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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.
 
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
 
Weierstrass' necessary condition: For the functional
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097400/w0974001.png" /></td> </tr></table>
+
$$
 +
J( x)  = \int\limits _ { t _ {0} } ^ { {t _ 1 } }
 +
L( t, x( t), \dot{x} ( t))  dt,\ \
 +
L: \mathbf R \times \mathbf R  ^ {n} \times \mathbf R  ^ {n} \rightarrow \mathbf R ,
 +
$$
  
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
+
to attain a strong local minimum on the extremal $  x _ {0} ( t) $,  
 +
it is necessary that the inequality
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097400/w0974003.png" /></td> </tr></table>
+
$$
 +
{\mathcal E} ( t, x _ {0} ( t), \dot{x} _ {0} ( t), \xi )  \geq  0,
 +
$$
  
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
+
where $  {\mathcal E} $
 +
is the [[Weierstrass E-function|Weierstrass $  {\mathcal E} $-
 +
function]], be satisfied for all $  t $,  
 +
$  t _ {0} \leq  t \leq  t _ {1} $,  
 +
and all $  \xi \in \mathbf C  ^ {n} $.  
 +
This condition may be expressed in terms of the function
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097400/w0974009.png" /></td> </tr></table>
+
$$
 +
\Pi ( t, x, p, u)  = ( p, u) - L ( t, x, u)
 +
$$
  
(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.
+
(cf. [[Pontryagin maximum principle|Pontryagin maximum principle]]). The Weierstrass condition ( $  {\mathcal E} \geq  0 $
 +
on the extremal $  x _ {0} ( t) $)  
 +
is equivalent to saying that the function $  \Pi ( t, x _ {0} ( t), p _ {0} ( t), u) $,  
 +
where $  p _ {0} ( t) = L _ {\dot{x} }  ( t, x _ {0} ( t), \dot{x} _ {0} ( t)) $,  
 +
attains a maximum in $  u $
 +
for $  u = {\dot{x} _ {0} } ( t) $.  
 +
Thus, Weierstrass' necessary condition is a special case of the Pontryagin maximum principle.
  
 
Weierstrass' sufficient condition: For the functional
 
Weierstrass' sufficient condition: For the functional
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097400/w09740016.png" /></td> </tr></table>
+
$$
 +
J ( x)  = \int\limits _ { t _ {0} } ^ { {t _ 1 } }
 +
L ( t, x( t), \dot{x} ( t))  dt,\ \
 +
L : \mathbf R \times \mathbf R  ^ {n} \times \mathbf R  ^ {n} \rightarrow \mathbf R ,
 +
$$
  
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
+
to attain a strong local minimum on the vector function $  x _ {0} ( t) $,  
 +
it is sufficient that there exists a vector-valued field slope function $  U( t, x) $(
 +
geodesic slope) (cf. [[Hilbert invariant integral|Hilbert invariant integral]]) in a neighbourhood $  G $
 +
of the curve $  x _ {0} ( t) $,  
 +
for which
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097400/w09740021.png" /></td> </tr></table>
+
$$
 +
\dot{x} _ {0} ( t)  = U( t, x _ {0} ( t))
 +
$$
  
 
and
 
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097400/w09740022.png" /></td> </tr></table>
+
$$
 +
{\mathcal E} ( t, x, U( t, x), \xi )  \geq  0
 +
$$
  
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" />.
+
for all $  ( t, x) \in G $
 +
and any vector $  \xi \in \mathbf R  ^ {n} $.
  
 
====References====
 
====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>
 
<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====
 
====Comments====

Latest revision as of 08:28, 6 June 2020


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

$$ J( x) = \int\limits _ { t _ {0} } ^ { {t _ 1 } } L( t, x( t), \dot{x} ( t)) dt,\ \ L: \mathbf R \times \mathbf R ^ {n} \times \mathbf R ^ {n} \rightarrow \mathbf R , $$

to attain a strong local minimum on the extremal $ x _ {0} ( t) $, it is necessary that the inequality

$$ {\mathcal E} ( t, x _ {0} ( t), \dot{x} _ {0} ( t), \xi ) \geq 0, $$

where $ {\mathcal E} $ is the Weierstrass $ {\mathcal E} $- function, be satisfied for all $ t $, $ t _ {0} \leq t \leq t _ {1} $, and all $ \xi \in \mathbf C ^ {n} $. This condition may be expressed in terms of the function

$$ \Pi ( t, x, p, u) = ( p, u) - L ( t, x, u) $$

(cf. Pontryagin maximum principle). The Weierstrass condition ( $ {\mathcal E} \geq 0 $ on the extremal $ x _ {0} ( t) $) is equivalent to saying that the function $ \Pi ( t, x _ {0} ( t), p _ {0} ( t), u) $, where $ p _ {0} ( t) = L _ {\dot{x} } ( t, x _ {0} ( t), \dot{x} _ {0} ( t)) $, attains a maximum in $ u $ for $ u = {\dot{x} _ {0} } ( t) $. Thus, Weierstrass' necessary condition is a special case of the Pontryagin maximum principle.

Weierstrass' sufficient condition: For the functional

$$ J ( x) = \int\limits _ { t _ {0} } ^ { {t _ 1 } } L ( t, x( t), \dot{x} ( t)) dt,\ \ L : \mathbf R \times \mathbf R ^ {n} \times \mathbf R ^ {n} \rightarrow \mathbf R , $$

to attain a strong local minimum on the vector function $ x _ {0} ( t) $, it is sufficient that there exists a vector-valued field slope function $ U( t, x) $( geodesic slope) (cf. Hilbert invariant integral) in a neighbourhood $ G $ of the curve $ x _ {0} ( t) $, for which

$$ \dot{x} _ {0} ( t) = U( t, x _ {0} ( t)) $$

and

$$ {\mathcal E} ( t, x, U( t, x), \xi ) \geq 0 $$

for all $ ( t, x) \in G $ and any vector $ \xi \in \mathbf R ^ {n} $.

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)
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=49189
This article was adapted from an original article by V.M. Tikhomirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article