Uniform complete controllability and global control over asymptotic invariants of linear systems in the Hessenberg form

We prove that a linear control system
\begin{equation} \dot x=A(t)x+B(t)u,\qquad t\in\mathbb R,\quad x\in\mathbb R^n,\quad u\in\mathbb R^m, \end{equation}
with matrix coefficients of the Hessenberg form is uniformly completely controllable in the sense of Kalman under rather weak conditions imposed on coefficients. It is shown that some obtained sufficient conditions are essential. Corollaries are derived for quasi-differential equations. We construct feedback control $u=Ux$ for the system (1) and study the problem of global control over asymptotic invariants of the closed-loop system
\begin{equation} \dot x=(A(t)+B(t)U)x,\qquad t\in\mathbb R,\quad x\in\mathbb R^n, \end{equation}
The conditions on coefficients are weakened in the known results of S. N. Popova. For the system (2) with matrix coefficients of the Hessenberg form, the obtained results and the results of S. N. Popova are used to receive sufficient conditions for global reducibility to systems of scalar type and for global control over Lyapunov exponents and moreover, for global Lyapunov reducibility in the case of periodic $A(\cdot)$ and $B(\cdot)$.