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''non-singular extremal''
 
''non-singular extremal''
  
An [[Extremal|extremal]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080710/r0807101.png" /> at all points of which the following condition holds:
+
An [[Extremal|extremal]] $  y ( x) $
 +
at all points of which the following condition holds:
  
<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/r/r080/r080710/r0807102.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
F _ {y  ^  \prime  y  ^  \prime  } ( x , y ( x) , y  ^  \prime  ( x) ) \neq  0 ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080710/r0807103.png" /> is the integrand appearing in a functional
+
where $  F ( x , y , y  ^  \prime  ) $
 +
is the integrand appearing in a 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/r/r080/r080710/r0807104.png" /></td> </tr></table>
+
$$
 +
= \int\limits _ { x _ {1} } ^ { {x _ 2 } }
 +
F ( x , y , y  ^  \prime  )  d x
 +
$$
  
 
which is to be minimized. Like any extremal, a regular extremal is, by definition, a smooth solution of the [[Euler equation|Euler equation]]
 
which is to be minimized. Like any extremal, a regular extremal is, by definition, a smooth solution of the [[Euler equation|Euler equation]]
  
<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/r/r080/r080710/r0807105.png" /></td> </tr></table>
+
$$
 +
F _ {y} -  
 +
\frac{d}{dx}
 +
F _ {y  ^  \prime  }  = 0 .
 +
$$
  
The points of an extremal at which (1) holds are called regular points. It is known that at every regular point, an extremal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080710/r0807106.png" /> has a continuous second-order derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080710/r0807107.png" />. On a regular extremal, the second-order derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080710/r0807108.png" /> is continuous. For a regular extremal the Euler equation
+
The points of an extremal at which (1) holds are called regular points. It is known that at every regular point, an extremal $  y ( x) $
 +
has a continuous second-order derivative $  y  ^ {\prime\prime} ( x) $.  
 +
On a regular extremal, the second-order derivative $  y  ^ {\prime\prime} ( x) $
 +
is continuous. For a regular extremal the Euler equation
  
<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/r/r080/r080710/r0807109.png" /></td> </tr></table>
+
$$
 +
F _ {y} - F _ {y  ^  \prime  x } -
 +
F _ {y  ^  \prime  y } y  ^  \prime  - F _ {y  ^  \prime  y  ^  \prime  } y  ^ {\prime\prime}  = 0
 +
$$
  
 
can be written in the following form (that is, solved for the highest derivative):
 
can be written in the following form (that is, solved for the highest derivative):
  
<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/r/r080/r080710/r08071010.png" /></td> </tr></table>
+
$$
 +
y  ^ {\prime\prime}  = f ( x , y , y  ^  \prime  ) .
 +
$$
  
 
The regularity property (1) is directly connected with the necessary [[Legendre condition|Legendre condition]] (in the strong form), according to which at all points of the extremal the following inequality holds:
 
The regularity property (1) is directly connected with the necessary [[Legendre condition|Legendre condition]] (in the strong form), according to which at all points of the extremal the following inequality holds:
  
<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/r/r080/r080710/r08071011.png" /></td> </tr></table>
+
$$
 +
F _ {y  ^  \prime  y  ^  \prime  } ( x , y ( x) , y  ^  \prime  ( x) )  < 0 .
 +
$$
  
Essential use is made of regularity when proving that an extremal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080710/r08071012.png" /> can be included in a field of extremals surrounding it. If condition (1) is violated at even one point, the extremal cannot always be included in a field. This condition for including the extremal in a field is one of the sufficient conditions for being an extremal.
+
Essential use is made of regularity when proving that an extremal $  y ( x) $
 +
can be included in a field of extremals surrounding it. If condition (1) is violated at even one point, the extremal cannot always be included in a field. This condition for including the extremal in a field is one of the sufficient conditions for being an extremal.
  
The above definition of a regular extremal is given for the simplest problem of the calculus of variations, which concerns functionals depending on one unknown function. For functionals depending on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080710/r08071013.png" /> unknown functions,
+
The above definition of a regular extremal is given for the simplest problem of the calculus of variations, which concerns functionals depending on one unknown function. For functionals depending on $  n $
 +
unknown functions,
  
<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/r/r080/r080710/r08071014.png" /></td> </tr></table>
+
$$
 +
= \int\limits _ { x _ {1} } ^ { {x _ 2 } }
 +
F ( x , y _ {1} \dots y _ {n} , y _ {1}  ^  \prime
 +
\dots y _ {n}  ^  \prime  )  d x ,
 +
$$
  
a regular extremal is an extremal for which at every point the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080710/r08071015.png" />-th order determinant
+
a regular extremal is an extremal for which at every point the $  n $-
 +
th order determinant
  
<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/r/r080/r080710/r08071016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
| F _ {y _ {i}  ^  \prime  y _ {j}  ^  \prime  } |  \neq  0 .
 +
$$
  
In some general problems of the calculus of variations on a conditional extremum (see [[Bolza problem|Bolza problem]]), a regular extremal is defined in a similar way, except that in (2) instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080710/r08071017.png" /> one must put the [[Lagrange function|Lagrange function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080710/r08071018.png" />.
+
In some general problems of the calculus of variations on a conditional extremum (see [[Bolza problem|Bolza problem]]), a regular extremal is defined in a similar way, except that in (2) instead of $  F $
 +
one must put the [[Lagrange function|Lagrange function]] $  L $.
  
 
An extremal for which the regularity condition ((1) or (2)) is violated at every point of some section is called a singular extremal, and the section is called a section of singular regime. For singular regimes there are necessary conditions supplementing the known classical necessary conditions for an extremum (see [[Optimal singular regime|Optimal singular regime]]).
 
