Weak relative minimum

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.

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 must hold on it. For a strong relative minimum, the more general 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 have a local maximum (absolute maximum) with respect to the control (in agreement with the 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

[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]	D.G. Luenberger, "Optimization by vector space methods" , Wiley (1969)
[a2]	L. Cesari, "Optimization - Theory and applications" , Springer (1983)