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One of the fundamental problems in the classical calculus of variations. It consists in minimizing the functional
 
One of the fundamental problems in the classical calculus of variations. It consists in minimizing 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/l/l057/l057210/l0572101.png" /></td> </tr></table>
+
$$
 +
J ( y)  = \int\limits _ { x _ {1} } ^ { {x _ 2 } } f
 +
( x , y , y  ^  \prime  )  d x ,\ \
 +
f : \mathbf R \times \mathbf R  ^ {n} \times \mathbf R  ^ {n}  \rightarrow  \mathbf R ,
 +
$$
  
 
in the presence of differential constraints of equality type:
 
in the presence of differential constraints of equality type:
  
<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/l/l057/l057210/l0572102.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
\phi ( x , y , y  ^  \prime  )  = 0 ,\ \
 +
\phi : \mathbf R \times \mathbf R  ^ {n} \times \mathbf R  ^ {n}  \rightarrow \
 +
\mathbf R  ^ {m} ,\  m < n ,
 +
$$
  
 
and boundary conditions
 
and boundary conditions
  
<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/l/l057/l057210/l0572103.png" /></td> </tr></table>
+
$$
 +
\psi ( x _ {1} , y ( x _ {1} ) , x _ {2} , y ( x _ {2} ) )  = 0 ,\ \
 +
\psi : \mathbf R \times \mathbf R  ^ {n} \times \mathbf R \times \mathbf R  ^ {n}
 +
\rightarrow  \mathbf R  ^ {p} ,
 +
$$
 +
 
 +
$$
 +
p  \leq  2 n + 2 .
 +
$$
  
<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/l/l057/l057210/l0572104.png" /></td> </tr></table>
+
The Lagrange problem is usually considered under the condition that the system (1) is regular, that is, the matrix  $  \| \partial  \phi / \partial  y  ^  \prime  \| $
 +
has maximal rank:
  
The Lagrange problem is usually considered under the condition that the system (1) is regular, that is, the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057210/l0572105.png" /> has maximal rank:
+
$$
 +
\mathop{\rm rank}  \left \|
 +
\frac{\partial  \phi }{\partial  y  ^  \prime  }
 +
\right \|  = m .
 +
$$
  
<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/l/l057/l057210/l0572106.png" /></td> </tr></table>
+
Under this condition the system (1) can be solved for part of the variables and, using a different notation ( $  t , x $
 +
instead of  $  x , y $),
 +
the Lagrange problem can be reduced to the form
  
Under this condition the system (1) can be solved for part of the variables and, using a different notation (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057210/l0572107.png" /> instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057210/l0572108.png" />), the Lagrange problem can be reduced to the form
+
$$ \tag{2 }
 +
\left .
 +
\begin{array}{c}
 +
\int\limits _ { t _ {0} } ^ { {t _ 1 } } F ( t , x , u ) \
 +
dt ,\  F : \mathbf R \times \mathbf R  ^ {n}
 +
\times \mathbf R  ^ {r}  \rightarrow  \mathbf R ,  \\
 +
\dot{x}  = \Phi ( t , x , u ) ,\  \Phi : \mathbf R \times
 +
\mathbf R  ^ {n} \times \mathbf R  ^ {r}  \rightarrow  \mathbf R  ^ {n} . \\
 +
\end{array}
  
<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/l/l057/l057210/l0572109.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
\right \}
 +
$$
  
The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057210/l05721010.png" /> and the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057210/l05721011.png" /> are usually assumed to be continuously differentiable. Problems of optimal control are often specified in the form (2) (the Pontryagin form), and restrictions are, moreover, imposed on the control <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057210/l05721012.png" />. Necessary conditions for a strong extremum for the problem (2) (for simplicity, with fixed left-hand end <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057210/l05721013.png" /> and free right-hand end <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057210/l05721014.png" />) have the following form. Let
+
The function $  F $
 +
and the mapping $  \Phi $
 +
are usually assumed to be continuously differentiable. Problems of optimal control are often specified in the form (2) (the Pontryagin form), and restrictions are, moreover, imposed on the control $  u \in U $.  
 +
Necessary conditions for a strong extremum for the problem (2) (for simplicity, with fixed left-hand end $  x _ {0} $
 +
and free right-hand end $  x _ {1} $)  
 +
have the following form. Let
  
