Dynamics of the spatial restricted three-body problem stabilized by Hamiltonian structure-preserving control

Abstract

The local invariant (stable, unstable, and center) manifolds of a saddle \(\times \) center \(\times \) center equilibrium point can be used to construct a three-dimensional Hamiltonian structure-preserving (HSP) controller. The linear stability of the controlled Hamiltonian system is verified when one of the proposed criteria is satisfied. The nonlinear dynamics of a linearly stabilized Hamiltonian system is analyzed by normalizing the Hamiltonian high-order perturbation terms using the Lie series method. The analytical solutions of the invariant tori are then obtained in trigonometric series form. A theorem for the Nekhoroshev stability of the controlled Hamiltonian system is provided. When the constructed HSP controller is applied to a photogravitational restricted three-body problem with oblateness, a hyperbolic artificial equilibrium point at which a criterion is satisfied can be stabilized, and bounded Lissajous trajectories near the equilibrium point are obtained. The allocation law demonstrates that the attitude angles and lightness number of the solar sail can serve as control parameters to provide the required acceleration of the HSP controller.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (11432001 and 11172024) and the Fundamental Research Funds for the Central Universities.

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Correspondence to Ming Xu.

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Luo, T., Xu, M. Dynamics of the spatial restricted three-body problem stabilized by Hamiltonian structure-preserving control. Nonlinear Dyn 94, 1889–1905 (2018). https://doi.org/10.1007/s11071-018-4463-z

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Keywords

  • Hamiltonian structure-preserving control
  • Spatial restricted three-body problem
  • Hyperbolic equilibrium point
  • Linear stability
  • Normal form
  • Nekhoroshev stability