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Numerical study of the controlled Van der Pol oscillator in Chebyshev series

  • René Van Dooren
Brief Reports

Summary

The optimal control of the Van der Pol oscillator is investigated by a numerical method based on the expansion of the state function and the control strategy in Chebyshev series. The optimal control problem is reduced to a parameter optimization problem.

Keywords

Control Problem Mathematical Method Parameter Optimization Optimal Control Problem State Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Resumé

On étudie le contrôle optimal de l'oscillateur de Van der Pol à l'aide d'une méthode numérique qui utilise le développement des fonctions de position et de contrôle en série de Chebyshev. Le problème de contrôle optimal est réduit à un problème d'optimisation de paramètres.

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References

  1. [1]
    J. C. Gille, P. Decaulne and M. Pelegrin,Systèmes asservis non linéaires. Bordas, Paris 1975.Google Scholar
  2. [2]
    A. H. Nayfeh and D. T. Mook,Nonlinear Oscillations. Wiley, New York 1979.Google Scholar
  3. [3]
    R. Van Dooren and J. Vlassenbroeck,A new look at the brachistochrone problem. Z. angew. Math. Phys.31, 785–790 (1980).Google Scholar
  4. [4]
    R. Van Dooren and J. Vlassenbroeck,Chebyshev series solution of the controlled Duffing oscillator. J. Comput. Phys.47, 321–329 (1982).Google Scholar
  5. [5]
    M. Urabe and A. Reiter,Numerical computation of nonlinear forced oscillations by Galerkin's procedure. J. Math. Anal. Appl.14, 107–140 (1966).Google Scholar
  6. [6]
    R. Van Dooren,Numerical computation of forced oscillations in coupled Duffing equations. Numer. Math.20, 300–311 (1973).Google Scholar
  7. [7]
    R. Van Dooren,Differential tones in a damped mechanical system with quadratic and cubic non-linearities. Int. J. Non-Linear Mech.8, 575–583 (1973).Google Scholar
  8. [8]
    R. Van Dooren,Orbit computation in Celestial mechanics by Urabe's method. Publ. Res. Inst. Math. Sci., Kyoto University9, 535–542 (1974).Google Scholar
  9. [9]
    R. Van Dooren,Two mode subharmonic vibrations of order 1/9 of a non-linear beam forced by a two mode harmonic load. J. Sound Vib.41, 133–142 (1975).Google Scholar
  10. [10]
    R. Van Dooren,Forced two mode subharmonic vibrations of a nonlinear beam by a new analytical method. Abhandlungen der Akademie der Wissenschaften der DDR. Akademie-Verlag, Berlin (1977), No. 5, pp. 253–260.Google Scholar
  11. [11]
    R. Van Dooren,Numerical computation of solutions to the KdV equation in double Chebyshev series. Z. angew. Math. Phys.34, 118–123 (1983).Google Scholar
  12. [12]
    G. Dahlquist and A. Bjorck,Numerical Methods. Prentice Hall, Englewood Cliffs, N. J. 1974.Google Scholar
  13. [13]
    L. Fox and I. B. Parker,Chebyshev Polynomials in Numerical Analysis. Oxford University Press, Oxford 1972.Google Scholar
  14. [14]
    M. Urabe,Numerical solution of boundary value problems in Chebyshev series. Lect. Notes in Math.109, Springer, Berlin (1969), pp. 40–86.Google Scholar
  15. [15]
    L. S. Pontryagin, V. Boltyanskii, R. Gamkrelidze and E. Mischenko,The Mathematical Theory of Optimal Processes. Interscience Publ., New York 1962.Google Scholar

Copyright information

© Birkhäuser Verlag 1987

Authors and Affiliations

  • René Van Dooren
    • 1
  1. 1.Dept. of Analytical MechanicsVrije Universiteit BrusselBrusselsBelgium

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