# 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)