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PreprintCC BY-NC-ND 4.0arXiv

Hypersensitivity and Turnpikes in Optimal Control of Inverted Pendulum: A Dynamical Systems Perspective

  • Mai Bando
  • Yuzuru Sato
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Abstract

The hypersensitivity and turnpike phenomena in the optimal control of an inverted pendulum are investigated from a dynamical-systems perspective. We show that, for a fixed terminal time and a fixed terminal state optimal control problem, (1) the hypersensitivity originates from the fractal structure of the set of initial adjoint variables in the associated Hamiltonian dynamics, (2) the turnpike arises from slow dynamics in the vicinity of a degenerate center manifold, and (3) the escape channels are formed by normally hyperbolic invariant manifolds (NHIMs). As a consequence, small perturbations in the initial adjoint variables lead to qualitatively distinct extremal trajectories, resulting in severe numerical instability in trajectory optimization. Both the fractal structure and the invariant sets are characterized numerically and analytically.

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Cite

BibTeXRIS
BibTeX
@misc{bando2026hypersensitivity,
  title = {{Hypersensitivity and Turnpikes in Optimal Control of Inverted Pendulum: A Dynamical Systems Perspective}},
  author = {Mai Bando and Yuzuru Sato},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.22801},
  eprint = {2609.22801},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.22801},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Hypersensitivity and Turnpikes in Optimal Control of Inverted Pendulum: A Dynamical Systems Perspective
AU  - Mai Bando
AU  - Yuzuru Sato
PY  - 2026
DA  - 2026/09/19
PB  - arXiv
DO  - 10.48550/arxiv.2609.22801
UR  - https://arxiv.org/abs/2609.22801
AB  - The hypersensitivity and turnpike phenomena in the optimal control of an inverted pendulum are investigated from a dynamical-systems perspective. We show that, for a fixed terminal time and a fixed terminal state optimal control problem, (1) the hypersensitivity originates from the fractal structure of the set of initial adjoint variables in the associated Hamiltonian dynamics, (2) the turnpike arises from slow dynamics in the vicinity of a degenerate center manifold, and (3) the escape channels are formed by normally hyperbolic invariant manifolds (NHIMs). As a consequence, small perturbations in the initial adjoint variables lead to qualitatively distinct extremal trajectories, resulting in severe numerical instability in trajectory optimization. Both the fractal structure and the invariant sets are characterized numerically and analytically.
ER  -