. V., Sumin V. I., “Ob osobykh upravleniyakh printsipa maksimuma v terminalnoi zadache optimizatsii sistemy Gursa—Darbu”, Vestn. Tambov. un-ta. Estestv. tekhn. nauki., 20:5 (2015), 1264–1274

[31] Litsyna E., “Lineinye operatory Volterra v nekotorykh metricheskikh prostranstvakh”, Tr. semin. im. I. G. Petrovskogo., 26:3 (2007), 223–240

[32] Mansimov K. B., “K teorii neobkhodimykh uslovii optimalnosti v odnoi zadache upravleniya sistemami s raspredelennymi parametrami”, Dokl. AN SSSR., 301:3 (1988), 546–550

[33] Mansimov K. B., “Osobye upravleniya v zadachakh upravleniya sistemami s raspredelennymi parametrami”, Sovr. mat. prilozh., 42 (2006), 39–83

[34] Mansimov K. B., Mardanov M. Dzh., Kachestvennaya teoriya optimalnogo upravleniya sistemami Gursa—Darbu, Elm, Baku, 2010

[35] Melikov T. K., Issledovanie osobykh protsessov v nekotorykh optimalnykh sistemakh, Diss. na soisk. uch. step. kand. fiz.-mat. nauk, Baku, 1976

[36] Mikhailov V. P., Differentsialnye uravneniya v chastnykh proizvodnykh, Nauka, M., 1976

[37] Mordukhovich B. Sh., Metody approksimatsii v zadachakh optimizatsii i upravleniya, Nauka, M., 1988 | MR

[38] Plotnikov V. I., “Neobkhodimye usloviya optimalnosti dlya upravlyaemykh sistem obschego vida”, Dokl. AN SSSR., 199:2 (1971), 275–278 | Zbl

[39] Plotnikov V. I., Sumin V. I., “Optimizatsiya raspredelennykh sistem v lebegovom prostranstve”, Sib. mat. zh., 22:6 (1981), 142–161 | MR | Zbl

[40] Srochko V. A., “Usloviya optimalnosti dlya odnogo klassa sistem s raspredelennymi parametrami”, Sib. mat. zh., 17:5 (1976), 1108–1115 | MR | Zbl

[41] Sumin V. I., Optimizatsiya upravlyaemykh obobschennykh volterrovykh sistem, Diss. na soisk. uch. step. kand. fiz.-mat. nauk, Gorkii, 1975 | Zbl

[42] Sumin V. I., “Funktsionalno-operatornye volterrovy uravneniya v teorii optimalnogo upravleniya raspredelennymi sistemami”, Dokl. AN SSSR., 305:5 (1989), 1056–1059 | Zbl

[43] Sumin V. I., Funktsionalnye volterrovy uravneniya v teorii optimalnogo upravleniya raspredelennymi sistemami. Ch. 1, Izd-vo NNGU, N. Novgorod, 1992

[44] Sumin V. I., “O funktsionalnykh volterrovykh uravneniyakh”, Izv. vuzov. Mat., 1995, no. 9, 67–77 | Zbl

[45] Sumin V. I., “O rasshirenii optimizatsionnykh zadach, svyazannykh s funktsionalnymi uravneniyami v prostranstvakh suschestvenno ogranichennykh funktsii”, Vestn. Nizhegorod. un-ta. Mat. model. Optim. upravl., 1998, no. 1 (18), 126–133 | Zbl

[46] Sumin V. I., “Problema ustoichivosti suschestvovaniya globalnykh reshenii upravlyaemykh kraevykh zadach i volterrovy funktsionalnye uravneniya”, Vestn. Nizhegorod. un-ta. Mat., 2003, no. 1, 91–107 | Zbl

[47] Sumin V. I., “Upravlyaemye funktsionalnye volterrovy uravneniya v lebegovykh prostranstvakh”, Vestn. Nizhegorod. un-ta. Mat. model. optim. upravl., 1998, no. 2 (19), 138–151

