Abstract
Many optimization problems in economic analysis, when cast as optimal control problems, are initial-value problems, not two-point boundary-value problems. While the proof of Pontryagin (Ref. 1) is valid also for initial-value problems, it is desirable to present the potential practitioner with a simple proof specially constructed for initial-value problems. This paper proves the Pontryagin maximum principle for an initial-value problem with bounded controls, using a construction in which all comparison controls remain feasible. The continuity of the Hamiltonian is an immediate corollary. The same construction is also shown to produce the maximum principle for the problem of Bolza.
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Pontryagin, L. S., Boltyanskii, V. G., Gamkrelidze, R. V., andMishchenko, E. F.,The Mathematical Theory of Optimal Processes, John Wiley and Sons (Interscience Publishers), New York, 1969.
Courant, R.,Differtial and Integral Calculus, Vol. 2, John Wiley and Sons (Interscience Publishers), New York, 1969.
Athans, M., andFalb, P. L.,Optimal Control, McGraw-Hill Book Company, New York, 1966.
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Communicated by G. B. Dantzig
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Hartberger, R.J. A proof of the Pontryagin maximum principle for initial-value problems. J Optim Theory Appl 11, 139–145 (1973). https://doi.org/10.1007/BF00935879
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DOI: https://doi.org/10.1007/BF00935879