Abstract
We explore an approach involving the use of calculus of variations techniques for discrete event dynamic system (DEDS) performance optimization problems. The approach is motivated by the observation that such problems can be described by separable cost functions and recursive dynamics of the same form as that used to describe conventional discrete-time continuous-variable optimal control problems. Three important difficulties are that DEDS are generally stochastic, their dynamics typically involve max and min operations, which are not everywhere differentiable, and the state variables are often discrete. We demonstrate how to overcome these difficulties by applying the approach to a transportation problem, modeled as a polling system, where we are able to derive an explicit and intuitive analytic expression for an optimal control policy.
Similar content being viewed by others
References
Ho, Y. C., and Cao, X. R., Perturbation Analysis of Discrete Event Dynamic Systems, Kluwer Academic Publishers, Boston, Massachusetts, 1991.
Cassandras, C. G., and Panayiotou, C., Concurrent Sample Path Estimation for Discrete Event Systems, Proceedings of the 35th IEEE Conference on Decision and Control, pp. 3332–3337, 1996.
Rubinstein, R. Y., and Shapiro, A., Discrete Event Systems: Sensitivity Analysis and Stochastic Optimization by the Score Function Method, John Wiley and Sons, New York, New York, 1993.
Ho, Y. C., Overview of Ordinal Optimization, Proceedings of the 33rd IEEE Conference on Decision and Control, pp. 1975–1977, 1993.
Ho, Y. C., and Deng, M., The Problem of Large Search Space in Stochastic Optimization, Proceedings of the 33rd IEEE Conference on Decision and Control, pp. 1470–1475, 1993.
Bryson, A. E. Jr., and Ho, Y. C., Applied Optimal Control: Optimization, Estimation, and Control, Hemisphere Publishing Company, Washington, DC, 1975.
Sage, A. P., and White, C. C., III, Optimum Systems Control, 2nd Edition, Prentice-Hall, Englewood Cliffs, New Jersey, 1977.
Whittle, P., Optimization over Time: Dynamic Programming and Stochastic Control, Vol. 1, John Wiley and Sons, New York, New York, 1982.
Whittle, P., Optimization over Time: Dynamic Programming and Stochastic Control, Vol. 2, John Wiley and Sons, New York, New York, 1983.
Whittle, P., Optimal Control: Basics and Beyond, John Wiley and Sons, New York, New York, 1996.
Gazarik, M., and Wardi, Y., Optimal Release Times in a Single Server: An Optimal Control Perspective, Proceedings of the 35th IEEE Conference on Decision and Control, pp. 3831–3836, 1996.
Pepyne, D. L., and Cassandras, C. G., Modeling, Analysis, and Optimal Control of a Class of Hybrid Systems, Journal of Discrete Event Dynamic Systems, Vol. 8, pp. 175–201, 1998.
Clarke, F. H., Methods of Dynamic and Nonsmooth Optimization, SIAM, Philadelphia, Pennsylvania, 1989.
Clarke, F. H., Optimization and Nonsmooth Analysis, Wiley-Interscience, New York, New York, 1983.
Rockafellar, R. T., The Theory of Subgradients and Its Application to Problems of Optimization: Convex and Nonconvex Functions, Heldermann Verlag, Berlin, Germany, 1981.
Garfinkel, R. S., and Nemhauser, G. L., Integer Programming, John Wiley and Sons, New York, New York, 1972.
Minoux, M., Mathematical Programming: Theory and Algorithms, John Wiley and Sons, New York, New York, 1986.
Taha, H. A., Integer Programming: Theory, Applications, and Computations, Academic Press, New York, New York, 1975.
Baccelli, F., Cohen, G., Olsder, G. J., and Quadrat, J. P., Synchronization and Linearity, John Wiley and Sons, New York, New York, 1992.
Cassandras, C. G., Discrete Event Systems: Modeling and Performance Analysis, Irwin and Aksen, Boston, Massachusetts, 1993.
Cassandras, C. G., Lafortune, S., and Olsder, G. J., Introduction to the Modelling, Control, and Optimization of Discrete Event Systems, Trends in Control, Edited by A. Isidori, Springer Verlag, Berlin, Germany, pp. 217–292, 1995.
Kleinrock, L., Queueing Systems, Vol. 1: Theory, Wiley-Interscience, New York, New York, 1975.
Murata, T., Petri Nets: Properties, Analysis, and Applications, Proceedings of the IEEE, Vol. 77, pp. 541–580, 1989.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Pepyne, D.L., Cassandras, C.G. Performance Optimization of a Class of Discrete Event Dynamic Systems Using Calculus of Variations Techniques. Journal of Optimization Theory and Applications 100, 599–622 (1999). https://doi.org/10.1023/A:1022690507461
Issue Date:
DOI: https://doi.org/10.1023/A:1022690507461