A problem of attainability with constraints of asymptotic nature

A. G. Chentsov ; A. P. Baklanov ; I. I. Savenkov

Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 47 (2016) no. 1, pp. 54-118.

Résumé

We construct and study properties of attainability sets for a linear control system with discontinuous coefficients of the control action. This problem is tightly connected with the issue of an attainability “on average” (i.e. the type of an attainability in the class of mathematical expectations of random vectors). Focusing on the case of constraints of an asymptotic character, we tackle the mentioned problems by means of the universal approach. These constraints correspond to a control regime in the class of “narrow” pulses. We consider a control problem in the class of pulse-like controls complying with the requirement of full consumption of energy resources.

Détail

Keywords: attraction set, topological space, ultrafilter.

Bibliographie

[1] Warga J., Optimal control of differential and functional equations, Academic Press, Inc., New York–London, 1972, xiii+531 pp. | MR | Zbl

[2] Kelley J. L., General Topology, D. Van Nostrand Company, Inc., Toronto–New York–London, 1955, xiv+298 pp. | MR | Zbl

[3] Bourbaki N., Topologie Générale (General Topology), Hermann, Paris, 1961

[4] Chentsov A. G., “Certain constructions of asymptotic analysis related to the Stone–Čech compactification”, Itogi Nauki Tekh. Ser. Sovrem. Mat. Prilozh. Temat. Obz., 26, 2005, 119–150 (in Russian) | MR

[5] Chentsov A. G., “Attraction sets in abstract attainability problems: Equivalent representations and basic properties”, Russian Mathematics, 57:11 (2013), 28–44 | DOI | MR | Zbl

[6] Engelking R., General Topology, Polish Scientific Publishers, Warsaw, 1977, 626 pp. | MR | Zbl

[7] Chentsov A. G., “Filters and ultrafilters in the constructions of attraction sets”, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2011, no. 1, 113–142 (in Russian)

[8] Chentsov A. G., “Tier mappings and ultrafilter-based transformations”, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 18, no. 4 (2012), 298–314 (in Russian)

[9] Chentsov A. G., “Representation of attraction elements in abstract attainability problems with asymptotic constraints”, Russian Mathematics, 56:10 (2012), 38–49 | DOI | MR | Zbl

[10] Chentsov A. G., “On the question of representation of ultrafilters and their application in extension constructions”, Proceedings of the Steklov Institute of Mathematics, 287, 2014, 29–48 | DOI | MR

[11] Chentsov A. G., “Ultrafilters of measurable spaces and their application in extension constructions”, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 20, no. 1, 2014, 285–304 (in Russian) | MR

[12] Chentsov A. G., “On one example of representing the ultrafilter space for an algebra of sets”, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 17, no. 4 (2011), 293–311 (in Russian)

[13] Chentsov A. G., “On the question of representation of ultrafilters in a product of measurable spaces”, Proceedings of the Steklov Institute of Mathematics, 284, 2014, 65–78 | DOI | MR

[14] Chentsov A. G., “On certain problems of the structure of ultrafilters related to extensions of abstract control problems”, Automation and Remote Control, 74:12 (2013), 2020–2036 | DOI | MR | Zbl

[15] Chentsov A. G., Finitely additive measures and relaxations of extremal problems, Plenum Publ. Co., New York etc., 1996, 244 pp. | MR | Zbl

[16] Chentsov A. G., Asymptotic attainability, Kluwer Acad. Publ., Dordrecht etc., 1997, 322 pp. | MR | Zbl

[17] Chentsov A. G., Morina S. I., Extensions and relaxations, Kluwer Acad. Publ., Dordrecht–Boston–London, 2002, 408 pp. | MR | Zbl

[18] Chentsov A. G., “Finitely additive measures and extensions of abstract control problems”, Journal of Mathematical Sciences, 133:2 (2006), 1045–1206 | DOI | MR | Zbl

[19] Chentsov A. G., “To question about representation of Stone compactums”, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2013, no. 4, 156–174 (in Russian)

[20] 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, 291, 2015, 40–55 | DOI | MR

[21] Krasovskii N. N., Subbotin A. I., Positional differential games, Nauka, M., 1974, 456 pp.

[22] Krasovskii N. N., Theory of control of motion, Nauka, M., 1968, 476 pp.

[23] Kuratowski K., Mostowski A., Set theory, North-Holland Publishing Company, Amsterdam, 1967, xii+417 pp. | MR

[24] Chentsov A. G., Elements of finitely additive measure theory, v. I, Ural State Technical University, Yekaterinburg, 2008, 389 pp.

[25] Bulinskii A. V., Shiryaev A. N., Theory of stochastic processes, Fizmatlit, M., 2005, 402 pp.

[26] Edwards R. E., Functional analysis. Theory and applications, Holt Rinehart and Winston, New York–Chicago–San Francisco–Toronto–London, 1965, xiii+781 pp. | MR | Zbl

[27] Dunford N. J., Schwartz J. T., Linear operators, v. I, General theory, Interscience Publ., New York–London, 1958, 874 pp. | MR | Zbl

[28] Chentsov A. G., “On the construction of correct extensions in the class of finitely additive measures”, Izv. Vyssh. Uchebn. Zaved. Mat., 2002, no. 2, 58–80 (in Russian) | MR

[29] Chentsov A. G., Elements of finitely additive measure theory, v. II, Ural State Technical University, Yekaterinburg, 2010, 542 pp.

[30] Skvortsova A. V., Chentsov A. G., “On the construction of an asymptotic analog of a pencil of trajectories for a linear system with a single-impulse control”, Differential Equations, 40:12 (2004), 1726–1739 | DOI | MR | Zbl