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 $$x(t+1)=A(t)x(t)+B(t)u(t), \quad t\in\mathbb{N}_0, \quad (x,u)\in\mathbb{R}n\times\mathbb{R}m. \qquad(1)$$ 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 $\alpha_1 I\leqslant W_1(\tau+\vartheta,\tau)\leqslant\beta_1 I$, $\alpha_2 I\leqslant W_2(\tau+\vartheta,\tau)\leqslant\beta_2 I$ hold for some positive $\alpha_i$ and $\beta_i$, where $$ W_1(t,\tau)\doteq\sum_{s=\tau}{t-1} X(t,s+1)B(s)B*(s)X*(t,s+1),\quad W_2(t,\tau)\doteq\sum_{s=\tau}{t-1} X(\tau,s+1)B(s)B*(s)X*(\tau,s+1). $$ 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 $t\in[\tau,\tau+\vartheta)$.