Uniform complete controllability of linear hybrid systems

We consider a non-stationary linear hybrid discrete-continuous control system \[ \begin{cases} \dot x(t)=A_{11}(t)x(t)+A_{12}(k)y(k)+B_{11}(t)u(t)+B_{12}(k)v(k),\\ y(k+1)=A_{21}(k)x(k)+A_{22}(k)y(k)+B_{21}(k)u(k)+B_{22}(k)v(k). \end{cases} \] The concepts of uniform complete controllability and the Kalman matrix for this system are introduced. It is proved that if there exist $\vartheta\in\mathbb N$ and $\gamma>0$ such, that for all $l\in\mathbb N_0$, for the Kalman matrix, an inequality $W(l,l+\vartheta)\geqslant\gamma I$ is valid, then the hybrid system is uniformly completely controllable.