A hopf-algebraic formula for compositions of noncommuting flows

Eric Gehrig, Matthias Kawski

Research output: Chapter in Book/Report/Conference proceedingConference contribution

6 Citations (Scopus)

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 languageEnglish (US)
Title of host publicationProceedings of the IEEE Conference on Decision and Control
Pages1569-1574
Number of pages6
DOIs
StatePublished - 2008
Event47th IEEE Conference on Decision and Control, CDC 2008 - Cancun, Mexico
Duration: Dec 9 2008Dec 11 2008

Other

Other47th IEEE Conference on Decision and Control, CDC 2008
CountryMexico
CityCancun
Period12/9/0812/11/08

Fingerprint

Factorization
Series
Chemical analysis
Algebra
Explicit Formula
Magnus Expansion
Iterated integral
Infinite product
Nonlinear Control
Coefficient
Hopf Algebra
Logarithm
Numerical integration
Express

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Modeling and Simulation
  • Control and Optimization

Cite this

Gehrig, E., & Kawski, M. (2008). A hopf-algebraic formula for compositions of noncommuting flows. In Proceedings of the IEEE Conference on Decision and Control (pp. 1569-1574). [4738914] https://doi.org/10.1109/CDC.2008.4738914

A hopf-algebraic formula for compositions of noncommuting flows. / Gehrig, Eric; Kawski, Matthias.

Proceedings of the IEEE Conference on Decision and Control. 2008. p. 1569-1574 4738914.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Gehrig, E & Kawski, M 2008, A hopf-algebraic formula for compositions of noncommuting flows. in Proceedings of the IEEE Conference on Decision and Control., 4738914, 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
Gehrig E, Kawski M. A hopf-algebraic formula for compositions of noncommuting flows. In Proceedings of the IEEE Conference on Decision and Control. 2008. p. 1569-1574. 4738914 https://doi.org/10.1109/CDC.2008.4738914
Gehrig, Eric ; Kawski, Matthias. / A hopf-algebraic formula for compositions of noncommuting flows. Proceedings of the IEEE Conference on Decision and Control. 2008. pp. 1569-1574
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