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 Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2017, Volume 27, Issue 2, Pages 178–192 (Mi vuu579)

MATHEMATICS

On uniform global attainability of two-dimensional linear systems with locally integrable coefficients

A. A. Kozlov, I. V. Ints

Polotsk State University, ul. Blokhina, 29, Novopolotsk, 211440, Belarus

Abstract: We consider a linear time-varying control system with locally integrable and integrally bounded coefficients
$$\dot x =A(t)x+ B(t)u, \quad x\in\mathbb{R}^n,\quad u\in\mathbb{R}^m,\quad t\geqslant 0. \tag{1}$$
We construct control of the system $(1)$ as a linear feedback $u=U(t)x$ with measurable and bounded function $U(t)$, $t\geqslant 0$. For the closed-loop system
$$\dot x =(A(t)+B(t)U(t))x, \quad x\in\mathbb{R}^n, \quad t\geqslant 0, \tag{2}$$
we study a question about the conditions for its uniform global attainability. The last property of the system (2) means existence of a matrix $U(t)$, $t\geqslant 0$, that ensure equalities $X_U((k+1)T,kT)=H_k$ for the state-transition matrix $X_U(t,s)$ of the system (2) with fixed $T>0$ and arbitrary $k\in\mathbb N$, $\det H_k>0$. The problem is solved under the assumption of uniform complete controllability of the system (1), corresponding to the closed-loop system (2), i.e. assuming the existence of such $\sigma>0$ and $\gamma>0,$ that for any initial time $t_0\geqslant 0$ and initial condition $x(t_0)=x_0\in \mathbb{R}^n$ of the system (1) on the segment $[t_0,t_0+\sigma]$ there exists a measurable and bounded vector control $u=u(t),$ $\|u(t)\|\leqslant\gamma\|x_0\|,$ $t\in[t_0,t_0+\sigma],$ that transforms a vector of the initial state of the system into zero on that segment. It is proved that in two-dimensional case, i.e. when $n=2,$ the property of uniform complete controllability of the system (1) is a sufficient condition of uniform global attainability of the corresponding system (2).

Keywords: linear control system, uniform complete controllability, uniform global attainability.

 Funding Agency Grant Number National Academy of Sciences of Belarus, Ministry of Education of the Republic of Belarus ïîäïðîãðàììà 1, çàäàíèå 1.2.01

DOI: https://doi.org/10.20537/vm170203

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Bibliographic databases:

UDC: 517.926, 517.977
MSC: 34D08, 34H05, 93C15

Citation: A. A. Kozlov, I. V. Ints, “On uniform global attainability of two-dimensional linear systems with locally integrable coefficients”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 27:2 (2017), 178–192

Citation in format AMSBIB
\Bibitem{KozInt17} \by A.~A.~Kozlov, I.~V.~Ints \paper On uniform global attainability of two-dimensional linear systems with locally integrable coefficients \jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki \yr 2017 \vol 27 \issue 2 \pages 178--192 \mathnet{http://mi.mathnet.ru/vuu579} \crossref{https://doi.org/10.20537/vm170203} \elib{http://elibrary.ru/item.asp?id=29410190} 

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This publication is cited in the following articles:
1. A. A. Kozlov, “Ob odnoi faktorizatsii kvadratnykh matrits s polozhitelnym opredelitelem”, Tr. In-ta matem., 25:1 (2017), 51–61
2. A. A. Kozlov, A. D. Burak, “Ob upravlenii otdelnymi asimptoticheskimi invariantami dvumernykh lineinykh upravlyaemykh sistem s nablyudatelem”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 28:4 (2018), 445–461
3. A. A. Kozlov, “Kriterii ravnomernoi globalnoi dostizhimosti lineinykh sistem”, Izv. IMI UdGU, 52 (2018), 47–58
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