### Abstract

In this report, a method for approximating the stabilizing solution of the Hamilton-Jacobi equation for integrable systems is proposed using symplectic geometry and a Hamiltonian perturbation technique. Using the fact that the Hamiltonian lifted system of an integrable system is also integrable, the Hamiltonian system (canonical equation) that is derived from the theory of 1-st order partial differential equations is considered as an integrable Hamiltonian system with a perturbation caused by control. Assuming that the approximating Riccati equation from the Hamilton-Jacobi equation at the origin has a stabilizing solution, we construct approximating behaviors of the Hamiltonian flows on a stable Lagrangian submanifold, from which an approximation to the stabilizing solution is obtained.

Original language | Undefined |
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Title of host publication | Proceedings of the 45th IEEE Conference on Decision and Control |

Place of Publication | USA |

Publisher | IEEE CONTROL SYSTEMS SOCIETY |

Pages | 5857-5862 |

Number of pages | 6 |

ISBN (Print) | 1-4244-0171-2 |

DOIs | |

Publication status | Published - 2006 |

Event | 45th IEEE Conference on Decision and Control, CDC 2006 - San Diego, United States Duration: 13 Dec 2006 → 15 Dec 2006 Conference number: 45 |

### Publication series

Name | |
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Number | 1636734 (P |

ISSN (Print) | 0191-2216 |

### Conference

Conference | 45th IEEE Conference on Decision and Control, CDC 2006 |
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Abbreviated title | CDC |

Country | United States |

City | San Diego |

Period | 13/12/06 → 15/12/06 |

### Keywords

- EWI-9202
- IR-66916
- METIS-237951

## Cite this

Sakamoto, N., & van der Schaft, A. (2006). An approximating method for the stabilizing solution of the Hamilton-Jacobi equation for integrable systems using Hamiltonian perturbation theory. In

*Proceedings of the 45th IEEE Conference on Decision and Control*(pp. 5857-5862). USA: IEEE CONTROL SYSTEMS SOCIETY. https://doi.org/10.1109/CDC.2006.377789