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Numerical Schemes of Higher Order for a Class of Nonlinear Control Systems

  • Lars Grüne
  • Peter E. Kloeden
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2542)

Abstract

We extend a systematic method for the derivation of high order schemes for affinely controlled nonlinear systems to a larger class of systems in which the control variables are allowed to appear nonlinearly in multiplicative terms. Using an adaptation of the stochastic Taylor expansion to control systems we construct Taylor schemes of arbitrary high order and indicate how derivative free Runge-Kutta type schemes can be obtained.

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References

  1. 1.
    L. Arnold, Random Dynamical Systems. Springer-Verlag, Heidelberg (1998)zbMATHGoogle Scholar
  2. 2.
    F. Colonius and W. Kliemann, The Dynamics of Control, Birkhäuser, Boston, 2000.Google Scholar
  3. 3.
    Cyganowski, S., Grüne, L., and Kloeden, P.E.: Maple for Stochastic Differential Equations. In: Blowey, J.F., Coleman, J.P., Craig, A.W. (eds.): Theory and Numerics of Differential Equations, Springer-Verlag, Heidelberg (2001) 127–178.Google Scholar
  4. 4.
    Deuflhard, P.: Stochastic versus Deterministic Numerical ODE Integration. In: Platen, E. (ed.): Proc. 1st Workshop on Stochastic Numerics, Berlin, WIAS Berlin, Preprint Nr. 21 (1992) 16–20.Google Scholar
  5. 5.
    Falcone, M. and Ferretti, R.: Discrete Time High-Order Schemes for Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Numer. Math., 67 (1994) 315–344.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Ferretti, R.: Higher-Order Approximations of Linear Control Systems via Runge-Kutta Schemes. Computing, 58 (1997) 351–364.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Grüne, L.: An Adaptive Grid Scheme for the Discrete Hamilton-Jacobi-Bellman Equation. Numer. Math., 75 (1997) 319–337.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Grüne, L.: Asymptotic Behavior of Dynamical and Control Systems under Perturbation and Discretization, Lecture Notes in Mathematics, 1783, Springer-Verlag, Heidelberg (2002)zbMATHGoogle Scholar
  9. 9.
    Grüne, L. and Kloeden, P.E.: Higher Order Numerical Schemes for Affinely Controlled Nonlinear Systems. Numer. Math., 89 (2001) 669–690.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Grüne, L. and Kloeden, P.E.: Pathwise Approximation of Random Ordinary Differential Equations. BIT, 41 (2001) 710–721.CrossRefGoogle Scholar
  11. 11.
    Isidori, A.: Nonlinear Control Systems. An Introduction. Second edition, Springer-Verlag, Heidelberg (1995)Google Scholar
  12. 12.
    Hairer, E., Norsett, S.P. and Wanner, G.: Solving Ordinary Differential Equations I. Springer-Verlag, Heidelberg (1988)Google Scholar
  13. 13.
    Kloeden, P.E. and Platen, E.: Numerical Solution of Stochastic Differential Equations. Springer-Verlag, Heidelberg (1992) (3rd revised and updated printing, 1999)zbMATHGoogle Scholar
  14. 14.
    Veliov, V.: On the Time Discretization of Control Systems. SIAM J. Control Optim., 35 (1997) 1470–1486.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Lars Grüne
    • 1
  • Peter E. Kloeden
    • 1
  1. 1.Fachbereich MathematikJ.W. Goethe-UniversitätFrankfurt am MainGermany

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