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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Geometric theory of extremals in optimal control problems. I. The fold and Maxwell case
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by I. Kupka PDF
Trans. Amer. Math. Soc. 299 (1987), 225-243 Request permission


The behavior of the extremal curves in optimal control theory is much more complex than that of their namesakes in the classical calculus of variations. Here we analyze the simplest instances of singular behavior of these extremals. Among others, in sharp contrast to the classical case, a ${C^0}$-limit of extremals may not be an extremal. In the simplest cases (elliptic fold and Maxwell points) of this occurrence, the limits are trajectories of a new vector field. A special case of this field showed up in some work of V. I. Arnold. Results related to ours have been obtained in low dimension by I. Ekeland.
  • V. I. Arnol′d, Lagrangian manifolds with singularities, asymptotic rays and the unfurled swallowtail, Funktsional. Anal. i Prilozhen. 15 (1981), no. 4, 1–14, 96 (Russian). MR 639196
  • Ivar Ekeland, Discontinuités de champs hamiltoniens et existence de solutions optimales en calcul des variations, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 5–32 (1978) (French). MR 493584, DOI 10.1007/BF02684338
  • R. Abraham and J. E. Marsden, Foundations of mechanics, Benjamin, New York, 1966.
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Additional Information
  • © Copyright 1987 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 299 (1987), 225-243
  • MSC: Primary 49A10; Secondary 49E99, 58C27, 58E30
  • DOI:
  • MathSciNet review: 869409