Consistency of discrete-time linear stationary control systems with an incomplete feedback of the special form for $n=5$

We consider a discrete-time linear control system with an incomplete feedback
\begin{gather*} x(t+1)=Ax(t)+Bu(t),\qquad y(t)=C^*x(t),\qquad u(t)=Uy(t),\\ t\in\mathbb Z,\qquad(x,u,y)\in\mathbb K^n\times\mathbb K^m\times\mathbb K^k, \end{gather*}
where $\mathbb K=\mathbb C$ or $\mathbb K=\mathbb R$. We introduce the concept of consistency for the closed-loop system Здесь $\mathbb K=\mathbb C$ или $\mathbb K=\mathbb R$. Для замкнутой системы
\begin{equation} x(t+1)=(A+BUC^*)x(t),\quad x\in\mathbb K^n. \label{eq1} \end{equation}
This concept is a generalization of the concept of complete controllability to systems with an incomplete feedback. We study the consistency of the system \eqref{eq1} in connection with the problem of arbitrary placement of eigenvalue spectrum which is to bring a characteristic polynomial of a matrix of the system \eqref{eq1} to any prescribed polynomial by means of the time-invariant control $U$. For the system \eqref{eq1} of the special form where the matrix $A$ is Hessenberg and the rows of the matrix $B$ before the $p$-th and the rows of the matrix $C$ after the $p$-th (not including $p$) are equal to zero, the property of consistency is the sufficient condition for arbitrary placement of eigenvalue spectrum. In previous studies it has been proved that the converse is true for $n<5$ and false for $n>5$. In this paper, an open question for $n=5$ is resolved. For the system \eqref{eq1} of the special form, it is proved that if $n=5$ then the property of consistency is a necessary condition for the arbitrary placement of eigenvalue spectrum. The proof is carried out by direct searching of all possible valid values of dimensions $m,k,p$. The property of consistency is equivalent to the property of complete controllability of a big system of dimension $n^2$. For the proof we construct the big system and the controllability matrix $K$ of this system of dimension $n^2\times n^2mk$. We show that the matrix $K$ has a nonzero minor of order $n^2=25$. We use Maple 15 to calculate the high-order determinants.