### Abstract

We present a global solution for an optimal feedback control problem of the underactuated Heisenberg system or nonholonomic integrator. Set in the general framework of the Hamilton-Jacobi theory, this work demonstrates the potential applicability of our methodology to general underactuated optimal control problems. We incorporate the Heisenberg system into a typical optimal control formulation called the hard constraint problem, and transform into a two point boundary value problem for a Hamiltonian system. It is viewed as a canonical transformation in itself, to which we apply our recently developed technique based on generating functions appearing in the Hamilton-Jacobi theory. It is first recognized that our previously developed procedure for solving fullyactuated optimal control problems is not directly applicable due to a singularity caused by underactuation. However, within the same framework of generating functions we are provided with a way to circumvent this singularity by algebraic manipulations linked with the underactuated coordinate. This results in a scalar transcendental equation whose solution ultimately leads to a nonlinear optimal feedback control law in an analytical form. We illustrate our solution by numerical examples.

Original language | English |
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Title of host publication | Proceedings of the 45th IEEE Conference on Decision and Control 2006, CDC |

Pages | 2687-2692 |

Number of pages | 6 |

Publication status | Published - 2006 Dec 1 |

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

### Publication series

Name | Proceedings of the IEEE Conference on Decision and Control |
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ISSN (Print) | 0191-2216 |

### Other

Other | 45th IEEE Conference on Decision and Control 2006, CDC |
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Country | United States |

City | San Diego, CA |

Period | 06/12/13 → 06/12/15 |

### All Science Journal Classification (ASJC) codes

- Control and Systems Engineering
- Modelling and Simulation
- Control and Optimization

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## Cite this

*Proceedings of the 45th IEEE Conference on Decision and Control 2006, CDC*(pp. 2687-2692). [4177499] (Proceedings of the IEEE Conference on Decision and Control).