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A minimal or maximal value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090550/s0905501.png" /> taken by a functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090550/s0905502.png" /> at a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090550/s0905503.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090550/s0905504.png" />, for which one of the inequalities
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<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/s/s090/s090550/s0905505.png" /></td> </tr></table>
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holds for all comparison curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090550/s0905506.png" /> in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090550/s0905507.png" />-neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090550/s0905508.png" />. The curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090550/s0905509.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090550/s09055010.png" /> must satisfy given boundary conditions.
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A minimal or maximal value  $  J ( \widetilde{y}  ) $
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taken by a functional  $  J ( y) $
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at a curve  $  \widetilde{y}  ( x) $,
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$  x _ {1} \leq  x \leq  x _ {2} $,
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for which one of the inequalities
  
Since maximization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090550/s09055011.png" /> is equivalent to minimization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090550/s09055012.png" />, instead of a strong maximum one often discusses only a strong minimum. The term "strong" emphasizes that only the condition of being <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090550/s09055013.png" />-near to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090550/s09055014.png" /> is imposed on the comparison curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090550/s09055015.png" />:
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$$
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J ( \widetilde{y} ) \leq  J ( y) \ \
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\textrm{ or } \ \
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J ( \widetilde{y}  )  \geq  J ( y)
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$$
  
<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/s/s090/s090550/s09055016.png" /></td> </tr></table>
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holds for all comparison curves  $  y ( x) $
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in an  $  \epsilon $-neighbourhood of  $  y ( x) $.  
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The curves  $  \widetilde{y}  ( x) $
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and  $  y ( x) $
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must satisfy given boundary conditions.
  
on the whole interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090550/s09055017.png" />, whereas the derivatives of the curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090550/s09055018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090550/s09055019.png" /> may differ as "strongly" as desired.
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Since maximization of  $  J ( y) $
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is equivalent to minimization of  $  - J ( y) $,
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instead of a strong maximum one often discusses only a strong minimum. The term  "strong" emphasizes that only the condition of being  $  \epsilon $-near to  $  \widetilde{y}  ( x) $
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is imposed on the comparison curves  $ y ( x) $:
  
However, the very definition of a strong extremum is of a relative rather than absolute nature, since it gives an extremum not on the whole class of admissible comparison curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090550/s09055020.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090550/s09055021.png" /> makes sense, but only relative to the subset of all admissible comparison curves belonging to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090550/s09055022.png" />-neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090550/s09055023.png" />. However, for brevity, the term  "relative"  is often omitted and one speaks of a strong extremum, meaning a strong relative extremum (see also [[Strong relative minimum|Strong relative minimum]]).
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$$
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| y ( x) - \widetilde{y}  ( x) |  \leq  \epsilon
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$$
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on the whole interval  $  [ x _ {1} , x _ {2} ] $,
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whereas the derivatives of the curves  $  y ( x) $
 +
and  $  \widetilde{y}  ( x) $
 +
may differ as  "strongly"  as desired.
 +
 
 +
However, the very definition of a strong extremum is of a relative rather than absolute nature, since it gives an extremum not on the whole class of admissible comparison curves $  y ( x) $
 +
for which $  J ( y) $
 +
makes sense, but only relative to the subset of all admissible comparison curves belonging to the $  \epsilon $-neighbourhood of $  \widetilde{y}  ( x) $.  
 +
However, for brevity, the term  "relative"  is often omitted and one speaks of a strong extremum, meaning a strong relative extremum (see also [[Strong relative minimum|Strong relative minimum]]).
  
 
====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">  V.I. Smirnov,  "A course of higher mathematics" , '''4''' , Addison-Wesley  (1964)  (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">  V.I. Smirnov,  "A course of higher mathematics" , '''4''' , Addison-Wesley  (1964)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Cesari,  "Optimization - Theory and applications" , Springer  (1983)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Cesari,  "Optimization - Theory and applications" , Springer  (1983)</TD></TR></table>

Latest revision as of 01:26, 5 March 2022


A minimal or maximal value $ J ( \widetilde{y} ) $ taken by a functional $ J ( y) $ at a curve $ \widetilde{y} ( x) $, $ x _ {1} \leq x \leq x _ {2} $, for which one of the inequalities

$$ J ( \widetilde{y} ) \leq J ( y) \ \ \textrm{ or } \ \ J ( \widetilde{y} ) \geq J ( y) $$

holds for all comparison curves $ y ( x) $ in an $ \epsilon $-neighbourhood of $ y ( x) $. The curves $ \widetilde{y} ( x) $ and $ y ( x) $ must satisfy given boundary conditions.

Since maximization of $ J ( y) $ is equivalent to minimization of $ - J ( y) $, instead of a strong maximum one often discusses only a strong minimum. The term "strong" emphasizes that only the condition of being $ \epsilon $-near to $ \widetilde{y} ( x) $ is imposed on the comparison curves $ y ( x) $:

$$ | y ( x) - \widetilde{y} ( x) | \leq \epsilon $$

on the whole interval $ [ x _ {1} , x _ {2} ] $, whereas the derivatives of the curves $ y ( x) $ and $ \widetilde{y} ( x) $ may differ as "strongly" as desired.

However, the very definition of a strong extremum is of a relative rather than absolute nature, since it gives an extremum not on the whole class of admissible comparison curves $ y ( x) $ for which $ J ( y) $ makes sense, but only relative to the subset of all admissible comparison curves belonging to the $ \epsilon $-neighbourhood of $ \widetilde{y} ( x) $. However, for brevity, the term "relative" is often omitted and one speaks of a strong extremum, meaning a strong relative extremum (see also Strong relative minimum).

References

[1] M.A. Lavrent'ev, L.A. Lyusternik, "A course in variational calculus" , Moscow-Leningrad (1950) (In Russian)
[2] V.I. Smirnov, "A course of higher mathematics" , 4 , Addison-Wesley (1964) (Translated from Russian)

Comments

References

[a1] L. Cesari, "Optimization - Theory and applications" , Springer (1983)
How to Cite This Entry:
Strong extremum. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Strong_extremum&oldid=11731
This article was adapted from an original article by I.B. Vapnyarskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article