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Local convergence of an algorithm for solving optimal control problems

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Abstract

In this paper, a method is proposed for the numerical solution of optimal control problems with terminal equality constraints. The multiplier method is employed to deal with the terminal equality constraints. It is shown that a sequence of control functions, which converges to the optimal control, is obtained by the alternate update of control functions and multipliers.

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References

  1. Hestenes, M. R.,Multiplier and Gradient Methods, Journal of Optimization Theory and Applications, Vol. 4, pp. 303–320, 1969.

    Google Scholar 

  2. Hestenes, M. R.,An Indirect Sufficiency Proof for the Problem of Bolza in Nonparametric Form, Transactions of the American Mathematical Society, Vol. 62, pp. 509–535, 1947.

    Google Scholar 

  3. Rupp, R. D.,A Method for Solving a Quadratic Optimal Control Problem, Journal of Optimization Theory and Applications, Vol. 9, pp. 238–250, 1972.

    Google Scholar 

  4. Rupp, R. D.,Approximation of the Classical Isoperimetric Problems, Journal of Optimization Theory and Applications, Vol. 9, pp. 251–264, 1972.

    Google Scholar 

  5. Glad, S. T.,A Combination of Penalty Function and Multiplier Methods for Solving Optimal Control Problems, Journal of Optimization Theory and Applications, Vol. 28, pp. 303–329, 1979.

    Google Scholar 

  6. Nahra, J. E.,Balance Function for the Optimal Control Problem, Journal of Optimization Theory and Applications, Vol. 8, pp. 35–48, 1971.

    Google Scholar 

  7. Rupp, R. D.,On the Combination of the Multiplier Method of Hestenes and Powell with Newton's Method, Journal of Optimization Theory and Applications, Vol. 15, pp. 167–187, 1975.

    Google Scholar 

  8. Sakawa, Y., andShindo, Y.,On Global Convergence of an Algorithm for Optimal Control, IEEE Transactions on Automatic Control, Vol. AC-25, pp. 1149–1153, 1980.

    Google Scholar 

  9. Järmark, B.,On Convergence Control in Differential Dynamic Programming Applied to Realistic Aircraft and Differential Game Problems, Proceedings of the 1977 IEEE Conference on Decision and Control, New Orleans, Louisiana, pp. 471–479, 1977.

  10. Järmark, B.,A New Convergence Control Technique in Differential Dynamic Programming, The Royal Institute of Technology, Stockholm, Sweden, Department of Automatic Control, Report No. TRITA-REG-7502, 1975.

  11. Sakawa, Y., Shindo, Y., andHashimoto, Y.,Optimal Control of a Rotary Crane, Journal of Optimization Theory and Applications, Vol. 35, pp. 535–557, 1981.

    Google Scholar 

  12. Sakawa, Y., andShindo, Y.,Optimal Control of Container Cranes, Automatica, Vol. 18, pp. 257–266, 1982.

    Google Scholar 

  13. Berkovitz, L. D.,Variational Methods in Problems of Control and Programming, Journal of Mathematical Analysis and Applications, Vol. 3, pp. 145–169, 1961.

    Google Scholar 

  14. Bliss, G. A.,Lectures on the Calculus of Variations, The University of Chicago Press, Chicago, Illinois, 1946.

    Google Scholar 

  15. Reid, W. T.,Riccati Differential Equations, Academic Press, New York, New York, 1972.

    Google Scholar 

  16. Coddington, E. A., andLevinson, N.,Theory of Ordinary Differential Equations, McGraw-Hill Book Company, New York, New York, 1955.

    Google Scholar 

  17. Walter, W.,Differential and Integral Inequalities, Springer-Verlag, Berlin, Germany, 1970.

    Google Scholar 

  18. Dyer, P., andMcReynolds, S. R.,The Computation and Theory of Optimal Control, Academic Press, New York, New York, 1970.

    Google Scholar 

  19. Urabe, M.,The Newton Method and Its Application to Boundary-Value Problems with Nonlinear Boundary Conditions, Proceedings of the United States-Japan Seminar on Differential and Functional Equations, Edited by W. A. Harris, Jr., and Y. Sibuya, W. A. Benjamin, New York, New York, 1967.

    Google Scholar 

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Communicated by M. R. Hestenes

The authors wish to thank Dr. N. Fujii for his most valuable comments and suggestions.

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Shindo, Y., Sakawa, Y. Local convergence of an algorithm for solving optimal control problems. J Optim Theory Appl 46, 265–293 (1985). https://doi.org/10.1007/BF00939285

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