On polynomial solutions of the linear stationary control system

It is known that the controllable system $x'=Bx+Du$, where the $x$ is the $n$-dimensional vector, can be transferred from an arbitrary initial state $x(0)=x^0$ to an arbitrary finite state $x(T)=x^T$ by the control function $u(t)$ in the form of the polynomial in degrees $t$. In this work, the minimum degree of the polynomial is revised: it is equal to $2p+1$, where the number $(p-1)$ is a minimum number of matrices in the controllability matrix (Kalman criterion), whose rank is equal to $n$. A simpler and a more natural algorithm is obtained, which first brings to the discovery of coefficients of a certain polynomial from the system of algebraic equations with the Wronskian and then, with the aid of differentiation, to the construction of functions of state and control.