View source for Weierstrass conditions (for a variational extremum)

<!--
w0974001.png
$#A+1 = 24 n = 0
$#C+1 = 24 : ~/encyclopedia/old_files/data/W097/W.0907400 Weierstrass conditions (for a variational extremum)
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

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

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

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 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

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

(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

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

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|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====
<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====
See also [[Weierstrass–Erdmann corner conditions|Weierstrass–Erdmann corner conditions]], for necessary conditions at a corner of an extremal.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.M. Gel'fand,   S.V. Fomin,   "Calculus of variations" , Prentice-Hall  (1963)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Cesari,   "Optimization - Theory and applications" , Springer  (1983)</TD></TR></table>