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Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, 2018, Volume 52, Pages 47–58
DOI: https://doi.org/10.20537/2226-3594-2018-52-04
(Mi iimi360)
 

This article is cited in 1 scientific paper (total in 1 paper)

The criterion of uniform global attainability of linear systems

A. A. Kozlov

Polotsk State University, ul. Blokhina, 29, Novopolotsk, 211440, Belarus
Full-text PDF (240 kB) Citations (1)
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Abstract: In this paper, we consider a linear time-varying control system with locally integrable and integrally bounded coefficients
\begin{equation} \dot x =A(t)x+ B(t)u, \quad x\in\mathbb{R}^n,\quad u\in\mathbb{R}^m,\quad t\geqslant 0. \tag{1} \end{equation}
We construct control of the system $(1)$ as a linear feedback $u=U(t)x$ with a measurable and bounded function $U(t)$, $t\geqslant 0$. For the closed-loop system
\begin{equation} \dot x =(A(t)+B(t)U(t))x, \quad x\in\mathbb{R}^n, \quad t\geqslant 0, \tag{2} \end{equation}
the criterion for its uniform global attainability is established. The latter property means the existence of $T>0$ such that for any positive $\alpha$ and $\beta$ there exists a $d=d(\alpha,\beta)>0$ such that for any $t_0\geqslant 0$ and for any $(n\times n)$-matrix $H$, $\|H\|\leqslant\alpha$, $\det H\geqslant\beta$, there exists a measurable on $[t_0,t_0+T]$ gain matrix function $U(\cdot)$ such that $\sup\limits_{t\in [t_0,t_0+T]}\|U(t)\|\leqslant d$ and $X_U(t_0+T,t_0)=H$, where $X_U$ is the state transition matrix for the system (2). The proof of the criterion is based on the theorem on the representation of an arbitrary $(n\times n)$-matrix with a positive determinant in the form of a product of nine upper and lower triangular matrices with positive diagonal elements and additional conditions on the norm and determinant of these matrices.
Keywords: linear control system, state-transition matrix, uniform global attainability.
Funding agency Grant number
National Academy of Sciences of Belarus, Ministry of Education of the Republic of Belarus подпрограмма 1, задание 1.2.01
The work was done within the framework of the State Program of Scientific Research of the Republic of Belarus “Convergence–2020” (subprogram 1, task 1.2.01).
Received: 01.07.2018
Bibliographic databases:
Document Type: Article
UDC: 517.926, 517.977
MSC: 34D08, 34H05, 93C15
Language: Russian
Citation: A. A. Kozlov, “The criterion of uniform global attainability of linear systems”, Izv. IMI UdGU, 52 (2018), 47–58
Citation in format AMSBIB
\Bibitem{Koz18}
\by A.~A.~Kozlov
\paper The criterion of uniform global attainability of linear systems
\jour Izv. IMI UdGU
\yr 2018
\vol 52
\pages 47--58
\mathnet{http://mi.mathnet.ru/iimi360}
\crossref{https://doi.org/10.20537/2226-3594-2018-52-04}
\elib{https://elibrary.ru/item.asp?id=36508455}
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    Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta
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