View source for Weierstrass conditions (for a variational extremum)

Necessary and (partially) sufficient conditions for a strong extremum in the classical calculus of variations (cf. [[Variational calculus|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097400/w0974002.png" />, it is necessary that the inequality

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097400/w0974004.png" /> is the [[Weierstrass E-function|Weierstrass <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097400/w0974005.png" />-function]], be satisfied for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097400/w0974006.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097400/w0974007.png" />, and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097400/w0974008.png" />. This condition may be expressed in terms of the function

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(cf. [[Pontryagin maximum principle|Pontryagin maximum principle]]). The Weierstrass condition (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097400/w09740010.png" /> on the extremal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097400/w09740011.png" />) is equivalent to saying that the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097400/w09740012.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097400/w09740013.png" />, attains a maximum in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097400/w09740014.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097400/w09740015.png" />. Thus, Weierstrass' necessary condition is a special case of the Pontryagin maximum principle.

Weierstrass' sufficient condition: For the functional

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to attain a strong local minimum on the vector function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097400/w09740017.png" />, it is sufficient that there exists a vector-valued field slope function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097400/w09740018.png" /> (geodesic slope) (cf. [[Hilbert invariant integral|Hilbert invariant integral]]) in a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097400/w09740019.png" /> of the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097400/w09740020.png" />, for which

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and

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for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097400/w09740023.png" /> and any vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097400/w09740024.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.A. Lavrent'ev,   L.A. Lyusternik,   "A course in variational calculus" , Moscow-Leningrad  (1950)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.A. Bliss,   "Lectures on the calculus of variations" , Chicago Univ. Press  (1947)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.S. Pontryagin,   V.G. Boltayanskii,   R.V. Gamkrelidze,   E.F. Mishchenko,   "The mathematical theory of optimal processes" , Wiley  (1962)  (Translated from Russian)</TD></TR></table>



====Comments====
See also [[Weierstrass–Erdmann corner conditions|Weierstrass–Erdmann corner conditions]], for necessary conditions at a corner of an extremal.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.M. Gel'fand,   S.V. Fomin,   "Calculus of variations" , Prentice-Hall  (1963)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Cesari,   "Optimization - Theory and applications" , Springer  (1983)</TD></TR></table>