Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta
RUS  ENG    JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB  
General information
Latest issue
Archive
Guidelines for authors

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Izv. IMI UdGU:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, 2020, Volume 56, Pages 102–121
DOI: https://doi.org/10.35634/2226-3594-2020-56-08
(Mi iimi405)
 

This article is cited in 2 scientific papers (total in 2 papers)

MATHEMATICS

On a linear autonomous descriptor equation with discrete time. II. Canonical representation and structural properties

V. E. Khartovskii

Y. Kupala State University of Grodno, ul. Ozheshko, 22, Grodno, 230023, Belarus
Full-text PDF (254 kB) Citations (2)
References:
Abstract: We consider a linear homogeneous autonomous descriptor equation with discrete time
$$B_0g(k+1)+\sum_{i=1}^mB_ig(k+1-i)=0,\,k=m,m+1,\ldots,$$
with rectangular (in general case) matrices $ B_i. $ Such an equation arises in the study of the most important control problems for systems with many commensurate delays in control: the $0$-controllability problem, the synthesis problem of the feedback-type regulator, which provides calming to the solution of the original system, the modal controllability problem (controllability to the coefficients of characteristic quasipolynomial), the spectral reduction problem and the synthesis problem observers for dual surveillance system. The main method of the presented study is based on replacing the original equation with an equivalent equation in the “expanded” state space, which allows one to match the new equation of the beam of matrices. This made it possible to study a number of structural properties of the original equation by using the canonical form of the beam of matrices, and express the results in terms of minimal indices and elementary divisors. In the article, a criterion is obtained for the existence of a nontrivial admissible initial condition for the original equation, the verification of which is based on the calculation of the minimum indices and elementary divisors of the beam of matrices. The following problem was studied: it is required to construct a solution to the original equation in the form $g (k + 1) = T \psi (k + 1)$, $k = 1,2 \ldots, $ where $ T $ is some matrix, the sequence of vectors $ \psi (k + 1)$, $k = 1,2, \ldots, $ satisfies the equation $ \psi (k + 1) = S \psi (k)$, $k = 1,2,\ldots,$ and the square matrix $ S $ has a predetermined spectrum (or part of the spectrum). The results obtained make it possible to construct solutions of the initial descriptor equation with predetermined asymptotic properties, for example, uniformly asymptotically stable.
Keywords: linear descriptor autonomous equation with discrete time, the subspace of initial conditions, representation of the solution, beam of matrixes.
Received: 10.10.2020
Bibliographic databases:
Document Type: Article
UDC: 517.977
MSC: 93B99, 93C55
Language: Russian
Citation: V. E. Khartovskii, “On a linear autonomous descriptor equation with discrete time. II. Canonical representation and structural properties”, Izv. IMI UdGU, 56 (2020), 102–121
Citation in format AMSBIB
\Bibitem{Kha20}
\by V.~E.~Khartovskii
\paper On a linear autonomous descriptor equation with discrete time. II.~Canonical representation and structural properties
\jour Izv. IMI UdGU
\yr 2020
\vol 56
\pages 102--121
\mathnet{http://mi.mathnet.ru/iimi405}
\crossref{https://doi.org/10.35634/2226-3594-2020-56-08}
Linking options:
  • https://www.mathnet.ru/eng/iimi405
  • https://www.mathnet.ru/eng/iimi/v56/p102
    Cycle of papers
    This publication is cited in the following 2 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta
    Statistics & downloads:
    Abstract page:243
    Full-text PDF :106
    References:46
     
      Contact us:
     Terms of Use  Registration to the website  Logotypes © Steklov Mathematical Institute RAS, 2024