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

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

where

and

be satisfied at the corner point . 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 is a strictly convex function of , the extremals cannot have corner points. Corner points appear if and consequently the Weierstrass -function, contains segments of . For the Lagrange problem with conditions and Lagrange multipliers , the in the Weierstrass–Erdmann corner conditions is replaced by .

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)