### Abstract

The Chen-Fliess series is known to be an exponential Lie series. Previously explicit formulas for the iterated integral coefficients were known only for its factorization into a directed infinite product of exponentials. This factorization uses Hall sets and the Zinbiel product. We use the underlying Hopf algebra structure to derive explicit formulas for the corresponding coefficients in the logarithm of the series. This allows one to express the series as a single exponential. This work is closely related to Fer and Magnus expansions, and has interpretations in terms of a continuous Campbell-Baker-Hausdorff formula. The result facilitates work in nonlinear control, numerical integration and various applications that involve compositions of noncommuting flows.

Original language | English (US) |
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Title of host publication | Proceedings of the 47th IEEE Conference on Decision and Control, CDC 2008 |

Pages | 1569-1574 |

Number of pages | 6 |

DOIs | |

State | Published - Dec 1 2008 |

Event | 47th IEEE Conference on Decision and Control, CDC 2008 - Cancun, Mexico Duration: Dec 9 2008 → Dec 11 2008 |

### Publication series

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

### Other

Other | 47th IEEE Conference on Decision and Control, CDC 2008 |
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Country | Mexico |

City | Cancun |

Period | 12/9/08 → 12/11/08 |

### ASJC Scopus subject areas

- Control and Systems Engineering
- Modeling and Simulation
- Control and Optimization

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## Cite this

*Proceedings of the 47th IEEE Conference on Decision and Control, CDC 2008*(pp. 1569-1574). [4738914] (Proceedings of the IEEE Conference on Decision and Control). https://doi.org/10.1109/CDC.2008.4738914