Weierstrass-Erdmann corner conditions

Necessary conditions for an extremum, additional to the Euler equation, specified at points at which the extremal has a corner. Let

$$ J ( x) = \int\limits L ( t, x, \dot{x} ) dt $$

be a functional of the classical calculus of variations (cf. Variational calculus), and let the extremal $ x _ {0} ( t) $ be continuously differentiable in a neighbourhood of the point $ \tau $ except at the point $ \tau $ itself, at which it has a corner. In this situation, for $ x _ {0} ( t) $ to be at least a weak local extremum for the functional $ J( x) $, it is necessary that the equations

$$ p( \tau - 0) = p ( \tau + 0), $$

$$ H( \tau - 0) = H ( \tau + 0), $$

where

$$ p( t) = \ \frac{\partial L( t , x _ {0} ( t), \dot{x} _ {0} ( t)) }{\partial \dot{x} } $$

and

$$ H( t) = \ ( \dot{x} _ {0} ( t), p( t))- L ( t, x _ {0} ( t), \dot{x} _ {0} ( t)), $$

be satisfied at the corner point $ \tau $. These equations are known as the corner conditions of K. Weierstrass (1865) and G. Erdmann (1877) [1].

The meaning of the Weierstrass–Erdmann corner conditions is that the canonical variables and the Hamiltonian are continuous at a corner point of the extremal; their meaning in classical mechanics is the continuity of momentum and of energy at a corner point.

In regular problems, when $ L $ is a strictly convex function of $ \dot{x} $, the extremals cannot have corner points. Corner points appear if $ L( t, x, \dot{x} ) $ and consequently the Weierstrass $ {\mathcal E} $- function, contains segments of $ \dot{x} $. For the Lagrange problem with conditions $ \phi _ {i} ( t, x, \dot{x} ) = 0 $ and Lagrange multipliers $ \lambda _ {i} ( t) $, the $ L $ in the Weierstrass–Erdmann corner conditions is replaced by $ \widetilde{L} = L+ \sum _ {i} \lambda _ {i} \phi _ {i} $.

References

[1]	G. Erdmann, "Ueber die unstetige Lösungen in der Variationsrechnung" J. Reine Angew. Math. , 82 (1877) pp. 21–30
[2]	O. Bolza, "Lectures on the calculus of variations" , Chelsea, reprint (1960) (Translated from German)
[3]	N.I. Akhiezer, "The calculus of variations" , Blaisdell (1962) (Translated from Russian)

Comments

See also Weierstrass conditions (for a variational extremum).

References

[a1]	L. Cesari, "Optimization - Theory and applications" , Springer (1983)
[a2]	G.M. Ewing, "Calculus of variations with applications" , Dover, reprint (1985)
[a3]	Yu.P. Petrov, "Variational methods in optimum control theory" , Acad. Press (1968) pp. Chapt. IV (Translated from Russian)