A control procedure for total set of Lyapunov invariants for linear systems in nondegenerate case

Let the differential system $\dot{x}=(A(t)+B(t)U(t))x$, $x\in\mathbb{R}^n$, $t\ge 0$ has bounded piecewise continuous square coefficient matrices $A$ and $B$ and let the control matrix $U$ be of the same type. It is proved that the total Lyapunov invariants set of this system is globolly controllable if there exist numbers $\sigma>0$ and $\alpha>0$ such that the inequality $\int_{t_0}^{t_0+\sigma}|{\det B(\tau)}|\,d\tau\ge\alpha$ holds for all $t_0\ge 0$.