# Weierstrass conditions (for a variational extremum)

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Necessary and (partially) sufficient conditions for a strong extremum in the classical calculus of variations (cf. Variational calculus). Proposed in 1879 by K. Weierstrass.

Weierstrass' necessary condition: For the functional to attain a strong local minimum on the extremal , it is necessary that the inequality where is the Weierstrass -function, be satisfied for all , , and all . This condition may be expressed in terms of the function (cf. Pontryagin maximum principle). The Weierstrass condition ( on the extremal ) is equivalent to saying that the function , where , attains a maximum in for . Thus, Weierstrass' necessary condition is a special case of the Pontryagin maximum principle.

Weierstrass' sufficient condition: For the functional to attain a strong local minimum on the vector function , it is sufficient that there exists a vector-valued field slope function (geodesic slope) (cf. Hilbert invariant integral) in a neighbourhood of the curve , for which and for all and any vector .

How to Cite This Entry:
Weierstrass conditions (for a variational extremum). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weierstrass_conditions_(for_a_variational_extremum)&oldid=17563
This article was adapted from an original article by V.M. Tikhomirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article