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

TL;DR

This paper provides an explicit analytic solution to a class of nonlinear time-optimal control problems by linking them to a truncated Hausdorff moment problem, revealing a novel mathematical connection.

Contribution

It introduces a new approach by connecting nonlinear control problems to the truncated Hausdorff moment problem, enabling explicit solutions for arbitrary system dimensions.

Findings

Analytic solutions for the time-optimal control of nonlinear systems of arbitrary size.

Identification of a surprising link between control problems and the truncated Hausdorff moment problem.

Methodology applicable to a broad class of nonlinear control systems.

Abstract

A complete analytic solution for the time-optimal control problem for nonlinear 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 to present the following surprising observation: this nonlinear optimality problem leads to a truncated Hausdorff moment problem, which is applied essentially for finding the optimal time and optimal controls.