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)

Comments

See also Weierstrass–Erdmann corner conditions, for necessary conditions at a corner of an extremal.

References

[a1]	I.M. Gel'fand, S.V. Fomin, "Calculus of variations" , Prentice-Hall (1963) (Translated from Russian)
[a2]	L. Cesari, "Optimization - Theory and applications" , Springer (1983)