Abstract
An optimal control problem is considered whose performance index is represented by anm-fold multiple integral, the state equations being given by a system of first-order partial differential equations. The concept of a field of optimal control variables with respect to an independent integral is introduced, the significance of these fields being due to the fact that a control pair is optimal whenever it is imbedded in such a field. Since there arem distinctm-fold independent integrals, it is possible to constructm distinct fields of this kind. For each of these, a Pontryagin function is defined, and it is shown that, if an optimal pair is embedded in one of these fields, it satisfies a corresponding Pontryagin maximum principle.
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Rund, H.,Sufficiency Conditions for Multiple Integral Control Problems, Journal of Optimization Theory and Applications, Vol. 13, No. 2, 1974.
Klötzler, R.,On Pontryagin's Maximum Principle for Multiple Integrals, Analysis-Tagung, Kühlungsborn, Germany (Preprint), 1973.
Rund, H.,The Hamilton-Jacobi Theory in the Calculus of Variations, Van Nostrand Reinhold Company, New York, New York, 1966; Revised and augmented reprint, Krieger Publishing Company, Huntington, New York, 1973.
Butkovskiy, A. G.,Distributed Control Systems, American Elsevier Publishing Company, New York, New York, 1969.
Leitmann, G.,Sufficiency Theorems for Optimal Control, Journal of Optimization Theory and Applications, Vol. 3, No. 1, 1969.
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Communicated by G. Leitmann
This research was supported in part by NSF Grant No. GP-32830
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Rund, H. Pontryagin functions for multiple integral control problems. J Optim Theory Appl 18, 511–520 (1976). https://doi.org/10.1007/BF00932659
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DOI: https://doi.org/10.1007/BF00932659