A hopf-algebraic formula for compositions of noncommuting flows

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A hopf-algebraic formula for compositions of noncommuting flows
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 CampbellBaker-Hausdorff formula. The result facilitates work in nonlinear control, numerical integration and various applications that involve compositions of noncommuting flows.
Eric Gehrig, Matthias Kawski
Added 29 May 2010
Updated 29 May 2010
Type Conference
Year 2008
Where CDC
Authors Eric Gehrig, Matthias Kawski
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