<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/l/l057/l057210/l05721015.png" /></td> </tr></table>
+
$$
 +
L ( t , x , \dot{x} , u , p ( t) )  = \
 +
( p ( t)  \mid  - \dot{x} + \Phi ( t , x , u )) - F ( t , x , u )
 +
$$
  
be the [[Lagrange function|Lagrange function]]. For a vector function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057210/l05721016.png" /> to be a strong minimum in the Lagrange problem (2) it is necessary that the following relations hold:
+
be the [[Lagrange function|Lagrange function]]. For a vector function $  ( x  ^ {*} ( t) , u  ^ {*} ( t) ) $
 +
to be a strong minimum in the Lagrange problem (2) it is necessary that the following relations hold:
  
<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/l/l057/l057210/l05721017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
\left .  
 +
\frac{\partial  L }{\partial  \dot{x} }
  
<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/l/l057/l057210/l05721018.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
\right | _ {( x  ^ {*}  , u  ^ {*} ) } +
 +
\int\limits _ { t _ {0} } ^ { {t _ 1 } }
 +
\left .
 +
\frac{\partial  L }{d x }
  
<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/l/l057/l057210/l05721019.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>
+
\right | _ {( x  ^ {*}  , u  ^ {*} ) }  d t  =  0 ,
 +
$$
  
<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/l/l057/l057210/l05721020.png" /></td> </tr></table>
+
$$ \tag{4 }
 +
p ( t _ {1} )  = 0 ,
 +
$$
  
<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/l/l057/l057210/l05721021.png" /></td> </tr></table>
+
$$ \tag{5 }
 +
{\mathcal E}  \equiv  L ( t , x  ^ {*} ( t) , \dot{x} , u , p ( t) ) +
 +
$$
  
<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/l/l057/l057210/l05721022.png" /></td> </tr></table>
+
$$
 +
- L ( t , x  ^ {*} ( t) , \dot{x}  ^ {*} ( t) , u  ^ {*} ( t) , p ( t) ) +
 +
$$
  
<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/l/l057/l057210/l05721023.png" /></td> </tr></table>
+
$$
 +
- ( ( \dot{x} - \dot{x}  ^ {*} ( t) )  \mid  L _ {\dot{x} }  ( t , x  ^ {*}
 +
( t ) , \dot{x}  ^ {*} ( t ) , u  ^ {*} ( t ) , p ( t )) ) =
 +
$$
  
for all possible admissible values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057210/l05721024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057210/l05721025.png" />.
+
$$
 +
= \
 +
( p ( t)  \mid  \Phi ( t , x  ^ {*} ( t) , u ) - F ( t , x  ^ {*} ( t) , u )) +
 +
$$
  
If one carries out differentiation in (3) with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057210/l05721026.png" /> and uses the notation
+
$$
 +
- ( p ( t)  \mid  \Phi ( t , x  ^ {*} ( t) , u  ^ {*} ( t) )
 +
+ F ( t , x  ^ {*} ( t) , u  ^ {*} ( t) )) \leq  0
 +
$$
  
<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/l/l057/l057210/l05721027.png" /></td> </tr></table>
+
for all possible admissible values of  $  \dot{x} $
 +
and  $  u $.
  
then a necessary condition for a strong minimum can be stated in the form of a maximum principle, in which the Euler equation (3), the transversality condition (4) and the Weierstrass condition (5) are combined. For a vector function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057210/l05721028.png" /> to be a strong minimum in the problem (2) with fixed left-hand end and free right-hand end it is necessary that there is a solution of the system
+
If one carries out differentiation in (3) with respect to $  t $
 +
and uses the notation
  