[48] Sumin V. I., Funktsionalnye volterrovy uravneniya v matematicheskoi teorii optimalnogo upravleniya raspredelennymi sistemami, Diss. na soisk. uch. step. dokt. fiz.-mat. nauk, Nizhegorod. un-t im. N. I. Lobachevskogo, N. Novgorod, 1998

[49] Sumin V. I., “Ravnostepennaya kvazinilpotentnost: opredeleniya, priznaki, primery primeneniya”, Vestn. Tambov. un-ta. Estestv. tekhn. nauki., 15:1 (2010), 453–466

[50] Sumin V. I., “Volterrovy funktsionalnye uravneniya v probleme ustoichivosti suschestvovaniya globalnykh reshenii raspredelennykh upravlyaemykh sistem”, Vestn. ross. un-tov. Mat., 25:132 (2020), 422–440 | Zbl

[51] Sumin V. I., “Ob obosnovanii gradientnykh metodov dlya raspredelennykh zadach optimalnogo upravleniya”, Zh. vychisl. mat. mat. fiz., 30:1 (1990), 3–21 | MR | Zbl

[52] Sumin V. I., “O dostatochnykh usloviyakh ustoichivosti suschestvovaniya globalnykh reshenii upravlyaemykh kraevykh zadach”, Differ. uravn., 26:12 (1990), 2097–2109 | MR | Zbl

[53] Sumin V. I., “Funktsionalno-operatornye uravneniya Volterra i ustoichivost suschestvovaniya globalnykh reshenii kraevykh zadach”, Ukr. mat. zh., 43:4 (1991), 555–561 | MR | Zbl

[54] Sumin V. I., “Ob ustoichivosti suschestvovaniya globalnogo resheniya pervoi kraevoi zadachi dlya upravlyaemogo parabolicheskogo uravneniya”, Differ. uravn., 22:9 (1986), 1587–1595 | MR

[55] Sumin V. I., “K probleme singulyarnosti raspredelennykh upravlyaemykh sistem, I”, Vestn. Nizhegorod. un-ta. Mat. model. optim. upravl., 1999, no. 2 (21), 145–155

[56] Sumin V. I., “K probleme singulyarnosti raspredelennykh upravlyaemykh sistem, II”, Vestn. Nizhegorod. un-ta. Mat. model. optim. upravl., 2001, no. 1 (23), 198–204 | Zbl

[57] Sumin V. I., “K probleme singulyarnosti raspredelennykh upravlyaemykh sistem, III”, Vestn. Nizhegorod. un-ta. Mat. model. optim. upravl., 2002, no. 1 (25), 164–174

[58] Sumin V. I., “Volterrovy funktsionalnye uravneniya i optimizatsiya raspredelennykh sistem”, Mat. Mezhdunaryu konf. «Sovremennye metody teorii kraevykh zadach. Voronezh. vesennyaya mat. shkola Pontryaginskie chteniya–XXXII» (Voronezh, 3–9 maya 2021 g.), Voronezh, 2021, 239–242

[59] Sumin V. I., “Volterrovy funktsionalno-operatornye uravneniya i raspredelennye zadachi optimizatsii”, Vestn. Tambov. un-ta. Estestv. tekhn. nauki., 23:124 (2018), 745–756

[60] Sumin V. I., “Upravlyaemye volterrovy funktsionalnye uravneniya i printsip szhimayuschikh otobrazhenii”, Tr. IMM UrO RAN., 25:1 (2019), 262–278 | MR

[61] Sumin V. I., Chernov A. V., “Operatory v prostranstvakh izmerimykh funktsii: volterrovost i kvazinilpotentnost”, Differ. uravn., 34:10 (1998), 1402–1411 | MR | Zbl

[62] Sumin V. I., “Usloviya ustoichivosti suschestvovaniya globalnykh reshenii upravlyaemykh kraevykh zadach dlya nelineinykh parabolicheskikh uravnenii”, Vestn. Tambov. un-ta. Estestv. tekhn. nauki., 5:4 (2000), 493–495

