On obeying constraints of asymptotic character

A. G. Chentsov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and topology. II, Tome 271 (2010), pp. 59-75.

Résumé

We consider an abstract attainability problem under constraints of asymptotic character and describe a general approach to constructing “nonsequential” attraction sets in the space of generalized elements formalized as finitely additive measures. We also study the existence and structure of an asymptotic formula universal in the range of “asymptotic constraints” and topologies of the space of generalized elements in the case when the space of ordinary solutions is not necessarily compactifiable.

Bibliographie

[1] Varga Dzh., Optimalnoe upravlenie differentsialnymi i funktsionalnymi uravneniyami, Nauka, M., 1977 | MR

[2] Gamkrelidze R.V., Osnovy optimalnogo upravleniya, Izd-vo Tbil. un-ta, Tbilisi, 1975 | MR

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

[4] Krasovskii N.N., Upravlenie dinamicheskoi sistemoi: Zadacha o minimume garantirovannogo rezultata, Nauka, M., 1985 | MR

[5] Subbotin A.I., Chentsov A.G., Optimizatsiya garantii v zadachakh upravleniya, Nauka, M., 1981 | MR | Zbl

[6] Krasovskii N.N., Teoriya upravleniya dvizheniem: Lineinye sistemy, Nauka, M., 1968 | MR

[7] Chentsov A.G., Konechno-additivnye mery i relaksatsii ekstremalnykh zadach, Nauka, Ekaterinburg, 1993

[8] Chentsov A.G., Asymptotic attainability, Kluwer, Dordrecht, 1997 | MR | Zbl

[9] Chentsov A.G., Morina S.I., Extensions and relaxations, Kluwer, Dordrecht, 2002 | MR | Zbl

[10] Chentsov A.G., Sovr. matematika i ee pril., 17, Inst. kibern. Akad. nauk Gruzii, Tbilisi, 2004 ; Chentsov A.G., “Finitely additive measures and extensions of abstract control problems”, J. Math. Sci., 133:2 (2006), 1045–1206 | DOI | MR | Zbl

[11] Chentsov A.G., “K voprosu o korrektnom rasshirenii odnoi zadachi o vybore plotnosti veroyatnosti pri ogranicheniyakh na sistemu matematicheskikh ozhidanii”, UMN, 50:5 (1995), 223–242 | MR | Zbl

[12] Chentsov A.G., “K voprosu o korrektnom rasshirenii nekotorykh neustoichivykh zadach upravleniya s integralnymi ogranicheniyami”, Izv. RAN. Ser. mat., 63:3 (1999), 185–223 | DOI | MR | Zbl

[13] Chentsov A.G., “K voprosu o postroenii korrektnykh rasshirenii v klasse konechno-additivnykh mer”, Izv. vuzov. Matematika, 2002, no. 2, 58–80 | MR | Zbl

[14] Chentsov A.G., “K voprosu o kompaktifikatsii puchka traektorii odnoi abstraktnoi upravlyaemoi sistemy”, Izv. vuzov. Matematika, 2006, no. 5, 55–66 | MR | Zbl

[15] Chentsov A.G., “Finitely additive measures and extension constructions”, Atti Sem. Mat. Fis. Univ. Modena, 49 (2001), 531–545 | MR | Zbl

[16] Chentsov A.G., “Asimptoticheski dostizhimye elementy i ikh obobschennoe predstavlenie v klasse konechno-additivnykh mer”, Tr. In-ta matematiki i mekhaniki UrO RAN, 3, 1995, 211–244 | MR | Zbl

[17] Chentsov A.G., “Vektornye konechno-additivnye mery i voprosy regulyarizatsii zadachi o postroenii mnozhestv asimptoticheskoi dostizhimosti”, Tr. In-ta matematiki i mekhaniki UrO RAN, 4, 1996, 266–295 | MR | Zbl

[18] Daffin R.Dzh., “Beskonechnye programmy”, Lineinye neravenstva i smezhnye voprosy, Izd-vo inostr. lit., M., 1959, 263–267

[19] Golshtein E.G., Teoriya dvoistvennosti v matematicheskom programmirovanii i ee prilozheniya, Nauka, M., 1971 | MR

[20] Kuratovskii K., Mostovskii A., Teoriya mnozhestv, Mir, M., 1970 | MR

[21] Burbaki N., Obschaya topologiya: Osnovnye struktury, Nauka, M., 1968 | MR

[22] Engelking R., Obschaya topologiya, Mir, M., 1986 | MR

[23] Kelli Dzh.L., Obschaya topologiya, Nauka, M., 1981 | MR

[24] Chentsov A.G., “Nesekventsialnye priblizhennye resheniya v abstraktnykh zadachakh upravleniya”, Teoriya upravleniya i teoriya obobschennykh reshenii uravnenii Gamiltona–Yakobi, Tr. Mezhdunar. sem., v. 1, Izd-vo Ural. un-ta, Ekaterinburg, 2006, 48–60 | MR

[25] Chentsov A.G., “Nekotorye konstruktsii asimptoticheskogo analiza, svyazannye s kompaktifikatsiei Stouna–Chekha”, Sovr. matematika i ee pril., 26 (2005), 119–150 | MR

[26] Chentsov A.G., Elementy konechno-additivnoi teorii mery, v. 1, UGTU–UPI, Ekaterinburg, 2008

[27] Danford N., Shvarts Dzh., Lineinye operatory: Obschaya teoriya, Izd-vo inostr. lit., M., 1962

[28] Nevë Zh., Matematicheskie osnovy teorii veroyatnostei, Mir, M., 1969 | MR | Zbl

[29] Melentsov A.A., Baidosov V.A., Zmeev G.M., Elementy teorii mery i integrala, Izd-vo Ural. gos. un-ta, Sverdlovsk, 1980

[30] Chentsov A.G., “Rasshireniya v klasse konechno-additivnykh mer i usloviya asimptoticheskoi nechuvstvitelnosti pri oslablenii chasti ogranichenii”, Vestn. Udm. un-ta. Matematika. Mekhanika. Kompyut. nauki, 2009, no. 1, 131–152

[31] Chentsov A.G., “Universalnaya asimptoticheskaya realizatsiya integralnykh ogranichenii i konstruktsii rasshireniya v klasse konechno-additivnykh mer”, Tr. In-ta matematiki i mekhaniki UrO RAN, 5, 1998, 328–356 | MR | Zbl

[32] Chentsov A.G., “Obobschennye mnozhestva prityazheniya i priblizhennye resheniya, ikh formiruyuschie”, Tr. In-ta matematiki i mekhaniki UrO RAN, 10, no. 2, 2004, 178–196 | Zbl

[33] Chentsov A.G., Tarasova S.I., “Extensions of abstract analogs of unstable control problems”, Funct. Diff. Equat., 16:2 (2009), 237–261 | MR | Zbl

[34] Chentsov A.G., “Rasshireniya abstraktnykh zadach o dostizhimosti: nesekventsialnaya versiya”, Tr. In-ta matematiki i mekhaniki UrO RAN, 13, no. 2, 2007, 184–217