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Transactions of the American Mathematical Society

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



The semigroup property of value functions in Lagrange problems

Author: Peter R. Wolenski
Journal: Trans. Amer. Math. Soc. 335 (1993), 131-154
MSC: Primary 49J52; Secondary 49K15, 49L05
MathSciNet review: 1156301
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Abstract: The Lagrange problem in the calculus of variations exhibits the principle of optimality in a particularly simple form. The binary operation of inf-composition applied to the value functions of a Lagrange problem equates the principle of optimality with a semigroup property. This paper finds the infinitesimal generator of the semigroup by differentiating at $ t = 0$. The type of limit is epigraphical convergence in a uniform sense. Moreover, the extent to which a semigroup is uniquely determined by its infinitesimal generator is addressed. The main results provide a new approach to existence and uniqueness questions in Hamilton-Jacobi theory. When $ L$ is in addition finite-valued, the results are given in terms of pointwise convergence.

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Keywords: Lagrange problems, principle of optimality, epigraphical convergence, Hamilton-Jacobi theory
Article copyright: © Copyright 1993 American Mathematical Society