Skip to main content
SpringerLink
Log in
Book cover

Control, Identification, and Input Optimization pp 77–161Cite as

Numerical Solutions for Nonlinear Two-Point Boundary-Value Problems

  • Chapter

  • 234 Accesses

Abstract

Optimal control problems for linear systems with quadratic performance criteria involve the solution of linear two-point boundary-value problems. The numerical solutions of these problems were discussed in the previous chapter. For nonlinear optimal control problems, iterative methods must be used to obtain the numerical solutions of the nonlinear two-point boundary-value problems. Two iterative methods for the numerical solutions are discussed in this chapter, (i) the method of quasilinearization and (ii) the Newton-Raphson method. These methods can also be used for linear or nonlinear optimal control problems subject to integral constraints.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bellman, R. E., and Kalaba, R. E., Quasilinearization and Nonlinear Boundary-Value Problems, American Elsevier Publishing Company, Inc., New York, p. 127, 1965.

    MATH  Google Scholar 

  2. Roberts, S. M., and Shipman, J. S., Two-Point Boundary-Value Problems: Shooting Methods, American Elsevier Publishing Company, Inc., New York, 1972.

    MATH  Google Scholar 

  3. Kalaba, R. E., and Spingarn, K., Optimization of functional subject to integral constraints, Journal of Optimization Theory and Applications, Vol. 24, No. 2, pp. 325–335, 1978.

    Article  MathSciNet  MATH  Google Scholar 

  4. Gottfried, B. S., and Weisman, J., Introduction to Optimization Theory, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1973.

    Google Scholar 

  5. Bellman, R., Introduction to the Mathematical Theory of Control Processes, Vol. 1, Academic Press, New York, p. 76, 1967.

    Google Scholar 

  6. Kalaba, R., and Ruspini, E., Identification of parameters in nonlinear boundary value problems, Journal of Optimization Theory and Applications, Vol. 4, No. 6, pp. 371–377, 1969.

    Article  MathSciNet  MATH  Google Scholar 

  7. Miele, A., and Levy, A. V., Modified quasilinearization and optimal initial choice of the multipliers, Part 1, Mathematical programming problems, Journal of Optimization Theory and Applications, Vol. 6, No. 5, pp. 364–380, 1970.

    Article  MathSciNet  MATH  Google Scholar 

  8. Miele, A., Iyer, R. R., and Well, K. H., Modified quasilinearization and optimal initial choice of the multipliers, Part 2, Optimal control problems, Journal of Optimization Theory and Applications, Vol. 6, No. 5, pp. 381–409, 1970.

    Article  MathSciNet  MATH  Google Scholar 

  9. Kalaba, R. E., and Spingarn, K., Design of linear regulators with energy constraints, IEEE Conference on Decision and Control, Albuquerque, pp. 301–305, December, 1980.

    Google Scholar 

  10. Kalman, R. E., Contributions to the theory of optimal control, Boletin Sociedad Matematica Mexicana, Vol. 5, No. 1, pp. 102–119, 1960.

    MathSciNet  Google Scholar 

  11. Athans, M., and Falb, P., Optimal Control, McGraw-Hill Book Company, New York, pp. 750–814, 1966.

    MATH  Google Scholar 

  12. Bryson, A. E., Jr, and Ho, Y., Applied Optimal Control, Blaisdell Publishing Co., Waltham, Massachusetts, 1969.

    Google Scholar 

  13. Sage, A. P., Optimum Systems Control, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1968.

    MATH  Google Scholar 

  14. Harvey, C. A., and Stein, G., Quadratic weights for asymptotic regulator properties, IEEE Transactions on Automatic Control, Vol. 23, No. 3, pp. 378–387, 1978.

    Article  MathSciNet  MATH  Google Scholar 

  15. Stein, G., Generalized quadratic weights for asymptotic regulator properties, IEEE Conference on Decision and Control, San Diego, California, pp. 831–837, 1978.

    Google Scholar 

  16. Kalaba, R. E., Spingarn, K., and Zagustin, E., On the integral equation method for buckling loads, Applied Mathematics and Computation, Vol. 1, No. 3, pp. 253–261, 1975.

    Article  MathSciNet  MATH  Google Scholar 

  17. Vol’mir, A. S., Stability of Elastic Systems, Fitmatgiz, Moscow, 1965.

    Google Scholar 

  18. Popov, E. P., Mechanics of Materials, Prentice-Hall, Inc., New York, 1952.

    Google Scholar 

  19. Kagiwada, H., and Kalaba, R., An Initial Value Method for Fredholm Resolvents, Biomedical Engineering Program, University of Southern California, Los Angeles, California, 1970.

    Google Scholar 

  20. Kalaba, R. E., and Spingarn, K., Numerical solution of the integral equations for optimal sequential filtering, International Journal for Numerical Methods in Engineering, Vol. 6, No. 3, pp. 451–454, 1973.

