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Trudy Mat. Inst. Steklova, 2015, Volume 291, Pages 292–311 (Mi tm3682)  

This article is cited in 7 scientific papers (total in 7 papers)

On an asymptotic analysis problem related to the construction of an attainability domain

A. G. Chentsovab, A. P. Baklanovcba

a Ural Federal University named after the First President of Russia B. N. Yeltsin, Ekaterinburg, Russia
b Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg, Russia
c International Institute for Applied Systems Analysis, Laxenburg, Austria

Abstract: Problems of constructing and analyzing the properties of attainability domains play an important role in control theory and its applications. In particular, this applies to control under impulse constraints that reflect the energetics of a process. The situation is complicated by the possible instability of the process under variation (in particular, under relaxation) of constraints related to boundary and intermediate conditions. Stability of the problem is also missing, in general, under relaxation of state constraints. In these cases, it is natural to focus on the asymptotic variant of the statement; this is especially expedient when one has to deal with initially asymptotic requirements. In all these cases, it seems expedient to use analogs of J. Warga's approximate solutions. At the same time, to seek necessary approximate (and, in fact, asymptotic) solutions, it is natural to use generalized modes. For problems with impulse constraints and discontinuity in the coefficients of control actions, such modes lead to phenomena described by products of discontinuous functions and generalized functions even in the class of linear systems. In a large series of his studies, to overcome the arising difficulties, one of the authors used constructions of extension in the class of finitely additive measures. The present paper follows this approach and is ideologically relevant to the engineering problem of controlling the thrust of an engine under conditions of a given program of variation of its orientation; it is postulated that energy resources are completely consumed in a natural (for a number of impulse control problems) mode of short-duration impulses: the set of time instants at which the instantaneous control is different from zero can be embedded in an interval of vanishingly small length. Within these short periods of time, the engine should consume all energy resources while obeying some other constraints (making the sense of moment constraints) to a high degree of accuracy. In addition, one should take into account the possible discontinuity of the functions defining the coefficients of control actions. As a natural analog of the attainability domain, we use an attraction set, whose construction, together with the subsequent study of its main properties, constitutes the goal of the present study.

Funding Agency Grant Number
Russian Foundation for Basic Research 12-01-00537а
Russian Academy of Sciences - Federal Agency for Scientific Organizations 12-П-1-1012
This work was supported by the Russian Foundation for Basic Research (project nos. 12-01-00537a, 13-01-90414-Ukr_f_a, and 13-01-00304a) and by the Presidium of the Russian Academy of Sciences within programs for basic research (project nos. 12-P-1-1012 and 12-P-1-1019).


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English version:
Proceedings of the Steklov Institute of Mathematics, 2015, 291, 279–298

Bibliographic databases:

UDC: 517.9
Received: October 15, 2014

Citation: A. G. Chentsov, A. P. Baklanov, “On an asymptotic analysis problem related to the construction of an attainability domain”, Optimal control, Collected papers. In commemoration of the 105th anniversary of Academician Lev Semenovich Pontryagin, Trudy Mat. Inst. Steklova, 291, MAIK Nauka/Interperiodica, Moscow, 2015, 292–311; Proc. Steklov Inst. Math., 291 (2015), 279–298

Citation in format AMSBIB
\by A.~G.~Chentsov, A.~P.~Baklanov
\paper On an asymptotic analysis problem related to the construction of an attainability domain
\inbook Optimal control
\bookinfo Collected papers. In commemoration of the 105th anniversary of Academician Lev Semenovich Pontryagin
\serial Trudy Mat. Inst. Steklova
\yr 2015
\vol 291
\pages 292--311
\publ MAIK Nauka/Interperiodica
\publaddr Moscow
\jour Proc. Steklov Inst. Math.
\yr 2015
\vol 291
\pages 279--298

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    This publication is cited in the following articles:
    1. A. G. Chentsov, I. I. Savenkov, Yu. V. Shapar, “Odna zadacha na programmnyi maksimin pri ogranicheniyakh impulsnogo kharaktera”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 28:1 (2018), 91–110  mathnet  crossref  elib
    2. E. G. Pytkeev, A. G. Chentsov, “Volmenovskii kompaktifikator i ego primenenie dlya issledovaniya abstraktnoi zadachi o dostizhimosti”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 28:2 (2018), 199–212  mathnet  crossref  elib
    3. A. G. Chentsov, “Maximal linked systems and ultrafilters in abstract attainability problem”, IFAC PAPERSONLINE, 51:32 (2018), 239–244  crossref  isi
    4. A. G. Chentsov, “Ultrafiltry i maksimalnye stseplennye sistemy: osnovnye sootnosheniya”, Izv. IMI UdGU, 53 (2019), 138–157  mathnet  crossref  elib
    5. A. G. Chentsov, E. G. Pytkeev, “Constraints of asymptotic nature and attainability problems”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 29:4 (2019), 569–582  mathnet  crossref
    6. A. G. Chentsov, “Some properties of ultrafilters of widely understood measurable spaces”, Dokl. Math., 99:3 (2019), 255–259  crossref  isi
    7. A. G. Chentsov, “Ultrafiltry i maksimalnye stseplennye sistemy”, Tr. IMM UrO RAN, 26, no. 1, 2020, 274–292  mathnet  crossref  elib
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