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A complete analytic solution for the time-optimal control problem fornonlinear control systems of the form $\dot x_1=u$, $\dot x_j=x_1^{j-1}$,$j=2,\ldots,n$, is obtained for arbitrary $n$. The main goal of the paper is topresent the following surprising observation: this nonlinear optimality problemleads to a truncated Hausdorff moment problem, which is applied essentially forfinding the optimal time and optimal controls.

Truncated Hausdorff Moment Problem and Analytic Solution of the Time-Optimal Problem for One Class of Nonlinear Control Systems