Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 50 min ago
PreprintCC BY-NC-ND 4.0arXiv

Jordan-Block Degeneracy and Cubic-Order Bifurcating Periodic Orbits in Minimum-Energy Optimal Control of Hamiltonian Equilibria

  • Mai Bando
  • Ayano Tsuruta
  • Shanshan Pan
  • Daniel J. Scheeres
Open at sourcePDFDOICite

Abstract

Equilibria of the Hamiltonian system associated with Pontryagin's minimum principle exhibit an exact doubling of the natural spectrum and, under a simple pairing condition, a Jordan block at every simple purely imaginary eigenvalue. Consequently, the classical Lyapunov Center Theorem does not apply to the augmented system, and no periodic orbit with nonzero optimal control bifurcates at linear order. We establish this mechanism in general and show that an optimal-control-induced periodic family emerges at cubic order in a Lindstedt--Poincaré expansion. The mechanism is illustrated in closed form for the pendulum and evaluated numerically for the planar L2 equilibrium of Hill's restricted three-body problem, where the third-order approximation is validated against an independently computed family of periodic orbits.

Related

  1. On the stability of equilibria in degenerate periodic Hamiltonian systems
    Nina Xue
    PLOS OnePeer-reviewedCC BY 4.0Full textAlgebra & number theoryMathematical physicsPLOS
  2. On the Minimum Number of Hamilton Cycles in Hamiltonian Threshold Graphs: A Simple Proof
    Chengli Li, Feng Liu
    The Electronic Journal of CombinatoricsPeer-reviewedCC BY 4.0Full textCombinatorics & discrete mathematicsCrossref
  3. Point vortex dynamics in quasi-periodic channels: transporting trajectories and ergodic distribution of equilibria
    Mohamed Ali, Taoufik Hmidi
    arXivPreprintCC0Full textAnalysis & PDEsDynamical systemsarXiv
  4. Hamiltonian graphs with prescribed minimum degree and no near-spanning cycles
    Xingzhi Zhan
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
  5. Bayesian Equilibria of Heterogeneous Non-Atomic Routing Games With Private Information
    Alexia Ambrogio, Leonardo Cianfanelli, Giacomo Como +1
    IEEE Control Systems LettersPeer-reviewedPublisher termsFull textGeneral mathematicsHAL

Cite

BibTeXRIS
BibTeX
@misc{bando2026jordanblock,
  title = {{Jordan-Block Degeneracy and Cubic-Order Bifurcating Periodic Orbits in Minimum-Energy Optimal Control of Hamiltonian Equilibria}},
  author = {Mai Bando and Ayano Tsuruta and Shanshan Pan and Daniel J. Scheeres},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.22800},
  eprint = {2609.22800},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.22800},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Jordan-Block Degeneracy and Cubic-Order Bifurcating Periodic Orbits in Minimum-Energy Optimal Control of Hamiltonian Equilibria
AU  - Mai Bando
AU  - Ayano Tsuruta
AU  - Shanshan Pan
AU  - Daniel J. Scheeres
PY  - 2026
DA  - 2026/09/19
PB  - arXiv
DO  - 10.48550/arxiv.2609.22800
UR  - https://arxiv.org/abs/2609.22800
AB  - Equilibria of the Hamiltonian system associated with Pontryagin's minimum principle exhibit an exact doubling of the natural spectrum and, under a simple pairing condition, a Jordan block at every simple purely imaginary eigenvalue. Consequently, the classical Lyapunov Center Theorem does not apply to the augmented system, and no periodic orbit with nonzero optimal control bifurcates at linear order. We establish this mechanism in general and show that an optimal-control-induced periodic family emerges at cubic order in a Lindstedt--Poincaré expansion. The mechanism is illustrated in closed form for the pendulum and evaluated numerically for the planar L_2 equilibrium of Hill's restricted three-body problem, where the third-order approximation is validated against an independently computed family of periodic orbits.
ER  -