View source for Weak relative minimum

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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|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|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====
<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====

====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>