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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.
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
Linking options:
https://www.mathnet.ru/eng/vuu533 https://www.mathnet.ru/eng/vuu/v26/i2/p221
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