# Weierstrass conditions (for a variational extremum)

Necessary and (partially) sufficient conditions for a strong extremum in the classical calculus of variations (cf. 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 ${\mathcal E}$- 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). 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) 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

 [1] M.A. Lavrent'ev, L.A. Lyusternik, "A course in variational calculus" , Moscow-Leningrad (1950) (In Russian) [2] G.A. Bliss, "Lectures on the calculus of variations" , Chicago Univ. Press (1947) [3] L.S. Pontryagin, V.G. Boltayanskii, R.V. Gamkrelidze, E.F. Mishchenko, "The mathematical theory of optimal processes" , Wiley (1962) (Translated from Russian)