A hopf-algebraic formula for compositions of noncommuting flows

Eric Gehrig; Matthias Kawski

doi:10.1109/CDC.2008.4738914

A hopf-algebraic formula for compositions of noncommuting flows

Eric Gehrig, Matthias Kawski

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

9 Scopus citations

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)
Title of host publication	Proceedings of the 47th IEEE Conference on Decision and Control, CDC 2008
Publisher	Institute of Electrical and Electronics Engineers Inc.
Pages	1569-1574
Number of pages	6
ISBN (Print)	9781424431243
DOIs	https://doi.org/10.1109/CDC.2008.4738914
State	Published - 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
ISSN (Print)	0743-1546
ISSN (Electronic)	2576-2370

Other

Other	47th IEEE Conference on Decision and Control, CDC 2008
Country/Territory	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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10.1109/CDC.2008.4738914

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A hopf-algebraic formula for compositions of noncommuting flows. / Gehrig, Eric; Kawski, Matthias.
Proceedings of the 47th IEEE Conference on Decision and Control, CDC 2008. Institute of Electrical and Electronics Engineers Inc., 2008. p. 1569-1574 4738914 (Proceedings of the IEEE Conference on Decision and Control).

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

Gehrig, E & Kawski, M 2008, A hopf-algebraic formula for compositions of noncommuting flows. in Proceedings of the 47th IEEE Conference on Decision and Control, CDC 2008., 4738914, Proceedings of the IEEE Conference on Decision and Control, Institute of Electrical and Electronics Engineers Inc., pp. 1569-1574, 47th IEEE Conference on Decision and Control, CDC 2008, Cancun, Mexico, 12/9/08. https://doi.org/10.1109/CDC.2008.4738914

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AB - 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.

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A hopf-algebraic formula for compositions of noncommuting flows

Abstract

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