Construction of controllability function as time of motion

This article is devoted to the controllability function method in admissible synthesis problems for linear canonical systems. The work considers methods of constructing such control so that the controllability function is time of motion of an arbitrary point to the origin. A canonical controlled system of linear equations $\dot{x}_i=x_{i+1}, i=\overline{1,n-1}, \dot{x}_n=u$ with control constraints $|u| \le d$ is considered. The controllability function $\Theta$ can be found as the only positive solution of the implicit equation $2a_0\Theta=(D(\Theta)FD(\Theta)x,x)$, where $D(\Theta)= diag(\Theta^ {-\frac{-2n-2i+1}{2}})_{i=1}^n$. Matrix $F=\{f_{ij}\}_{i,j=1}^n$ is positive definite and $a_0>0$ is chosen so that the control constraints are satisfied. The controllability function is motion time if $\dot{\Theta}= -1$. From this condition, an equation is obtained, the solution of which is considered in this work. Unlike previous works on this topic, no additional restrictions are imposed on the appearance of matrix $F$. The task of this article is to find the parameters set of the matrix $F$ and the column vector $a$, which satisfy the obtained equation and for which the controllability function is the time of movement from the point $x$ to the origin. In this way, we get a family of controls depending on this parameters such that the trajectory of system steers the origin in finite time. In general case, difficulties may arise when finding the solution of Cauchy problem of the corresponding system. Canonical system can be reduced to Euler's equation, for which a characteristic equation can be found, and therefore a trajectory in an explicit form. Two-dimensional, three-dimensional and four-dimensional canonical systems are considered. In each case, the matrix equation is solved and sets of parameters for which the controllability functions value will be the time of movement of an arbitrary point to the origin are found. Conditions on parameters are obtained from positive definiteness of the matrix $F$. Some parameters and an arbitrary initial point are chosen and the solution of Cauchy problem in analytical form is found.