A complete analytic solution for the time-optimal control problem nonlinear systems of form $\dot x_1=u$, x_j=x_1^{j-1}$, $j=2,\ldots,n$, is obtained arbitrary~$n$. In paper we present following surprising observation: this optimality leads to a truncated Hausdorff moment problem, which gives tools finding optimal time and controls. Being homogeneous, mentioned system approximates certain class...