### Abstract

Families of differential operators, like those defining affine, generally nonlinear, control systems are known to have natural Hopf algebra structures. These provide deeper insight into relationships and properties of such objects as the Chen-Fliess series and state space realizations of systems defined by input-output operators. A starting point for this work are representations of control objects by formal powerseries in, generally noncommuting, indeterminates. Such series are elements of a free associative algebra which contains several rich substructure that are equipped with various products. This article summarizes correspondences between objects from control theory and algebraic combinatorics, with special focus on elements of the two Hopf algebra structures on this algebra of formal power series, including the antipode and two convolution products. Newly found relationships are discussed, and open questions are posed. Recent work has demonstrated that these abstract combinatorial algebra structures help obtain effective solution formulas for various control problems including questions about controllability and algorithms for path planning. They also find applications in the field of geometric numerical integration algorithms.

Original language | English (US) |
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Title of host publication | Proceedings of the IEEE Conference on Decision and Control |

Pages | 7503-7508 |

Number of pages | 6 |

DOIs | |

State | Published - 2009 |

Event | 48th IEEE Conference on Decision and Control held jointly with 2009 28th Chinese Control Conference, CDC/CCC 2009 - Shanghai, China Duration: Dec 15 2009 → Dec 18 2009 |

### Other

Other | 48th IEEE Conference on Decision and Control held jointly with 2009 28th Chinese Control Conference, CDC/CCC 2009 |
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Country | China |

City | Shanghai |

Period | 12/15/09 → 12/18/09 |

### Fingerprint

### ASJC Scopus subject areas

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

### Cite this

*Proceedings of the IEEE Conference on Decision and Control*(pp. 7503-7508). [5400441] https://doi.org/10.1109/CDC.2009.5400441

**Control interpretations of products in the Hopf algebra.** / Kawski, Matthias.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings of the IEEE Conference on Decision and Control.*, 5400441, pp. 7503-7508, 48th IEEE Conference on Decision and Control held jointly with 2009 28th Chinese Control Conference, CDC/CCC 2009, Shanghai, China, 12/15/09. https://doi.org/10.1109/CDC.2009.5400441

}

TY - GEN

T1 - Control interpretations of products in the Hopf algebra

AU - Kawski, Matthias

PY - 2009

Y1 - 2009

N2 - Families of differential operators, like those defining affine, generally nonlinear, control systems are known to have natural Hopf algebra structures. These provide deeper insight into relationships and properties of such objects as the Chen-Fliess series and state space realizations of systems defined by input-output operators. A starting point for this work are representations of control objects by formal powerseries in, generally noncommuting, indeterminates. Such series are elements of a free associative algebra which contains several rich substructure that are equipped with various products. This article summarizes correspondences between objects from control theory and algebraic combinatorics, with special focus on elements of the two Hopf algebra structures on this algebra of formal power series, including the antipode and two convolution products. Newly found relationships are discussed, and open questions are posed. Recent work has demonstrated that these abstract combinatorial algebra structures help obtain effective solution formulas for various control problems including questions about controllability and algorithms for path planning. They also find applications in the field of geometric numerical integration algorithms.

AB - Families of differential operators, like those defining affine, generally nonlinear, control systems are known to have natural Hopf algebra structures. These provide deeper insight into relationships and properties of such objects as the Chen-Fliess series and state space realizations of systems defined by input-output operators. A starting point for this work are representations of control objects by formal powerseries in, generally noncommuting, indeterminates. Such series are elements of a free associative algebra which contains several rich substructure that are equipped with various products. This article summarizes correspondences between objects from control theory and algebraic combinatorics, with special focus on elements of the two Hopf algebra structures on this algebra of formal power series, including the antipode and two convolution products. Newly found relationships are discussed, and open questions are posed. Recent work has demonstrated that these abstract combinatorial algebra structures help obtain effective solution formulas for various control problems including questions about controllability and algorithms for path planning. They also find applications in the field of geometric numerical integration algorithms.

UR - http://www.scopus.com/inward/record.url?scp=77950873197&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77950873197&partnerID=8YFLogxK

U2 - 10.1109/CDC.2009.5400441

DO - 10.1109/CDC.2009.5400441

M3 - Conference contribution

AN - SCOPUS:77950873197

SN - 9781424438716

SP - 7503

EP - 7508

BT - Proceedings of the IEEE Conference on Decision and Control

ER -