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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Optimal arcs and the minimum value function in problems of Lagrange

Author: R. Tyrrell Rockafellar
Journal: Trans. Amer. Math. Soc. 180 (1973), 53-83
MSC: Primary 49A10
MathSciNet review: 0320852
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Abstract: Existence theorems are proved for basic problems of Lagrange in the calculus of variations and optimal control theory, in particular problems for arcs with both endpoints fixed. Emphasis is placed on deriving continuity and growth properties of the minimum value of the integral as a function of the endpoints of the arc and the interval of integration. Control regions are not required to be bounded. Some results are also obtained for problems of Bolza. Conjugate convex functions and duality are used extensively in the development, but the problems themselves are not assumed to be especially “convex". Constraints are incorporated by the device of allowing the Lagrangian function to be extended-real-valued. This necessitates a new approach to the question of what technical conditions of regularity should be imposed that will not only work, but will also be flexible and general enough to meet the diverse applications. One of the underlying purposes of the paper is to present an answer to this question.

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Keywords: Optimal control, problems of Lagrange, problems of Bolza, existence of solutions, minimum value function, nonfinite Lagrangians, conjugate convex functions, unbounded controls
Article copyright: © Copyright 1973 American Mathematical Society