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

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Numerical Methods and Applications (NMA 2002)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2542))

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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. L. Arnold, Random Dynamical Systems. Springer-Verlag, Heidelberg (1998)

    MATH  Google Scholar 

  2. F. Colonius and W. Kliemann, The Dynamics of Control, Birkhäuser, Boston, 2000.

    Google Scholar 

  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. 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. Falcone, M. and Ferretti, R.: Discrete Time High-Order Schemes for Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Numer. Math., 67 (1994) 315–344.

    Article  MATH  MathSciNet  Google Scholar 

  6. Ferretti, R.: Higher-Order Approximations of Linear Control Systems via Runge-Kutta Schemes. Computing, 58 (1997) 351–364.

    Article  MATH  MathSciNet  Google Scholar 

  7. Grüne, L.: An Adaptive Grid Scheme for the Discrete Hamilton-Jacobi-Bellman Equation. Numer. Math., 75 (1997) 319–337.

    Article  MATH  MathSciNet  Google Scholar 

  8. Grüne, L.: Asymptotic Behavior of Dynamical and Control Systems under Perturbation and Discretization, Lecture Notes in Mathematics, 1783, Springer-Verlag, Heidelberg (2002)

    MATH  Google Scholar 

  9. Grüne, L. and Kloeden, P.E.: Higher Order Numerical Schemes for Affinely Controlled Nonlinear Systems. Numer. Math., 89 (2001) 669–690.

    Article  MATH  MathSciNet  Google Scholar 

  10. Grüne, L. and Kloeden, P.E.: Pathwise Approximation of Random Ordinary Differential Equations. BIT, 41 (2001) 710–721.

    Article  Google Scholar 

  11. Isidori, A.: Nonlinear Control Systems. An Introduction. Second edition, Springer-Verlag, Heidelberg (1995)

    Google Scholar 

  12. Hairer, E., Norsett, S.P. and Wanner, G.: Solving Ordinary Differential Equations I. Springer-Verlag, Heidelberg (1988)

    Google Scholar 

  13. Kloeden, P.E. and Platen, E.: Numerical Solution of Stochastic Differential Equations. Springer-Verlag, Heidelberg (1992) (3rd revised and updated printing, 1999)

    MATH  Google Scholar 

  14. Veliov, V.: On the Time Discretization of Control Systems. SIAM J. Control Optim., 35 (1997) 1470–1486.

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Fachbereich Mathematik, J.W. Goethe-Universität, Postfach 111932, 60054, Frankfurt am Main, Germany

    Lars Grüne & Peter E. Kloeden

Authors
  1. Lars Grüne
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  2. Peter E. Kloeden
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Editor information

Editors and Affiliations

  1. Central Laboratory for Parallel Processing, Bulgarian Academy of Sciences, Acad. G. Bonchev, bl. 25A, 1113, Sofia, Bulgaria

    Ivan Dimov , Ivan Lirkov  & Svetozar Margenov ,  & 

  2. National Environmental Research Institute, Frederiksborgvej 399, P.O. Box 358, 4000, Roskilde, Denmark

    Zahari Zlatev

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Grüne, L., Kloeden, P.E. (2003). Numerical Schemes of Higher Order for a Class of Nonlinear Control Systems. In: Dimov, I., Lirkov, I., Margenov, S., Zlatev, Z. (eds) Numerical Methods and Applications. NMA 2002. Lecture Notes in Computer Science, vol 2542. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36487-0_23

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  • DOI: https://doi.org/10.1007/3-540-36487-0_23

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-00608-4

  • Online ISBN: 978-3-540-36487-0

  • eBook Packages: Springer Book Archive

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