On the property of uniform complete controllability of a discrete-time linear control system

We study the property of uniform complete controllability (according to Kalman) for a discrete-time linear control system
\begin{equation} x(t+1)=A(t)x(t)+B(t)u(t),\quad t\in\mathbb N_0,\quad (x,u)\in\mathbb R^n\times\mathbb R^m. \end{equation}
We prove that if the system (1) is uniformly completely controllable, then the matrix $A(\cdot)$ is completely bounded on $\mathbb N_0$ (i.e. $\sup_{t\in\mathbb N_0}(|A(t)|+|A^{-1}(t)|)<+\infty$) and the matrix $B(\cdot)$ is bounded on $\mathbb N_0$. We prove that the system (1) is uniformly completely controllable if and only if there exists a $\vartheta\in\mathbb N$ such that for all $\tau\in\mathbb N_0$ the inequalities $\alpha_1I\leqslant W_1(\tau+\vartheta,\tau)\leqslant\beta_1I$, $\alpha_2I\leqslant W_2(\tau+\vartheta,\tau)\leqslant\beta_2I$ hold for some positive $\alpha_i$ and $\beta_i$, where
\begin{gather*} W_1(t,\tau)\doteq\sum_{s=\tau}^{t-1}X(t,s+1)B(s)B^*(s)X^*(t,s+1),\\ W_2(t,\tau)\doteq\sum_{s=\tau}^{t-1}X(\tau,s+1)B(s)B^*(s)X^*(\tau,s+1). \end{gather*}
On the basis of this statement, we prove the following criterion for uniform complete controllability of the system (1), which is similar to the Tonkov criterion of uniform complete controllability for continuous-time systems: the system (1) is $\vartheta$-uniformly completely controllable if and only if the matrix $A(\cdot)$ is completely bounded on $\mathbb N_0$; the matrix $B(\cdot)$ is bounded on $\mathbb N_0$; there exists an $\ell=\ell(\vartheta)>0$ such that for every $\tau\in\mathbb N_0$ and for any $x_1\in\mathbb R^n$ there exists a control function $u(t)$, $t\in[\tau,\tau+\vartheta)$, which transfers the solution of the system (1) from the state $x(\tau)=0$ to the state $x(\tau+\vartheta)=x_1$, and the inequality $|u(t)|\leqslant\ell|x_1|$ holds for all $t\in[\tau,\tau+\vartheta)$.