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A minimal value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097220/w0972201.png" /> attained by a functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097220/w0972202.png" /> on a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097220/w0972203.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097220/w0972204.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097220/w0972205.png" /> for all comparison curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097220/w0972206.png" /> satisfying the first-order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097220/w0972208.png" />-proximity condition
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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/w/w097/w097220/w0972209.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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throughout the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097220/w09722010.png" />. It is assumed that the curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097220/w09722011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097220/w09722012.png" /> satisfy the given boundary conditions.
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A minimal value  $  J ( \widetilde{y}  ) $
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attained by a functional  $  J ( y) $
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on a curve  $  \widetilde{y}  ( x) $,
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$  x _ {1} \leq  x \leq  x _ {2} $,
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such that $  J ( \widetilde{y}  ) \leq  J ( y) $
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for all comparison curves $  y ( x) $
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satisfying the first-order  $  \epsilon $-proximity condition
  
If in (1) one disregards the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097220/w09722013.png" />-proximity condition on the derivative, then this leads to the zero-order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097220/w09722015.png" />-proximity condition. The minimal value of the functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097220/w09722016.png" /> in a zero-order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097220/w09722017.png" />-neighbourhood is called a [[Strong relative minimum|strong relative minimum]].
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$$ \tag{1 }
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| y ( x) - \widetilde{y}  ( x) |  \leq  \epsilon ,\ \
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| y  ^  \prime  ( x) - \widetilde{y}  ^  \prime  ( x) | \leq  \epsilon
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$$
  
Since the zero-order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097220/w09722018.png" />-proximity condition distinguishes a wider class of curves than the first-order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097220/w09722019.png" />-proximity condition, every strong relative minimum is also a weak relative minimum, but not conversely.
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throughout the interval  $  [ x _ {1} , x _ {2} ] $.  
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It is assumed that the curves $  \widetilde{y}  ( x) $,
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$  y ( x) $
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satisfy the given boundary conditions.
  
For an extremal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097220/w09722020.png" /> to give a weak relative minimum, the [[Legendre condition|Legendre condition]] must hold on it. For a strong relative minimum, the more general [[Weierstrass conditions (for a variational extremum)|Weierstrass conditions (for a variational extremum)]] must hold. In terms of optimal control theory, the distinction between these necessary conditions means that for a weak (strong) relative minimum, it is necessary that, at the points of the extremal, the [[Hamilton function|Hamilton function]] have a local maximum (absolute maximum) with respect to the control (in agreement with the [[Pontryagin maximum principle|Pontryagin maximum principle]]).
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If in (1) one disregards the $  \epsilon $-proximity condition on the derivative, then this leads to the zero-order  $  \epsilon $-proximity condition. The minimal value of the functional  $  J ( y) $
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in a zero-order  $  \epsilon $-neighbourhood is called a [[Strong relative minimum|strong relative minimum]].
  
Sufficient conditions for a weak relative minimum impose certain requirements only on the extremal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097220/w09722021.png" />, whereas in the case of a strong minimum, one requires conditions similar in the sense to hold not only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097220/w09722022.png" />, but also in a certain zero-order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097220/w09722023.png" />-neighbourhood of it. An extremal will be a weak relative minimum if the strong Legendre and strong Jacobi conditions hold along it. An extremal will be a strong relative minimum if it can be imbedded in a field of extremals at all points of which the Weierstrass function is non-negative.
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Since the zero-order  $  \epsilon $-proximity condition distinguishes a wider class of curves than the first-order  $  \epsilon $-proximity condition, every strong relative minimum is also a weak relative minimum, but not conversely.
 +
 
