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
be satisfied at the corner point . These equations are known as the corner conditions of K. Weierstrass (1865) and G. Erdmann (1877) .
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 .
|||G. Erdmann, "Ueber die unstetige Lösungen in der Variationsrechnung" J. Reine Angew. Math. , 82 (1877) pp. 21–30|
|||O. Bolza, "Lectures on the calculus of variations" , Chelsea, reprint (1960) (Translated from German)|
|||N.I. Akhiezer, "The calculus of variations" , Blaisdell (1962) (Translated from Russian)|
|[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)|
Weierstrass-Erdmann corner conditions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weierstrass-Erdmann_corner_conditions&oldid=16105