The criterion of uniform global attainability of periodic systems

We consider a linear time-varying control system
\begin{equation} \dot x =A(t)x+ B(t)u, \quad x\in\mathbb{R}^n,\quad u\in\mathbb{R}^m,\quad t\in \mathbb{R} \end{equation}
with piecewise continuous and bounded $\omega$-periodic coefficient matrices $A (\cdot)$ and $B (\cdot).$ We construct control of the system $(1)$ as a linear feedback $u=U(t)x$ with piecewise continuous and bounded matrix function $U(t)$, $t\in \mathbb{R}$. For the closed-loop system
\begin{equation} \dot x =(A(t)+B(t)U(t))x, \quad x\in\mathbb{R}^n, \quad t\in \mathbb{R}, \end{equation}
the conditions of its uniform global attainability are studied. The latest property of the system (2) means existence of matrix $U(t)$, $t\in \mathbb{R}$, 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 $k\in\mathbb{Z}$, $\det H_k>0$. The problem is solved under the assumption of uniform complete controllability (by Kalman) of the system (1), corresponding to the closed-loop system (2), i.e. assuming the existence of such numbers $\sigma>0$ and $\alpha_i>0,$ $i=\overline{1,4}$, that for any number $t_0\in\mathbb{R}$ and vector $\xi\in \mathbb{R}^n$ the following inequalities hold:
$$\alpha_1\|\xi\|^2\leqslant \xi^*\int\nolimits_{t_0}^{t_0+\sigma}X(t_0,s)B(s)B^*(s)X^*(t_0,s)\,ds\,\xi\leqslant\alpha_2\|\xi\|^2,$$

$$\alpha_3\|\xi\|^2\leqslant\xi^*\int\nolimits_{t_0}^{t_0+\sigma}X(t_0+\sigma,s)B(s)B^*(s)X^*(t_0+\sigma,s)\,ds\,\xi\leqslant\alpha_4 \|\xi\|^2,$$
where $X(t,s)$ is the state-transition matrix of linear system (1) with $u(t)\equiv0.$ It is proved that the property of uniform complete controllability (by Kalman) of the periodic system (1) is a necessary and sufficient condition of uniform global attainability of the corresponding system (2).