 +
For an extremal  $  \widetilde{y}  ( x) $
 +
to give a weak relative minimum, the [[Legendre condition|Legendre condition]] must hold on it. For a strong relative minimum, the more general [[Weierstrass conditions (for a variational extremum)|Weierstrass conditions (for a variational extremum)]] must hold. In terms of optimal control theory, the distinction between these necessary conditions means that for a weak (strong) relative minimum, it is necessary that, at the points of the extremal, the [[Hamilton function|Hamilton function]] have a local maximum (absolute maximum) with respect to the control (in agreement with the [[Pontryagin maximum principle|Pontryagin maximum principle]]).
 +
 
 +
Sufficient conditions for a weak relative minimum impose certain requirements only on the extremal $  \widetilde{y}  ( x) $,  
 +
whereas in the case of a strong minimum, one requires conditions similar in the sense to hold not only on $  \widetilde{y}  ( x) $,  
 +
but also in a certain zero-order $  \epsilon $-neighbourhood of it. An extremal will be a weak relative minimum if the strong Legendre and strong Jacobi conditions hold along it. An extremal will be a strong relative minimum if it can be imbedded in a field of extremals at all points of which the Weierstrass function is non-negative.
  
 
====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">  D.G. Luenberger,  "Optimization by vector space methods" , Wiley  (1969)</TD></TR><TR><TD valign="top">[a2]</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">  D.G. Luenberger,  "Optimization by vector space methods" , Wiley  (1969)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Cesari,  "Optimization - Theory and applications" , Springer  (1983)</TD></TR></table>

Latest revision as of 08:14, 4 March 2022


A minimal value $ J ( \widetilde{y} ) $ attained by a functional $ J ( y) $ on a curve $ \widetilde{y} ( x) $, $ x _ {1} \leq x \leq x _ {2} $, such that $ J ( \widetilde{y} ) \leq J ( y) $ for all comparison curves $ y ( x) $ satisfying the first-order $ \epsilon $-proximity condition

$$ \tag{1 } | y ( x) - \widetilde{y} ( x) | \leq \epsilon ,\ \ | y ^ \prime ( x) - \widetilde{y} ^ \prime ( x) | \leq \epsilon $$

throughout the interval $ [ x _ {1} , x _ {2} ] $. It is assumed that the curves $ \widetilde{y} ( x) $, $ y ( x) $ satisfy the given boundary conditions.

If in (1) one disregards the $ \epsilon $-proximity condition on the derivative, then this leads to the zero-order $ \epsilon $-proximity condition. The minimal value of the functional $ J ( y) $ in a zero-order $ \epsilon $-neighbourhood is called a strong relative minimum.

Since the zero-order $ \epsilon $-proximity condition distinguishes a wider class of curves than the first-order $ \epsilon $-proximity condition, every strong relative minimum is also a weak relative minimum, but not conversely.

For an extremal $ \widetilde{y} ( x) $ to give a weak relative minimum, the Legendre condition must hold on it. For a strong relative minimum, the more general Weierstrass conditions (for a variational extremum) must hold. In terms of optimal control theory, the distinction between these necessary conditions means that for a weak (strong) relative minimum, it is necessary that, at the points of the extremal, the Hamilton function have a local maximum (absolute maximum) with respect to the control (in agreement with the Pontryagin maximum principle).

Sufficient conditions for a weak relative minimum impose certain requirements only on the extremal $ \widetilde{y} ( x) $, whereas in the case of a strong minimum, one requires conditions similar in the sense to hold not only on $ \widetilde{y} ( x) $, but also in a certain zero-order $ \epsilon $-neighbourhood of it. An extremal will be a weak relative minimum if the strong Legendre and strong Jacobi conditions hold along it. An extremal will be a strong relative minimum if it can be imbedded in a field of extremals at all points of which the Weierstrass function is non-negative.

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] D.G. Luenberger, "Optimization by vector space methods" , Wiley (1969)
[a2] L. Cesari, "Optimization - Theory and applications" , Springer (1983)
How to Cite This Entry:
Weak relative minimum. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weak_relative_minimum&oldid=12410
This article was adapted from an original article by I.B. Vapnyarskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article