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## 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 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 Engelking R. General topology, Warszawa: Państwowe Wydawnictwo Naukowe, 1985. Translated under the title Obshchaya topologiya, Moscow: Mir, 1986, 752 p. Arhangel'skii A.V. 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Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2015, vol. 25, issue 2, pp. 212-229 (in Russian). DOI: 10.20537/vm150206 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 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