An extremal for which the regularity condition ((1) or (2)) is violated at every point of some section is called a singular extremal, and the section is called a section of singular regime. For singular regimes there are necessary conditions supplementing the known classical necessary conditions for an extremum (see [[Optimal singular regime|Optimal singular regime]]).
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.A. Bliss,  "Lectures on the calculus of variations" , Chicago Univ. Press  (1947)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.A. Lavrent'ev,  L.A. Lyusternik,  "A course in variational calculus" , Moscow-Leningrad  (1950)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.A. Bliss,  "Lectures on the calculus of variations" , Chicago Univ. Press  (1947)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.A. Lavrent'ev,  L.A. Lyusternik,  "A course in variational calculus" , Moscow-Leningrad  (1950)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
A family of curves in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080710/r08071019.png" /> is called a field of curves if for every point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080710/r08071020.png" /> there is exactly one member of the family passing through it. For an account of the role of field theory in the calculus of variations and fields of extremals cf. [[#References|[a2]]] and [[Extremal field|Extremal field]].
+
A family of curves in a domain $  D $
 +
is called a field of curves if for every point of $  D $
 +
there is exactly one member of the family passing through it. For an account of the role of field theory in the calculus of variations and fields of extremals cf. [[#References|[a2]]] and [[Extremal field|Extremal field]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Cesari,  "Optimization - Theory and applications" , Springer  (1983)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  Yu.P. Petrov,  "Variational methods in optimum control theory" , Acad. Press  (1968)  pp. Chapt. IV  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Cesari,  "Optimization - Theory and applications" , Springer  (1983)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  Yu.P. Petrov,  "Variational methods in optimum control theory" , Acad. Press  (1968)  pp. Chapt. IV  (Translated from Russian)</TD></TR></table>

Latest revision as of 08:10, 6 June 2020


non-singular extremal

An extremal $ y ( x) $ at all points of which the following condition holds:

$$ \tag{1 } F _ {y ^ \prime y ^ \prime } ( x , y ( x) , y ^ \prime ( x) ) \neq 0 , $$

where $ F ( x , y , y ^ \prime ) $ is the integrand appearing in a functional

$$ J = \int\limits _ { x _ {1} } ^ { {x _ 2 } } F ( x , y , y ^ \prime ) d x $$

which is to be minimized. Like any extremal, a regular extremal is, by definition, a smooth solution of the Euler equation

$$ F _ {y} - \frac{d}{dx} F _ {y ^ \prime } = 0 . $$

The points of an extremal at which (1) holds are called regular points. It is known that at every regular point, an extremal $ y ( x) $ has a continuous second-order derivative $ y ^ {\prime\prime} ( x) $. On a regular extremal, the second-order derivative $ y ^ {\prime\prime} ( x) $ is continuous. For a regular extremal the Euler equation

$$ F _ {y} - F _ {y ^ \prime x } - F _ {y ^ \prime y } y ^ \prime - F _ {y ^ \prime y ^ \prime } y ^ {\prime\prime} = 0 $$

can be written in the following form (that is, solved for the highest derivative):

$$ y ^ {\prime\prime} = f ( x , y , y ^ \prime ) . $$

The regularity property (1) is directly connected with the necessary Legendre condition (in the strong form), according to which at all points of the extremal the following inequality holds:

$$ F _ {y ^ \prime y ^ \prime } ( x , y ( x) , y ^ \prime ( x) ) < 0 . $$

Essential use is made of regularity when proving that an extremal $ y ( x) $ can be included in a field of extremals surrounding it. If condition (1) is violated at even one point, the extremal cannot always be included in a field. This condition for including the extremal in a field is one of the sufficient conditions for being an extremal.

The above definition of a regular extremal is given for the simplest problem of the calculus of variations, which concerns functionals depending on one unknown function. For functionals depending on $ n $ unknown functions,

$$ J = \int\limits _ { x _ {1} } ^ { {x _ 2 } } F ( x , y _ {1} \dots y _ {n} , y _ {1} ^ \prime \dots y _ {n} ^ \prime ) d x , $$

a regular extremal is an extremal for which at every point the $ n $- th order determinant

$$ \tag{2 } | F _ {y _ {i} ^ \prime y _ {j} ^ \prime } | \neq 0 . $$

In some general problems of the calculus of variations on a conditional extremum (see Bolza problem), a regular extremal is defined in a similar way, except that in (2) instead of $ F $ one must put the Lagrange function $ L $.

An extremal for which the regularity condition ((1) or (2)) is violated at every point of some section is called a singular extremal, and the section is called a section of singular regime. For singular regimes there are necessary conditions supplementing the known classical necessary conditions for an extremum (see Optimal singular regime).

References

[1] G.A. Bliss, "Lectures on the calculus of variations" , Chicago Univ. Press (1947)
[2] M.A. Lavrent'ev, L.A. Lyusternik, "A course in variational calculus" , Moscow-Leningrad (1950) (In Russian)

Comments

A family of curves in a domain $ D $ is called a field of curves if for every point of $ D $ there is exactly one member of the family passing through it. For an account of the role of field theory in the calculus of variations and fields of extremals cf. [a2] and Extremal field.

References

[a1] L. Cesari, "Optimization - Theory and applications" , Springer (1983)
[a2] Yu.P. Petrov, "Variational methods in optimum control theory" , Acad. Press (1968) pp. Chapt. IV (Translated from Russian)
How to Cite This Entry:
Regular extremal. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Regular_extremal&oldid=17757
This article was adapted from an original article by I.B. Vapnyarskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article