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Direct method for solving optimal control problems with kinks

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Abstract

In this paper, we introduce a direct solution concept applicable to a slightly restricted class of optimal control problems with kinks. The method treats the optimal trajectory when it lies on a kink for an interval of time (a kink arc) in a special manner. The nonkink arc must satisfy the classical set of necessary conditions of the maximum principle, excluding from that set terminal equations on the kink terminal (the junction between this nonkink arc and a kink arc). At this junction, certain conditions, which compensate for the missing terminal conditions, have to be satisfied. In this paper, the method is applied to solve a fairly realistic model for an important inventory problem, namely, the problem of scheduling the production of a commodity.

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References

  1. 1.

    Luenberger, D. G.,Control Problems with Kinks, IEEE Transactions on Automatic Control, Vol. AC-15, pp. 570–575, 1970.

  2. 2.

    Vincent, T. L., andMason, J. D.,Disconnected Optimal Trajectories, Journal of Optimization Theory and Applications, Vol. 3, pp. 263–281, 1969.

  3. 3.

    Ghanem, M. Z.,Optimal Control Problems with Nondifferential Cost Functionals, Stanford University, Ph.D. Thesis, 1971.

  4. 4.

    Arrow, K. J., Karlin, S., andScarf, H.,Studies in Mathematical Theory of Inventory and Production, Stanford University Press, Stanford, California, 1959.

Additional Bibliography

  1. 5.

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

  2. 6.

    Ghanam, M. Z.,Optimal Entry-Optimal Exit Method for Solving Optimal Control Problems with Kinks, Fourth Asilomar Conference on Circuits and Systems, Pacific Grove, California, 1970.

  3. 7.

    Bryson, A. E., andHo, Y. C.,Applied Optimal Control: Optimization, Estimation, and Control, Blaisdell Publishing Company, Waltham, Massachusetts, 1969.

  4. 8.

    Athans, M., andFalb, P. L.,Optimal Control, McGraw-Hill Book Company, New York, New York, 1966.

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Additional information

This research was supported by the National Science Foundation, Grant No. GK-16125. The author is indebted to Professor D. Luenberger of Stanford University for many discussions which led to this research effort.

Communicated by G. Leitmann

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Ghanem, M.Z. Direct method for solving optimal control problems with kinks. J Optim Theory Appl 14, 405–417 (1974). https://doi.org/10.1007/BF00933306

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Key Words

  • Maximum principle
  • control theory
  • inventory theory
  • optimization theorems