A method for solving two-point boundary-value problems of difference equations

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

Abstract

The paper proposes an iterative solution method for discrete-time, nonlinear, two-point boundary-value problems (TPBVP) of the form:
$$\begin{gathered} x(k) - x(k - 1) = f(k, x(k - 1), p(k)), \hfill \\ p(k) - p(k - 1) = g(k, x(k - 1), p(k)), \hfill \\ \end{gathered} $$
subject to
$$h(x(0), p(0)) = 0,e(x(N), p(N)) = 0.$$
It is a counterpart of a method recently proposed by the authors for similar continuous-time TPBVPs with ordinary differential equations. The method, based on invariant imbedding and a generalized Riccati transformation, reduces the TPBVP to a pair of approximate initial-value problems with ordinary difference equations. Numerical tests are run on two examples originating in optimal control problems.

Key Words

Two-point boundary-value problems difference equations invariant imbedding Riccati transformation iterative methods 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Orava, P. J., andLautala, P. A. J.,Back-and-Forth Shooting Method for Solving Two-Point Boundary-Value Problems, Journal of Optimization Theory and Applications, Vol. 18, No. 4, 1976.Google Scholar
  2. 2.
    Orava, P. J., andLautala, P. A. J.,Interval Length Continuation Method for Solving Two-Point Boundary-Value Problems, Journal of Optimization Theory and Applications, Vol. 23, No. 2, 1977.Google Scholar
  3. 3.
    Falb, P. L., andDeJong, J. L.,Some Successive Approximation Methods in Control and Oscillation Theory, Academic Press, New York, New York, 1969.Google Scholar
  4. 4.
    Canon, M. D., Cullum, C. D., andPolak, E.,Theory of Optimal Control and Mathematical Programming, McGraw-Hill Book Company, New York, New York, 1970.Google Scholar
  5. 5.
    Sage, A. P., andMelsa, J. L.,Estimation Theory with Applications to Communications and Control, McGraw-Hill Book Company, New York, New York, 1971.Google Scholar
  6. 6.
    Dorato, P., andLevis, A. H.,Optimal Linear Regulators: The Discrete-Time Case, IEEE Transactions on Automatic Control, Vol. AC-16, No. 6, 1971.Google Scholar
  7. 7.
    Roberts, S. M., andShipman, J. S.,Two-Point Boundary-Value Problems: Shooting Methods, American Elsevier Publishing Company, New York, New York, 1972.Google Scholar
  8. 8.
    Polak, E.,Computational Methods in Optimization, Academic Press, New York, New York, 1971.Google Scholar
  9. 9.
    Schley, C. H., andLee, I.,Optimal Control Computation by the Newton-Raphson Method and the Riccati Transformation, IEEE Transactions on Automatic Control, Vol. AC-12, No. 2, 1967.Google Scholar
  10. 10.
    Bashein, G., andEnns, M.,Computation of Optimal Controls by a Method Combining Quasi-Linearization and Quadratic Programming, International Journal of Control, Vol. 16, No. 1, 1972.Google Scholar
  11. 11.
    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
  12. 12.
    Meyer, G. H.,Initial-Value Methods for Boundary-Value Problems, Academic Press, New York, New York, 1973.Google Scholar

Copyright information

© Plenum Publishing Corporation 1978

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

Personalised recommendations