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An instability result of Hamiltonian systems related to optimal swing-up control of a pendulum

Noboru Sakamoto

Abstract

This paper presents an instability result of Hamiltonian systems associated with optimal swing-up control for a pendulum. The systems possess weak (higher-order) instability at the initial point of the swing-up control, the analysis for which requires techniques from celestial mechanics. The obtained result may have relationships with the previously obtained numerical studies for the existence of multiple locally optimal solutions and the non-existence conjecture of the optimal control.

An instability result of Hamiltonian systems related to optimal swing-up control of a pendulum

Abstract

This paper presents an instability result of Hamiltonian systems associated with optimal swing-up control for a pendulum. The systems possess weak (higher-order) instability at the initial point of the swing-up control, the analysis for which requires techniques from celestial mechanics. The obtained result may have relationships with the previously obtained numerical studies for the existence of multiple locally optimal solutions and the non-existence conjecture of the optimal control.
Paper Structure (10 sections, 43 equations, 3 figures, 1 table)

This paper contains 10 sections, 43 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem setting
  3. Stability analysis
  4. Transformation of the linear part
  5. Higher order computations with scaling
  6. Computation of normal forms with Lie transformation
  7. Proof of instability
  8. Conclusions
  9. Hamiltonian normal forms and Lie series
  10. Chetayev's instability theorem

Figures (3)

  • Figure 1: A swing-up trajectory with 107 swings in horibe-masterE
  • Figure 2: 2-dimensional pendulum of length $2l$
  • Figure 3: A numerical simulation of (\ref{['eqn:hamsys_z_normalform']}) with $\bm{z}(0)=(0,10^{-6},10^{-6},0)$.