phone +7 (3412) 91 60 92
mail imi@udsu.ru
ru-RU Russian
Home » Issues » Archive of Issues » 2018 - 2 issue » pp. 199-212

Archive of Issues

Russia Yekaterinburg
Year
2018
Volume
28
Issue
2
Pages
199-212
<<
>>
Section Mathematics
Title The Wallman compactifier and its application for investigation of the abstract attainability problem
Author(-s) Pytkeev E.G.ab, Chentsov A.G.ab
Affiliations Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciencesa, Ural Federal Universityb
Abstract The attainability problem with asymptotic constraints is considered. Such constraints can arise under weakening of constraints that are standard in control theory: phase constraints, boundary and intermediate conditions; trajectories of a system must satisfy these constraints. But asymptotic constraints can arise from the beginning as a characterization of trends in the implementation of desired behavior. For example, one can speak of implementation of powerful control impulses with vanishingly small duration. In this case, it is hard to tell whether any standard constraints are weakened. So, we have a set of complicating conditions with each of which we can juxtapose some analog of the attainability domain in control theory and (more precisely) the image of a subset of the usual solution space under the action of a given operator. In this paper, we investigate questions concerning the structure of an attraction set arising as an analog of the attainability domain. The investigation scheme is based on the application of a special way of extending solution space which admits a natural analogy with Wallman extension used in general topology. Then it is natural to suppose that the space of usual solutions is endowed with a topology (usually, it is a $T_1$-space that is explored in this case). In this connection, questions concerning the replacement of sets forming asymptotic constraints by closures and interiors are addressed. Partially, questions associated with representation of the interior of the set of admissible generalized elements that form an auxiliary attraction set are discussed.
Keywords asymptotic constraints, extension of a problem, topology
UDC 519.6
MSC 28A33
DOI 10.20537/vm180206
Received 19 March 2018
Language Russian
Citation Pytkeev E.G., Chentsov A.G. The Wallman compactifier and its application for investigation of the abstract attainability problem, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2018, vol. 28, issue 2, pp. 199-212.
References
  1. Chentsov A.G. Compactifiers in extension constructions for reachability problems with constraints of asymptotic nature, Proceedings of the Steklov Institute of Mathematics, 2017, vol. 296, suppl. 1, pp. 102-118. DOI: 10.1134/S0081543817020109
  2. Engelking R. General topology, Warszawa: Państwowe Wydawnictwo Naukowe, 1985. Translated under the title Obshchaya topologiya, Moscow: Mir, 1986, 752 p.
  3. Arhangel'skii A.V. Compactness, General Topology II, Enсyсlopaedia Math. Sсi., vol. 50, Berlin: Springer-Verlag, 1996, pp. 1-117.
  4. Chentsov A.G. Filters and ultrafilters in the constructions of attraction sets, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2011, issue 1, pp. 113-142 (in Russian). DOI: 10.20537/vm110112
  5. Chentsov A.G., Pytkeev E.G. Some topological structures of extensions of abstract reachability problems, Proceedings of the Steklov Institute of Mathematics, 2016, vol. 292, suppl. 1, pp. 36-54. DOI: 10.1134/S0081543816020048
  6. Warga J. Optimal control of differential and functional equations, New York: Academic Press, 1972, 546 p. DOI: 10.1016/C2013-0-11669-8 Translated under the title Optimal'noe upravlenie differentsial'nymi i funktsional'nymi uravneniyami, Moscow: Nauka, 1977, 624 p.
  7. Gamkrelidze R.V. Osnovy optimal'nogo upravleniya (Foundations of optimal control), Tbilisi: Tbilisi University, 1975, 230 p.
  8. Krasovskii N.N. Teoriya upravleniya dvizheniem (Theory of motion control), Moscow: Nauka, 1968, 476 p.
  9. Chentsov A.G. Finitely additive measures and relaxations of extremal problems, New York: Springer US, 1996, XII, 244 p.
  10. Chentsov A.G. Asymptotic attainability, Dordrecht: Springer Netherlands, 1997, XIV, 322 p. DOI: 10.1007/978-94-017-0805-0
  11. Chentsov A.G. Finitely additive measures and extensions of abstract control problems, Journal of Mathematical Sciences, 2006, vol. 133, no. 2, pp. 1045-1206. DOI: 10.1007/s10958-006-0030-0
  12. Chentsov A.G., Morina S.I. Extensions and relaxations, Dordrecht: Springer Netherlands, 2002, XIV, 408 p. DOI: 10.1007/978-94-017-1527-0
  13. Chentsov A.G., Baklanov A.P. On the question of construction of an attraction set under constraints of asymptotic nature, Proceedings of the Steklov Institute of Mathematics, 2015, vol. 291, suppl. 1, pp. 40-55. DOI: 10.1134/S0081543815090035
  14. Chentsov A.G., Baklanov A.P. On an asymptotic analysis problem related to the construction of an attainability domain, Proceedings of the Steklov Institute of Mathematics, 2015, vol. 291, issue 1, pp. 279-298. DOI: 10.1134/S0081543815080222
  15. Chentsov A.G., Baklanov A.P., Savenkov I.I. A problem of reachability with asymptotic constraints, Izv. Inst. Mat. Inform. Udmurt. Gos. Univ., 2016, issue 1 (47), pp. 54-118 (in Russian).
  16. Kuratowski K., Mostowski A. Set theory, Warszawa: PWN, 1967, vii, 417 p. Translated under the title Teoriya mnozhestv, Moscow: Mir, 1970, 416 p.
  17. Chentsov A.G. Attraction sets in abstract attainability problems: equivalent representations and basic properties, Russian Mathematics, 2013, vol. 57, no. 11, pp. 28-44. DOI: 10.3103/S1066369X13110030
  18. Bourbaki N. Topologie Generale, Paris: Hermann, 1961, 263 p. Translated under the title Obshchaya topologiya, Moscow: Nauka, 1968, 272 p.
  19. Chentsov A.G. To question about realization of attraction elements in abstract attainability problems, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2015, vol. 25, issue 2, pp. 212-229 (in Russian). DOI: 10.20537/vm150206
  20. Chentsov A.G. To the validity of constraints in the class of generalized elements, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2014, issue 3, pp. 90-109 (in Russian). DOI: 10.20537/vm140309
  21. Brodskaya L.I., Chentsov A.G. Nekotorye primery neustoichivyh zadach upravleniya (Some examples of unstable control problems), Yekaterinburg: Ural Federal University, 2014, 101 p.
Full text
/doc/issues-2018/18-02-06.pdf
<< Previous article
Next article >>