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. $\sup_{t\in\mathbb{N}_0}(|A(t)|+|A{-1}(t)|)0$ 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 $t\in[\tau,\tau+\vartheta)$.