Weierstrass conditions (for a variational extremum)
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
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to attain a strong local minimum on the extremal , it is necessary that the inequality
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where is the Weierstrass
-function, be satisfied for all
,
, and all
. This condition may be expressed in terms of the function
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(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
.
References
[1] | M.A. Lavrent'ev, L.A. Lyusternik, "A course in variational calculus" , Moscow-Leningrad (1950) (In Russian) |
[2] | G.A. Bliss, "Lectures on the calculus of variations" , Chicago Univ. Press (1947) |
[3] | L.S. Pontryagin, V.G. Boltayanskii, R.V. Gamkrelidze, E.F. Mishchenko, "The mathematical theory of optimal processes" , Wiley (1962) (Translated from Russian) |
Comments
See also Weierstrass–Erdmann corner conditions, for necessary conditions at a corner of an extremal.
References
[a1] | I.M. Gel'fand, S.V. Fomin, "Calculus of variations" , Prentice-Hall (1963) (Translated from Russian) |
[a2] | L. Cesari, "Optimization - Theory and applications" , Springer (1983) |
Weierstrass conditions (for a variational extremum). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weierstrass_conditions_(for_a_variational_extremum)&oldid=49189