    Article  MathSciNet  Google Scholar 

  21. Kalaba, R. E., Spingarn, K., and Zagustin, E., An imbedding method for buckling loads, Computers and Mathematics with Applications, Vol. 1, No. 3/4, pp. 277–284, 1975.

    Google Scholar 

  22. Casti, J., and Kalaba, R., Imbedding Methods in Applied Mathematics, Addison-Wesley Publishing Company, Reading, Massachusetts, 1973.

    MATH  Google Scholar 

  23. Alspaugh, D., Kagiwada, H., and Kalaba, R., Applications of invariant imbedding to the buckling of columns, Journal of Computational Physics, Vol. 5, No. 1, pp. 56–69, February 1970.

    Article  MathSciNet  Google Scholar 

  24. Kalaba, R. E., and Spingarn, K., Numerical solution of a nonlinear two-point boundary value problem by an imbedding method, Nonlinear Analysis, Theory, Methods, & Applications, Vol. 1, No. 2, pp. 129–133, 1977.

    Article  MathSciNet  MATH  Google Scholar 

  25. Kagiwada, H., and Kalaba, R., Integral Equations via Imbedding Methods, Addison-Wesley Publishing Company, Reading, Massachusetts, 1974.

    MATH  Google Scholar 

  26. Kalaba, R., and Zagustin, E., Reduction of a nonlinear two-point boundary value problem with nonlinear boundary conditions to a Cauchy system, International Journal of Nonlinear Mechanics, Vol. 9, pp. 221–228, 1974.

    Article  MathSciNet  MATH  Google Scholar 

  27. Huss, R., and Kalaba, R., Invariant imbedding and the numerical determination of Green’s functions, Journal of Optimization Theory and Applications, Vol. 6, pp. 415–423, 1970.

    Article  MathSciNet  MATH  Google Scholar 

  28. Mikhlin, S., and Smolitskiy, K., Approximate Methods for Solution of Differential and Integral Equations, American Elsevier Publishing Company, Inc., New York, 1967.

    MATH  Google Scholar 

  29. Kalaba, R. E., and Spingarn, K., Post buckling beam configurations via an imbedding method, Computers and Mathematics with Applications, Vol. 4, No. 1, pp. 1–10, 1978.

    Article  MathSciNet  MATH  Google Scholar 

  30. Kalaba, R. E., Spingarn, K., and Zagustin, E., Imbedding methods for bifurcation problems and post-buckling behavior of nonlinear columns, in Nonlinear Systems and Applications, edited by V. Lakshmikantham, Academic Press, New York, pp. 173–187, 1977.

    Google Scholar 

  31. Kalaba, R., Dynamic programming and the variational principles of classical and statistical mechanics, Developments in Mechanics, Vol. 1, Plenum Press, New York, pp. 1–9, 1961.

    Google Scholar 

  32. Abramowitz, M., and Stegum, I. A., editors, Handbook of Mathematical Functions, National Bureau of Standards, Applied Mathematics Series-55, U. S. Government Printing Office, Washington, D.C., 1964.

    MATH  Google Scholar 

  33. Kalaba, R., Spingarn, K., and Tesfatsion, L., A sequential method for nonlinear filtering: Numerical implementation and comparisons, Journal of Optimization Theory and Applications, Vol. 34, No. 4, pp. 543–561, 1981.

    Article  MathSciNet  Google Scholar 

  34. Bellman, R., Kagiwada, H., Kalaba, R., and Sridhar, R., Invariant imbedding and nonlinear filtering theory, Journal of the Astronautical Sciences, Vol. 13, pp. 110–115, 1966.

    MathSciNet  Google Scholar 

  35. Detchmendy, D., and Sridhar, R., Sequential estimation of states and parameters in noisy nonlinear dynamical systems, Transactions of the ASME, pp. 362–368, 1966.

    Google Scholar 

  36. Kagiwada, H., Kalaba, R., Schumitsky, A., and Sridhar, R., Invariant imbedding and sequential interpolating filters for nonlinear processes, Journal of Basic Engineering, Vol. 91, pp. 195–200, 1969.

    Article  Google Scholar 

  37. Sugisaka, M., and Sagara, S., A non-linear filter with higher-order weight functions functions via invariant imbedding, International Journal of Control, Vol. 21, pp. 801–823, 1975.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Southern California, Los Angeles, California, USA

    Robert Kalaba

  2. Hughes Aircraft Company, Los Angeles, California, USA

    Karl Spingarn

Authors
  1. Robert Kalaba
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Karl Spingarn
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 1982 Plenum Press, New York

About this chapter

Cite this chapter

Kalaba, R., Spingarn, K. (1982). Numerical Solutions for Nonlinear Two-Point Boundary-Value Problems. In: Control, Identification, and Input Optimization. Mathematical Concepts and Methods in Science and Engineering. Springer, Boston, MA. https://doi.org/10.1007/978-1-4684-7662-0_4

Download citation

  • DOI: https://doi.org/10.1007/978-1-4684-7662-0_4

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4684-7664-4

  • Online ISBN: 978-1-4684-7662-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation