The criterion of uniform global attainability of linear systems

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.