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Back-and-forth shooting method for solving two-point boundary-value problems

  • P. J. Orava
  • P. A. J. Lautala
Contributed Papers

Abstract

The paper proposes a special iterative method for a nonlinear TPBVP of the form\(\dot x\)(t)=f(t, x(t),p(t)),\(\dot p\)(t)=g(t, x(t),p(t)), subject toh(x(0),p(0))=0,e(x(T),p(T))=0. Certain stability properties of the above differential equations are taken into consideration in the method, so that the integration directions associated with these equations respectively are opposite to each other, in contrast with the conventional shooting methods. Via an embedding and a Riccati-type transformation, the TPBVP is reduced to consecutive initial-value problems of ordinary differential equations. A preliminary numerical test is given by a simple example originating in an optimal control problem.

Key Words

Two-point boundary-value problems shooting methods invariant embedding Riccati transformation iterative numerical methods 

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References

  1. 1.
    Falb, P. L., andDeJong, J. L.,Some Successive Approximation Methods in Control and Oscillation Theory, Academic Press, New York, New York, 1969.Google Scholar
  2. 2.
    Miele, A., Naqvi, S., Levy, A. V., andIyer, R. R.,Numerical Solution of Nonlinear Equations and Nonlinear Two-Point Boundary-Value Problems, Advances in Control Systems, Vol. 8, Edited by C. T. Leondes, Academic Press, New York, New York, 1971.Google Scholar
  3. 3.
    Roberts, S. M., andShipman, J. S.,Two-Point Boundary Value Problems: Shooting Methods, American Elsevier Publishing Company, New York, New York, 1972.Google Scholar
  4. 4.
    Miele, A., andIyer, R. R.,General Technique for Solving Nonlinear, Two-Point Boundary-Value Problems Via the Method of Particular Solutions, Journal of Optimization Theory and Applications, Vol. 5, No. 5, 1970.Google Scholar
  5. 5.
    Keller, H. B.,Shooting and Embedding for Two-Point Boundary Value Problems, Journal of Mathematical Analysis and Applications, Vol. 36, No. 3, 1971.Google Scholar
  6. 6.
    Mataušek, M. R.,Direct Shooting Method for the Solution of Boundary-Value Problems, Journal of Optimization Theory and Applications, Vol. 12, No. 2, 1973.Google Scholar
  7. 7.
    Lastman, G. J.,Obtaining Starting Values for the Shooting Method Solution of a Class of Two-Point Boundary-Value Problems, Journal of Optimization Theory and Applications, Vol. 14, No. 3, 1974.Google Scholar
  8. 8.
    Georganas, N. D., andChatterjee, A.,Invariant Imbedding and the Continuation Method: A Comparison, International Journal of Systems Science, Vol. 6, No. 3, 1975.Google Scholar
  9. 9.
    Meyer, G. H.,Initial Value Methods for Boundary Value Problems, Academic Press, New York, New York, 1973.Google Scholar
  10. 10.
    Mufti, I. H., Chow, C. K., andStock, F. T.,Solution of Ill-Conditioned Linear Two-Point Boundary Value Problems by the Riccati Transformation, SIAM Review, Vol. 11, No. 4, 1969.Google Scholar

Copyright information

© Plenum Publishing Corporation 1976

Authors and Affiliations

  • P. J. Orava
    • 1
  • P. A. J. Lautala
    • 2
  1. 1.Systems Theory LaboratoryHelsinki University of TechnologyOtaniemiFinland
  2. 2.Control Engineering LaboratoryHelsinki University of TechnologyOtaniemiFinland

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