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

Omri Dalin, Alexander Ovseevich, Michael Margaliot

TL;DR

This work links time-optimal control of single-input linear systems with purely imaginary spectra to the classical mean motion problem. By representing the switching function as a sum of complex exponentials and invoking Weyl's mean motion framework, the authors establish a linear-in-$T$ lower bound on the number of switchings, $N(T)\ge cT+o(T)$, for generic adjoint/initial data with a positive density $c$. The constant $c$ is computable via the mean motion $\Omega=\sum_k \lambda_k V_k$, where $V_k$ are BBW-measured volumes on an $(n-1)$-torus and $\Omega$ can be expressed through integrals of Bessel functions using the BWW formula. The results illuminate the density of switchings in time-optimal controls under purely imaginary spectra and open avenues for extending the approach to more general spectral structures and explicit computation of the density via Bessel integral representations.

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.

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

TL;DR

This work links time-optimal control of single-input linear systems with purely imaginary spectra to the classical mean motion problem. By representing the switching function as a sum of complex exponentials and invoking Weyl's mean motion framework, the authors establish a linear-in- lower bound on the number of switchings, , for generic adjoint/initial data with a positive density . The constant is computable via the mean motion , where are BBW-measured volumes on an -torus and can be expressed through integrals of Bessel functions using the BWW formula. The results illuminate the density of switchings in time-optimal controls under purely imaginary spectra and open avenues for extending the approach to more general spectral structures and explicit computation of the density via Bessel integral representations.

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.

Paper Structure

This paper contains 8 sections, 2 theorems, 70 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The problem of mean motion
  4. Bessel functions
  5. The Bohl--Weyl--Wintner formula
  6. Solution of the mean motion problem
  7. Main result
  8. Discussion

Key Result

Theorem 1

Suppose that $A$ satisfies Assumption assump:dominanat and that the pair $(A,b)$ is controllable. Then, for generic vectors $p,b\in\mathbb R^{n}\setminus\{0\}$ there exists $c>0$ such that for any $T$ large enough the number of zeros of the switching function $m$ in eq:abst_m on the interval $[0,T]$

Figures (3)

  • Figure 1: Geometric representation of $z(\phi)=z(\phi_1,\phi_2)$.
  • Figure 2: Plotting $z=a_1e^{i\phi_1}+a_2e^{i\phi_2}$, with $\phi_1=0$, in the complex plane.
  • Figure 3: Plotting $\frac{\arg(z(T)) }{T}$ as a function of $T$.

Theorems & Definitions (10)

  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 1
  • Example 5
  • Proposition 1