An application of the mean motion problem to time-optimal control

TL;DR

This paper explores the behavior of time-optimal controls for linear systems with purely imaginary eigenvalues, linking the problem to classical mean motion theory and establishing bounds on switching points.

Contribution

It introduces a novel connection between the mean motion problem and time-optimal control for systems with imaginary eigenvalues, providing bounds on switching points.

Findings

Number of switching points grows linearly with time interval length for systems with imaginary eigenvalues.

Established a lower bound on switching points in the generic case.

Linked optimal control switching behavior to classical mean motion problem.

Abstract

We consider time-optimal controls of a controllable linear system with a scalar control on a long time interval. It is well-known that if all the eigenvalues of the matrix describing the linear system dynamics are real then any time-optimal control has a bounded number of switching points, where the bound does not depend on the length of the time interval. We consider the case where the governing matrix has purely imaginary eigenvalues, and show that then, in the generic case, the number of switching points is bounded from below by a linear function of the length of the time interval. The proof is based on relating the switching function in the optimal control problem to the mean motion problem that dates back to Lagrange and was solved by Hermann Weyl.