Journal of Dynamical and Control Systems

, Volume 18, Issue 4, pp 551–571

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


DOI: 10.1007/s10883-012-9159-y

Cite this article as:
Pietrzkowski, G. J Dyn Control Syst (2012) 18: 551. doi:10.1007/s10883-012-9159-y


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 algebrashuffe productspecial linear algebraRiccati equationLie-Sheffers system

2010 Mathematics Subject Classification


Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Institute of MathematicsPolish Academy of SciencesWarszawaPoland