Optimal Control

  • Harriet Kagiwada
  • Robert Kalaba
  • Nima Rasakhoo
  • Karl Spingarn
Part of the Mathematical Concepts and Methods in Science and Engineering book series (MCSENG, volume 31)


The automatic solution of a certain class of optimal control problems is described in this chapter. Optimal control problems involve the solution of two-point boundary value problems. The derivatives in the two-point boundary value equations are evaluated automatically and the complete solution of the optimal control problem is obtained. None of the derivatives usually associated with the Euler-Lagrange equations, Pontryagin–s maximum principle, the Newton-Raphson method, or the gradient method need be calculated by hand. Depending on the problem and the method of solution used, the user of the program need only specify the initial conditions and the terminal time, and input one of the following by calling the appropriate FORTRAN subroutines: (i) the integrand of the cost functional and the differential constraints if applicable, (ii) the integrand of the cost functional and the Hamiltonian function, or (iii) only the Hamiltonian function.


Optimal Control Problem Hamiltonian Function Terminal Time Grid Interval Automatic Solution 
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.


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Chapter 3

  1. 1.
    Kalaba, R., and Spingarn, K., Control, Identification, and Input Optimization ,Plenum Press, New York, 1982.MATHGoogle Scholar
  2. 2.
    Athans, M., and Falb, P., Optimal Control ,McGraw-Hill, New York, 1966.MATHGoogle Scholar
  3. 3.
    Bryson, A. E., Jr., and Ho, Y., Applied Optimal Control ,Blaisdell, Waltham, Massachusetts, 1969.Google Scholar
  4. 4.
    Sage, A. P., and White, C. C., III., Optimum Systems Control ,Prentice-Hall, Engle wood Cliffs, New Jersey, 1977.Google Scholar
  5. 5.
    Gottfried, B. S., and Weissman, J., Introduction to Optimization Theory ,Prentice Hall, Englewood Cliffs, New Jersey, 1973.Google Scholar
  6. 6.
    Pontryagin, L. S., The Mathematical Theory of Optimal Processes ,Wiley Interscience, New York, 1962.MATHGoogle Scholar
  7. 7.
    GelfandI. M., and FominS. V.Calculus of Variations ,Prentice-Hall, Englewood Cliffs, New Jersey, 1963.Google Scholar
  8. 8.
    Courant, R., and HilbertD.Methods of Mathematical Physics ,Vol. 1, Wiley Interscience, New York, 1953.Google Scholar
  9. 9.
    Miele, A., Introduction to the calculus of variations in one independent variable, in Theory of Optimum Aerodynamic Shapes ,edited by A. Miele, Academic, New York, pp. 3–19, 1965.Google Scholar
  10. 10.
    Leondes, C. T., editor of the series in Control and Dynamic Systems, Advances in Theory and Applications ,Academic, New York, Volumes 1–21, 1965–1984.Google Scholar
  11. 11.
    Wengert, R., A simple automatic derivative evaluation program, Communications of the ACM ,Vol. 7, pp. 463–464, 1964.MATHCrossRefGoogle Scholar
  12. 12.
    Bellman, R., Kagiwada, H., and Kalaba, R., Wengert’s numerical method for partial derivatives, orbit determination, and quasilinearization, Communications of the ACM ,Vol. 8, pp. 231–232, 1965.MathSciNetGoogle Scholar
  13. 13.
    Kalaba, R., and Spingarn, K., Automatic solution of optimal control problems, I. Simplest problem in the calculus of variations, Applied Mathematics and Computation ,Vol. 13, pp. 131–148, February 1984.MathSciNetGoogle Scholar
  14. 14.
    Kalaba, R., and Spingarn, K., Automatic solution of optimal control problems, II. First-order nonlinear systems, American Control Conference, San Francisco, California, June 1983.Google Scholar
  15. 15.
    Kalaba, R., and Spingarn, K., Automatic solution of optimal control problems, III. Differential and integral constraints, IEEE Control Systems Magazine ,Vol. 4, pp. 3–8, February 1984.Google Scholar
  16. 16.
    Kalaba, R., and Spingarn, K., Automatic solution of optimal control problems, IV. Gradient method, Applied Mathematics and Computation ,Vol. 14, pp. 289–300, April 1984.MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Kalaba, R., and Spingarn, K., Automatic solution of optimal control problems, V. Second-order nonlinear systems, Seventeenth Asilomar Conference on Circuits, Systems, and Computers, Pacific Grove, California, November 1983.Google Scholar
  18. 18.
    Kalaba, R., and Spingarn, K., Automatic solution of Nth-order optimal control problems, IEEE Transactions on Aerospace and Electronic Systems ,Vol. 21, pp. 345–350, May 1985.MathSciNetCrossRefGoogle Scholar
  19. 19.
    Bellman, R. E., and Kalaba, R. EQuasilinearization and Nonlinear Boundary-Value Problems ,Elsevier, New York, 1965.MATHGoogle Scholar
  20. 20.
    Mullins, E. R., Jr., and Rosen, D., Probability and Calculus ,Bogden & Quigley, Tarrytown-on-the-Hudson, New York, p. 66, 1971.Google Scholar

Copyright information

© Plenum Press, New York 1986

Authors and Affiliations

  • Harriet Kagiwada
    • 1
  • Robert Kalaba
    • 1
  • Nima Rasakhoo
    • 1
  • Karl Spingarn
    • 1
  1. 1.Hughes Aircraft CompanyEl SegundoUSA

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