[63] Sumin V. I., “Silnoe vyrozhdenie osobykh upravlenii v raspredelennykh zadachakh optimizatsii”, Dokl. AN SSSR., 320:2 (1991), 295–299 | MR | Zbl

[64] Sumin V. I., “Silnoe vyrozhdenie osobykh upravlenii v zadachakh optimizatsii raspredelennykh sistem”, Optimizatsiya. Sb. nauchn. tr., 1993, no. 52 (69), 74–94, In-t matematiki SO RAN, Novosibirsk

[65] Sumin V. I., “Ob osobykh upravleniyakh potochechnogo printsipa maksimuma v raspredelennykh zadachakh optimizatsii”, Vestn. Udmurt. un-ta. Ser. Mat. Mekh. Komp. nauki., 2010, no. 3, 70–80

[66] Sumin V. I., Chernov A. V., Volterrovy operatornye uravneniya v banakhovykh prostranstvakh: ustoichivost suschestvovaniya globalnykh reshenii, Dep. v VINITI 25.04.2000. No 1198-V00

[67] Sumin V. I., Chernov A. V., “O dostatochnykh usloviyakh ustoichivosti suschestvovaniya globalnykh reshenii volterrovykh operatornykh uravnenii”, Vestn. Nizhegorod. un-ta. Mat. model. optim. upravl., 2003, no. 1 (26), 39–49

[68] Sumin V. I., Chernov A. V., “Volterrovy funktsionalno-operatornye uravneniya v teorii optimizatsii raspredelennykh sistem”, Tr. Mezhdunar. konf. «Dinamika sistem i protsessy upravleniya», posv. 90-letiyu so dnya rozhd. akad. N. N. Krasovskogo (Ekaterinburg, 2014 g.), Izd-vo IMM UrO RAN, Ekaterinburg, 2015, 293–300

[69] Tikhonov A. N., “O funktsionalnykh uravneniyakh tipa Volterra i ikh primeneniyakh k nekotorym zadacham matematicheskoi fiziki”, Byull. MGU. Sekts. A., 1:8 (1938), 1–25

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

[71] Krein S. G. (red.)., Funktsionalnyi analiz, Nauka, M., 1972 | MR

[72] Chernov A. V., Volterrovy operatornye uravneniya i ikh primenenie v teorii optimizatsii giperbolicheskikh sistem, Diss. na soisk. uch. step. kand. fiz.-mat. nauk, Nizhnii Novgorod, 2000

[73] Chernov A. V., “O sokhranenii razreshimosti polulineinogo uravneniya globalnoi elektricheskoi tsepi”, Zh. vychisl. mat. mat. fiz., 58:12 (2018), 2095–2111 | Zbl

[74] Shefer Kh., Topologicheskie vektornye prostranstva, Mir, M., 1971

[75] Shragin I. V., “Abstraktnye operatory Nemytskogo — lokalno opredelennye operatory”, Dokl. AN SSSR., 227:1 (1976), 47–49 | MR | Zbl

[76] Chernov A. V. On Volterra functional operator games on a given set, Automat. Remote Control., 75:4 (2014), 787–803 | DOI | MR

[77] Corduneanu C., Integral Equations and Applications, Cambridge Univ. Press, Cambridge, 1991 | MR | Zbl

[78] Drakhlin M. E., Ponosov A., Stepanov E., “On some classes of operators determined by the structure of their memory”, Proc. Edinburgh Math. Soc., 45:2 (2002), 467–490 | DOI | MR | Zbl

[79] Drakhlin M. E., Litsyn E., “Volterra operator: Back to the future”, J. Nonlin. Conv. Anal., 6:3 (2005), 375–391 | MR | Zbl

[80] Sumin V. Volterra functional-operator equations in the theory of optimal control of distributed systems, IFAC-PapersOnLine., 51:32 (2018), 759–764 | DOI

[81] Tonelli L., “Sulle equazioni funzionali di Volterra”, Bull. Calcutta Math. Soc., 20 (1929), 31–48

[82] Väth M., “Abstract Volterra equations of the second kind”, J. Equat. Appl., 10:9 (1998), 125–144 | MR