Journal of Dynamical and Control Systems

, Volume 18, Issue 4, pp 551–571 | Cite as

Explicit solutions of the \( {\mathfrak{a}_1} \)-type lie-Scheffers system and a general Riccati equation

Article

Abstract

For a general differential system \( \dot{x}(t) = \sum\nolimits_{d = 1}^3 {u_d } (t){X_d} \), where Xd generates the simple Lie algebra of type \( {\mathfrak{a}_1} \), we compute the explicit solution in terms of iterated integrals of products of ud’s. As a byproduct we obtain the solution of a general Riccati equation by infinite quadratures.

Key words and phrases

Free Lie algebra shuffe product special linear algebra Riccati equation Lie-Sheffers system 

2010 Mathematics Subject Classification

17B80 34A05 34A26 

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Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Institute of MathematicsPolish Academy of SciencesWarszawaPoland

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