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Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2016, Volume 26, Issue 2, Pages 221–230 (Mi vuu533)  

MATHEMATICS

On the sufficient condition of global scalarizability of linear control systems with locally integrable coefficients

A. A. Kozlov

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
\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 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}
a definition of uniform global quasi-attainability is introduced. This notion is a weakening of the property of uniform global attainability. The last property means existence of matrix $U(t)$, $t\geqslant 0$, ensuring 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 $k\in\mathbb N$, $\det H_k>0$. We prove that uniform global quasi-attainability implies global scalarizability. The last property means that for any given locally integrable and integrally bounded scalar function $p=p(t)$, $t\geqslant0$, there exists a measurable and bounded function $U=U(t)$, $t\geqslant 0$, which ensures asymptotic equivalence of the system $(2)$ and the system of scalar type $\dot z=p(t)z$, $z\in\mathbb{R}^n$, $t\geqslant0$.

Keywords: linear control system, Lyapunov exponents, global scalarizability.

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


DOI: https://doi.org/10.20537/vm160208

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Bibliographic databases:

UDC: 517.926, 517.977
MSC: 34D08, 34H05, 93C15
Received: 04.04.2016

Citation: A. A. Kozlov, “On the sufficient condition of global scalarizability of linear control systems with locally integrable coefficients”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 26:2 (2016), 221–230

Citation in format AMSBIB
\Bibitem{Koz16}
\by A.~A.~Kozlov
\paper On the sufficient condition of global scalarizability of linear control systems with locally integrable coefficients
\jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki
\yr 2016
\vol 26
\issue 2
\pages 221--230
\mathnet{http://mi.mathnet.ru/vuu533}
\crossref{https://doi.org/10.20537/vm160208}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3522926}
\elib{http://elibrary.ru/item.asp?id=26244781}


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  • Вестник Удмуртского университета. Математика. Механика. Компьютерные науки
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