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



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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  1. 1.
    J. Wei and E. Norman, On global representations of the solutions of linear differential equations as a product of exponentials. Proc. Amer. Math. Soc. (1964), Vol. 15, 327–334.MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    J. F. Cariñena and J. de Lucas, Integrability of Lie systems through Riccati equations. Journal of Nonlinear Mathematical Physics (2011), Vol. 18, No. 1, 29–54.MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    F. Cariñena José, L. Javier de, and A. Ramos, A geometric approach to integrability conditions for Riccati equations. Electron. J. Differential Equations (2007), No. 122.Google Scholar
  4. 4.
    J. F. Cariñena, L. Javier de, and M. F. Rañada, Recent applications of the theory of Lie systems in Ermakov systems. SIGMA Symmetry Integrability Geom. Methods Appl. (2008), Vol. 4.Google Scholar
  5. 5.
    J. F. Cariñena, L. Javier de, and A. Ramos, A geometric approach to integrability conditions for Riccati equations. Electron. J. Differential Equations (2007), No. 122.Google Scholar
  6. 6.
    J. F. Cariñena, L. Javier de, and M. F. Rañada, Lie systems and integrability conditions for t-dependent frequency harmonic oscillators. Int. J. Geom. Methods Mod. Phys. (2010), Vol. 7, No. 2, 289–310.Google Scholar
  7. 7.
    J. F. Cariñena, L. Javier de, Applications of Lie systems in dissipative Milne-Pinney equations. Int. J. Geom. Methods Mod. Phys. (2009), Vol. 6, No. 4, 683–699.Google Scholar
  8. 8.
    R. M. Redheffer, The Riccati equation: Initial values and inequalities. Math. Ann. (1957), Vol. 133, 235–250.MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    R. M. Redheffer, On solutions of Riccati’s equation as functions of the initial values. J. Rational Mech. Anal. (1956), Vol. 5, 835–848.MathSciNetMATHGoogle Scholar
  10. 10.
    A. I. Shirshov, On the bases of a free Lie algebra. Algebra Logika (1962), Vol. 1, 14–19.MATHGoogle Scholar
  11. 11.
    G. Viennot, Algèbres de Lie libres et monoïdes libres. Lecture Notes in Mathematics (1978), Vol. 691.Google Scholar
  12. 12.
    M. Fliess, Fonctionnelles causales non linéaires et indéterminées non commutatives. Bull. Soc. Math. France (1981), Vol. 109, No. 1, 3–40.MathSciNetMATHGoogle Scholar
  13. 13.
    K.-T. Chen, Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula. Ann. of Math. 2 (1957), Vol. 65, 163–178.MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    K.-T. Chen, Iterated integrals and exponential homomorphisms. Proc. London Math. Soc. 3 (1954), Vol. 4, 502–512.MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    D. D’Alessandro, Introduction to quantum control and dynamics. Chapman & Hall/CRC Applied Mathematics and Nonlinear Science Series (2008).Google Scholar
  16. 16.
    A. Agrachev and Y. L. Sachkov, Control theory from the geometric viewpoint. Encyclopaedia of Mathematical Sciences (2004), Vol. 87.Google Scholar
  17. 17.
    K.-T. Chen, Algebraic paths. J. Algebra (1968), Vol. 10, 8–36.MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    M. Kawski, The combinatorics of nonlinear controllability and noncommuting flows. Mathematical control theory, Part 1, 2 (Trieste, 2001) (2002), 223–311.Google Scholar
  19. 19.
    G. Melançon and Ch. Reutenauer, Lyndon words, free algebras and shuffles. Canad. J. Math. 4 (1989), Vol. 41, No. 4, 577–591.MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Ch. Reutenauer, Free Lie algebras. London Mathematical Society Monographs. New Series (1193), Vol. 7.Google Scholar
  21. 21.
    M. Kawski and H. J. Sussmann, Noncommutative power series and formal Lie-algebraic techniques in nonlinear control theory. Operators, systems, and linear algebra (Kaiserslautern) (1997), 111–128.Google Scholar
  22. 22.
    H. J. Sussmann, A product expansion for the Chen series. Theory and applications of nonlinear control systems (1986), 323–335.Google Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Institute of MathematicsPolish Academy of SciencesWarszawaPoland

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