Ultrafilters of measurable spaces as generalized solutions in abstract attainability problems

A. G. Chentsov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 17 (2011) no. 1, pp. 268-293

Résumé

We consider problems of asymptotic analysis that arise, in particular, in the formalization of effects related to an approximate observation of constraints. We study nonsequential (generally speaking) variants of asymptotic behavior that can be formalized in the class of ultrafilters of an appropriate measurable space. We construct attraction sets in a topological space that are realized in the class of ultrafilters of the corresponding measurable space and specify conditions under which ultrafilters of a measurable space are sufficient for constructing the complete attraction set corresponding to applying ultrafilters of the family of all subsets of the space of ordinary solutions. We study a compactification of this space that is constructed in the class of Stone ultrafilters (ultrafilters of a measurable space with an algebra of sets) such that the attraction set is realized as a continuous image of the compact set of generalized solutions; we also study the structure of this compact set in terms of free ultrafilters and ordinary solutions that observe the constraints of the problem exactly. We show that, in the case when exact ordinary solutions are absent, this compact set consists of free ultrafilters only; i.e., it is contained in the remainder of the compactificator (an example is given that shows the possibility of the absence of the similar property for other variants of extending the original problem).

Détail

Keywords: attraction set, extension, ultrafilter.

Bibliographie

[1] Chentsov A. G., “Nekotorye konstruktsii asimptoticheskogo analiza, svyazannye s kompaktifikatsiei Stouna–Chekha”, Sovremennaya matematika i ee prilozheniya, 26, In-t kibernetiki AN Gruzii, Tbilisi, 2005, 119–150 | MR

[2] 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 | MR | Zbl

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

[4] Chentsov A. G., “Nesekventsialnye priblizhennye resheniya v abstraktnykh zadachakh o dostizhimosti”, Trudy Instituta matematiki i mekhaniki UrO RAN, 12, no. 1, 2006, 216–241 | MR | Zbl

[5] Varga Dzh., Optimalnoe upravlenie differentsialnymi i funktsionalnymi uravneniyami, Nauka, M., 1977, 624 pp. | MR

[6] Daffin R. Dzh., “Beskonechnye programmy”, Lineinye neravenstva i smezhnye voprosy, IL, M., 1959, 263–267

[7] Golshtein E. G., Teoriya dvoistvennosti v matematicheskom programmirovanii i ee prilozheniya, Nauka, M., 1971, 35 pp. | MR

[8] Chentsov A. G., Finitely additive measures and relaxations of extremal problems, Plenum Publishing Corporation, New York–London–Moscow, 1996, 244 pp. | MR | Zbl

[9] Chentsov A. G., Asymptotic attainability, Kluwer Academic Publishers, Dordrecht–Boston–London, 1997, 322 pp. | MR | Zbl

[10] Cech E., Topological spaces, Academia, Prague, 1966, 893 pp. | MR | Zbl

[11] Engelking R., Obschaya topologiya, Mir, M., 1986, 751 pp. | MR

[12] Kelli Dzh. L., Obschaya topologiya, Nauka, M., 1981, 431 pp. | MR

[13] Burbaki N., Obschaya topologiya, Nauka, M., 1968, 272 pp. | MR

[14] Krasovskii N. N., Teoriya upravleniya dvizheniem, Nauka, M., 1968, 475 pp. | MR

[15] Krasovskii N. N., Subbotin A. I., Pozitsionnye differentsialnye igry, Nauka, M., 1974, 456 pp. | MR | Zbl

[16] Krasovskii N. N., Upravlenie dinamicheskoi sistemoi. Zadacha o minimume garantirovannogo rezultata, Nauka, M., 1985, 518 pp. | MR

[17] Chentsov A. G., “Some questions of asymptotic analysis: approximate solutions and extension constructions”, Func. Diff. Equations, 12 (2005), 119–148 | MR | Zbl

[18] Chentsov A. G., “Some properties of generalized attraction sets”, Func. Diff. Equations, 13 (2006), 381–415 | MR

[19] Neve Zh., Matematicheskie osnovy teorii veroyatnostei, Mir, M., 1969, 309 pp. | MR | Zbl

[20] Chentsov A. G., Morina S. I., Extensions and relaxations, Kluwer Academic Publishers, Dordrecht–Boston–London, 2002, 408 pp. | MR

[21] Chentsov A. G., “Two-valued measures and zero-dimensional topologies”, Funct. Diff. Equations, 9 (2002), 71–89 | MR | Zbl

[22] Chentsov A. G., “Nekotorye svoistva konechno-additivnykh mer i ikh preobrazovanie na osnove gomomorfizmov izmerimykh struktur”, Dokl. RAN, 370:4 (2000), 449–452 | MR | Zbl

[23] Semadeni Z., Banach spaces of continuous functions, PWN, Warszava, 1971, 411 pp. | Zbl

[24] Chentsov A. G., “The nonsequential approximate solutions in problems of asymptotic analysis”, Soochow J. Math., 32:3 (2006), 441–475 | MR | Zbl