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Erschienen in:
2013 | OriginalPaper | Buchkapitel
18. Nonsmooth Lagrangians
verfasst von : Francis Clarke
Erschienen in: Functional Analysis, Calculus of Variations and Optimal Control
Verlag: Springer London
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Abstract
We now take a step in a direction never envisaged by the classical theory: the introduction of nonsmooth Lagrangians (as opposed to nonsmooth solutions). This is a modern issue that stems from new applications in such disciplines as engineering, economics, mechanics, and operations research. At the same time, this factor will play an essential role in the proof of such theorems as the multiplier rules of the preceding chapter, even when the underlying problems have smooth data. The new necessary conditions differ most from the earlier ones in the Euler equation, which now becomes an inclusion: We give examples to show why it can be useful to have weakened the regularity hypotheses on the Lagrangian, as is done here.
$$ p\,' (t)\,\in\, {\text {co}}\,\big\{ \, \omega: \big(\omega , p(t)\big)\in\, \partial_{ L}\,\Lambda \big(t, x_{ *}(t), x_* '(t)\big)\big\} {~\,\text {a.e.}}\; t\in\, [\,a ,b\,] . $$