The solution of one time-optimal problem on the basis of the Markov moment min-problem with even gaps

The time-optimal problem for a linear system which matrix has the spectrum $\sigma(A)=\{(2k-1)\lambda\}_{k=1}^n$ is considered. This problem is reduced to the power Markov moment min-problem with even gaps. The new generating function is suggested for finding the optimal time. The explicit form of the polynomial for which the set of nonnegative roots coincides with the set of switchings of the time-optimal control is given. The analytical form of the 0-controllability set at time $\Theta$ is obtained.