Conditionally stable control systems (concerning a certain type of optimum controlled systems)

The paper deals with a strictly linear system of the automatic control that is asymptotically stable with regard to the obvious solution $x\equiv0$. The state of the system is described by the vector $x=x(x_0,t)$.
The parameters of the system are supposed to be selected and fixed in accordance with any existing idea of optimum linear systems.
Let us assume $t^*$ to be the time of conditional damping of the transient, i. e. $t^*$ is a value for which the condition $\vert x(x_0,t)\vert\le x_0e^{-\pi}$ is satisfied for any $t\ge t^*$.
The fact is stated that there is such a value $t_*(0<t_*<t^*)$ for which the switching of the speed $\dot{\xi}$ of a servomotor $c+\dot{\xi}$ (for $t\le t_*$) to the speed -$\dot{\xi}$ (for $t\ge t_*$) transfers the initial system into anotherone that is stable (for $t\ge t_*$). The transient of the new system is damped at the first stroke and the time of conditional damping of the transient is essentially less than $t_*$.
The paper introduces a control method that includes determining $t_*$ as dependent on system initial state $x_0=x(x_0,0)$ and on system parameters. The determination of $t_*$ is performed by a special computer, its structure being analyzed in the paper.