<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/l/l057/l057210/l05721029.png" /></td> </tr></table>
+
$$
 +
{\mathcal H} ( t , x , u , p )  = ( p  \mid  \Phi ) - F ,
 +
$$
 +
 
 +
then a necessary condition for a strong minimum can be stated in the form of a maximum principle, in which the Euler equation (3), the transversality condition (4) and the Weierstrass condition (5) are combined. For a vector function  $  ( x  ^ {*} , u  ^ {*} ) $
 +
to be a strong minimum in the problem (2) with fixed left-hand end and free right-hand end it is necessary that there is a solution of the system
 +
 
 +
$$
 +
\dot{p} ( t)  = -  
 +
\frac{\partial  {\mathcal H} ( t , x  ^ {*} , u  ^ {*} , p ) }{\partial  x }
 +
,\  p ( t _ {1} )  = 0 ,
 +
$$
  
 
for which
 
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/l/l057/l057210/l05721030.png" /></td> </tr></table>
+
$$
 +
{\mathcal H} ( t , x  ^ {*} ( t) , u  ^ {*} ( t) , p ( t))  = \max _
 +
{u \in U }  {\mathcal H} ( t , x  ^ {*} ( t) , u , p ( t) ) .
 +
$$
  
 
J.L. Lagrange considered similar problems in connection with studies in mechanics (in the second half of the 18th century).
 
J.L. Lagrange considered similar problems in connection with studies in mechanics (in the second half of the 18th century).
  
 
For references see [[Variational calculus|Variational calculus]].
 
For references see [[Variational calculus|Variational calculus]].
 
 
  
 
====Comments====
 
====Comments====
The notation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057210/l05721031.png" /> denotes the [[Inner product|inner product]] of the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057210/l05721032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057210/l05721033.png" />.
+
The notation $  ( a  \mid  b) $
 +
denotes the [[Inner product|inner product]] of the vectors $  a $
 +
and $  b $.

Latest revision as of 22:15, 5 June 2020


One of the fundamental problems in the classical calculus of variations. It consists in minimizing the functional

$$ J ( y) = \int\limits _ { x _ {1} } ^ { {x _ 2 } } f ( x , y , y ^ \prime ) d x ,\ \ f : \mathbf R \times \mathbf R ^ {n} \times \mathbf R ^ {n} \rightarrow \mathbf R , $$

in the presence of differential constraints of equality type:

$$ \tag{1 } \phi ( x , y , y ^ \prime ) = 0 ,\ \ \phi : \mathbf R \times \mathbf R ^ {n} \times \mathbf R ^ {n} \rightarrow \ \mathbf R ^ {m} ,\ m < n , $$

and boundary conditions

$$ \psi ( x _ {1} , y ( x _ {1} ) , x _ {2} , y ( x _ {2} ) ) = 0 ,\ \ \psi : \mathbf R \times \mathbf R ^ {n} \times \mathbf R \times \mathbf R ^ {n} \rightarrow \mathbf R ^ {p} , $$

$$ p \leq 2 n + 2 . $$

The Lagrange problem is usually considered under the condition that the system (1) is regular, that is, the matrix $ \| \partial \phi / \partial y ^ \prime \| $ has maximal rank:

$$ \mathop{\rm rank} \left \| \frac{\partial \phi }{\partial y ^ \prime } \right \| = m . $$

Under this condition the system (1) can be solved for part of the variables and, using a different notation ( $ t , x $ instead of $ x , y $), the Lagrange problem can be reduced to the form

$$ \tag{2 } \left . \begin{array}{c} \int\limits _ { t _ {0} } ^ { {t _ 1 } } F ( t , x , u ) \ dt ,\ F : \mathbf R \times \mathbf R ^ {n} \times \mathbf R ^ {r} \rightarrow \mathbf R , \\ \dot{x} = \Phi ( t , x , u ) ,\ \Phi : \mathbf R \times \mathbf R ^ {n} \times \mathbf R ^ {r} \rightarrow \mathbf R ^ {n} . \\ \end{array} \right \} $$

