Strong extremum

A minimal or maximal value $ J ( \widetilde{y} ) $ taken by a functional $ J ( y) $ at a curve $ \widetilde{y} ( x) $, $ x _ {1} \leq x \leq x _ {2} $, for which one of the inequalities

$$ J ( \widetilde{y} ) \leq J ( y) \ \ \textrm{ or } \ \ J ( \widetilde{y} ) \geq J ( y) $$

holds for all comparison curves $ y ( x) $ in an $ \epsilon $-neighbourhood of $ y ( x) $. The curves $ \widetilde{y} ( x) $ and $ y ( x) $ must satisfy given boundary conditions.

Since maximization of $ J ( y) $ is equivalent to minimization of $ - J ( y) $, instead of a strong maximum one often discusses only a strong minimum. The term "strong" emphasizes that only the condition of being $ \epsilon $-near to $ \widetilde{y} ( x) $ is imposed on the comparison curves $ y ( x) $:

$$ | y ( x) - \widetilde{y} ( x) | \leq \epsilon $$

on the whole interval $ [ x _ {1} , x _ {2} ] $, whereas the derivatives of the curves $ y ( x) $ and $ \widetilde{y} ( x) $ may differ as "strongly" as desired.

However, the very definition of a strong extremum is of a relative rather than absolute nature, since it gives an extremum not on the whole class of admissible comparison curves $ y ( x) $ for which $ J ( y) $ makes sense, but only relative to the subset of all admissible comparison curves belonging to the $ \epsilon $-neighbourhood of $ \widetilde{y} ( x) $. However, for brevity, the term "relative" is often omitted and one speaks of a strong extremum, meaning a strong relative extremum (see also Strong relative minimum).

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)

References

[a1]	L. Cesari, "Optimization - Theory and applications" , Springer (1983)