Abstract
Multiprocess problems are dynamic optimization problems in which there is a collection of control systems coupled through constraints in the endpoints of the constituent trajectories and through the cost function. Optimality conditions for such problems posed over a finite interval have already been derived. However, multiprocess problems arise, for example in the mathematical economics literature, in which one of the component processes operates over an infinite horizon. We give a proof of the relevant necessary conditions in the form of a maximum principle under mild and verifiable hypotheses and apply the theory to a two-stage problem in which an explicit dependence on the intermediate time (taken as a choice variable) is incorporated in the integrands of the cost functional.
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Pontryagin, L. S., Boltyanskii, V. G., Gamkrelidze, R. V., andMishchenko, E. F.,The Mathematical Theory of Optimal Processes, English Translation Edited by L. W. Neustadt, Wiley-Interscience, New York, New York, 1962.
Halkin, H.,Necessary Conditions for Optimal Control Problems with Infinite Horizons, Econometrica, Vol. 42, pp. 267–272, 1974.
Carlson, D. A., andHaurie, A.,Infinite-Horizon Optimal Control: Theory and Applications, Lecture Notes in Economics and Mathematical Systems, Springer Verlag, Berlin, Germany, Vol. 270, 1987.
Stern, L. E.,Criteria of Optimality in the Infinite-Time Optimal Control Problem, Journal of Optimization Theory and Applications, Vol. 44, pp. 497–508, 1984.
Weitzman, M. L.,Duality Theory for Infinite-Horizon Convex Models, Management Science, Vol. 19, pp. 783–789, 1973.
Aubin, J. P., andClarke, F. H.,Shadow Prices and Duality for a Class of Optimal Control Problems, SIAM Journal on Control and Optimization, Vol. 17, pp. 567–586, 1979.
Gray, J. A., andSalant, S. W.,Transversality Conditions in Infinite-Horizon Models, International Finance Discussion Paper No. 172, Board of Governors of U.S. Federal Reserve, 1981.
Benveniste, L. M., andScheinkman, J. A.,Duality Theory for Dynamic Optimization Models of Economics: The Continuous-Time Case, Journal of Economic Theory, Vol. 27, pp. 1–19, 1982.
Michel, P.,On the Transversality Condition in Infinite-Horizon Optimal Problems, Econometrica, Vol. 50, pp. 975–985, 1982.
Araujo, A., andScheinkman, J. A.,Maximum Principle and Transversality Condition for Concave Infinite-Horizon Economic Models, Journal of Economic Theory, Vol. 30, pp. 1–16, 1983.
Michel, P.,Some Clarifications on the Transversality Condition, Econometrica, Vol. 58, pp. 705–723, 1990.
Clarke, F. H., andVinter, R. B.,Applications of Optimal Multiprocesses, SIAM Journal on Control and Optimization, Vol. 27, pp. 1048–1071, 1989.
Clarke, F. H., andVinter, R. B.,Optimal Multiprocesses, SIAM Journal on Control and Optimization, Vol. 27, pp. 1072–1091, 1989.
Tomiyama, K., andRossana, R. J.,Two-Stage Optimal Control Problems with an Explicit Switch-Point Dependence: Optimality Criteria and an Example of Delivery Lags and Investment, Journal of Economic Dynamics and Control, Vol. 13, pp. 319–337, 1989.
Maccini, L. J.,Delivery Lags and the Demand for Investment, Review of Economic Studies, Vol. 40, pp. 269–281, 1973.
Hoel, M.,Resource Extraction When a Future Substitute Has an Uncertain Cost, Review of Economic Studies, Vol. 45, pp. 637–644, 1978.
Dasgupta, P., Gilbert, R. J., andStiglitz, J. E.,Invention and Innovation under Alternative Market Structures: The Case of Natural Resources, Review of Economic Studies, Vol. 49, pp. 567–582, 1982.
Rossana, R. J.,Delivery Lags and Buffer Stocks in the Theory of Investment by the Firm, Journal of Economic Dynamics and Control, Vol. 9, pp. 153–193, 1985.
Clarke, F. H.,Optimization and Nonsmooth Analysis, Wiley-Interscience, New York, New York, 1983.
Bellman, R.,Dynamic Programming, Princeton University Press, Princeton, New Jersey, 1957.
Babad, H. R.,Optimality Conditions and Sensitivity Relations in Dynamic Optimization, PhD Thesis, Imperial College of Science, Technology, and Medicine, London University, London, England, 1991.
Fleming, W. H., andRishel, R. W.,Deterministic and Stochastic Optimal Control, Springer Verlag, Berlin, Germany, 1975.
Royden, H. L.,Real Analysis, Collier-Macmillan, New York, New York, 1988.
Loewen, P. D., Clarke, F. H., andVinter, R. B.,Differential Inclusions with Free Time, Annales de l'Institut Henri Poincaré, Vol. 5, pp. 573–593, 1988.
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Communicated by L. D. Berkovitz
This research was carried out while the author was a Graduate Student at the Department of Electrical Engineering, Imperial College of Science, Technology, and Medicine, London, England. The author is grateful to Professor R. B. Vinter for his advice and helpful discussions.
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Babad, H.R. An infinite-horizon multistage dynamic optimization problem. J Optim Theory Appl 86, 529–552 (1995). https://doi.org/10.1007/BF02192158
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DOI: https://doi.org/10.1007/BF02192158