The function $ F $ and the mapping $ \Phi $ are usually assumed to be continuously differentiable. Problems of optimal control are often specified in the form (2) (the Pontryagin form), and restrictions are, moreover, imposed on the control $ u \in U $. Necessary conditions for a strong extremum for the problem (2) (for simplicity, with fixed left-hand end $ x _ {0} $ and free right-hand end $ x _ {1} $) have the following form. Let

$$ L ( t , x , \dot{x} , u , p ( t) ) = \ ( p ( t) \mid - \dot{x} + \Phi ( t , x , u )) - F ( t , x , u ) $$

be the Lagrange function. For a vector function $ ( x ^ {*} ( t) , u ^ {*} ( t) ) $ to be a strong minimum in the Lagrange problem (2) it is necessary that the following relations hold:

$$ \tag{3 } \left . \frac{\partial L }{\partial \dot{x} } \right | _ {( x ^ {*} , u ^ {*} ) } + \int\limits _ { t _ {0} } ^ { {t _ 1 } } \left . \frac{\partial L }{d x } \right | _ {( x ^ {*} , u ^ {*} ) } d t = 0 , $$

$$ \tag{4 } p ( t _ {1} ) = 0 , $$

$$ \tag{5 } {\mathcal E} \equiv L ( t , x ^ {*} ( t) , \dot{x} , u , p ( t) ) + $$

$$ - L ( t , x ^ {*} ( t) , \dot{x} ^ {*} ( t) , u ^ {*} ( t) , p ( t) ) + $$

$$ - ( ( \dot{x} - \dot{x} ^ {*} ( t) ) \mid L _ {\dot{x} } ( t , x ^ {*} ( t ) , \dot{x} ^ {*} ( t ) , u ^ {*} ( t ) , p ( t )) ) = $$

$$ = \ ( p ( t) \mid \Phi ( t , x ^ {*} ( t) , u ) - F ( t , x ^ {*} ( t) , u )) + $$

$$ - ( p ( t) \mid \Phi ( t , x ^ {*} ( t) , u ^ {*} ( t) ) + F ( t , x ^ {*} ( t) , u ^ {*} ( t) )) \leq 0 $$

for all possible admissible values of $ \dot{x} $ and $ u $.

If one carries out differentiation in (3) with respect to $ t $ and uses the notation

$$ {\mathcal H} ( t , x , u , p ) = ( p \mid \Phi ) - F , $$

then a necessary condition for a strong minimum can be stated in the form of a maximum principle, in which the Euler equation (3), the transversality condition (4) and the Weierstrass condition (5) are combined. For a vector function $ ( x ^ {*} , u ^ {*} ) $ to be a strong minimum in the problem (2) with fixed left-hand end and free right-hand end it is necessary that there is a solution of the system

$$ \dot{p} ( t) = - \frac{\partial {\mathcal H} ( t , x ^ {*} , u ^ {*} , p ) }{\partial x } ,\ p ( t _ {1} ) = 0 , $$

for which

$$ {\mathcal H} ( t , x ^ {*} ( t) , u ^ {*} ( t) , p ( t)) = \max _ {u \in U } {\mathcal H} ( t , x ^ {*} ( t) , u , p ( t) ) . $$

J.L. Lagrange considered similar problems in connection with studies in mechanics (in the second half of the 18th century).

For references see Variational calculus.

Comments

The notation $ ( a \mid b) $ denotes the inner product of the vectors $ a $ and $ b $.

How to Cite This Entry:
Lagrange problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lagrange_problem&oldid=47559
This article was adapted from an original article by I.B. VapnyarskiiV.M. Tikhomirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article