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.
UDC:
517.926,
517.977
MSC: 34D08,
34H05,
93C15 Received: 04.04.2016
DOI:
10.20537/vm160208