Weierstrass E-function

in the classical calculus of variations

A function which isolates the main part of the increment of a functional as the extremal is varied, using a local (needle-shaped) variation for a given value of its derivative, at a given point of the extremal. In the case of the functional

$$ J(x) \ = \ \int\limits _ {t _ 0} ^ {t _ 1} L(t,\ x,\ \dot{x} ) \ dt,\ \ L : \ \mathbf R \times \mathbf R ^{n} \times \mathbf R ^{n} \rightarrow \mathbf R , $$

the $ {\mathcal E} $- function has the form

$$ \tag{1} {\mathcal E} (t,\ x,\ \dot{x} ,\ x ^ \prime )\ = $$

$$ = \ L ( t,\ x,\ x ^ \prime ) - L ( t,\ x,\ \dot{x} ) - ( x ^ \prime - \dot{x} ,\ L _{ {\dot{x} }} ( t,\ x,\ \dot{x} )). $$

If one introduces the function

$$ \Pi (t,\ x,\ p,\ x ^ \prime ) \ = \ (p,\ x ^ \prime ) - L ( t,\ x,\ x ^ \prime ) $$

(cf. Legendre transform; Pontryagin maximum principle), the $ {\mathcal E} $- function assumes the form

$$ {\mathcal E} (t,\ x,\ \dot{x} ,\ x ^ \prime ) \ = \ \Pi ( t,\ x,\ p,\ \dot{x} ) - \Pi ( t,\ x,\ p,\ x ^ \prime ), $$

where $ p= L _{ {\dot{x} }} (t,\ x,\ \dot{x} ) $. The general construction of functions analogous to the $ {\mathcal E} $- function (1) consists of the following. Let $ f $ be a differentiable or convex function, defined on a Banach space $ X $, and let $ X ^{*} $ be the dual space. If the function $ \Pi : \ X ^{*} \times X \rightarrow \mathbf R $ is defined by the equation

$$ \Pi (x ^{*} ,\ x ^ \prime ) \ = \ \langle x ^{*} ,\ x ^ \prime \rangle - f ( x ^ \prime ) , $$

where $ x ^{*} $ is the derivative $ f ^ {\ \prime} (x) $ of $ f $ at $ x $( or the subdifferential element if $ f $ is convex), the function

$$ {\mathcal E} ( x,\ x ^ \prime ,\ x ^{*} ) \ = \ \Pi ( x ^{*} ,\ x ) - \Pi ( x ^{*} ,\ x ^ \prime ) $$

is the $ {\mathcal E} $- function constructed from $ f $. If $ f $ is differentiable,

$$ \tag{2} {\mathcal E} ( x ,\ x ^ \prime ) \ = \ f ( x ^ \prime ) - f ( x ) - < x ^ \prime - x ,\ f ^ {\ \prime} (x) > , $$

i.e. the $ {\mathcal E} $- function is the difference at $ x ^ \prime $ between $ f $ and the linear function tangent to $ f $ at $ x $. A comparison of formulas (1) and (2) shows that in the classical calculus of variations the $ {\mathcal E} $- function is obtained from the construction (2) with respect to the variables related to derivatives, while the variables $ t,\ x $ play the role of parameters.

In the case of a functional

$$ \int\limits _{T} L (t,\ x,\ \dot{x} ) \ dt,\ \ t \ = \ (t _{1} \dots t _{n} ) , \ \ T \ \subset \ \mathbf R ^{m} , $$

$$ L : \ \mathbf R ^{m} \times \mathbf R \times \mathbf R ^{m} \ \rightarrow \ \mathbf R ,\ \ \dot{x} \ = \ \left ( \frac{\partial x}{\partial t _ 1} \dots \frac{\partial x}{\partial t _ m} \right ) , $$

in a multi-dimensional variational problem, the $ {\mathcal E} $- function has the following form:

$$ {\mathcal E} ( t,\ x,\ z,\ z ^ \prime ) \ = \ L (t,\ x,\ z ^ \prime ) - L (t,\ x,\ z ) - $$

$$ - (z ^ \prime - z ,\ L _{z} ( t, x, z ) ). $$

In the case of the Lagrange problem with boundaries $ \phi _{i} (t,\ x,\ \dot{x} ) = 0 $ and Lagrange multipliers $ \lambda _{i} (t) $, $ i=1 \dots s $, the $ {\mathcal E} $- function has the form (1), in which

$$ \widetilde{L} \ = \ L + \sum _{i=1} ^ s \lambda _{i} \phi _{i} $$

has been substituted for $ L $.

The $ {\mathcal E} $- function, first introduced in 1879 by K. Weierstrass [1], lies at the foundation of the theory of the calculus of variations (cf. Variational calculus). It is used in the formulation of necessary and (partially) sufficient conditions for an extremum (cf. Weierstrass conditions (for a variational extremum)), and serves to express the increment of a functional $ J $ on an extremal in the form of a finite integral (cf. Weierstrass formula).

An especially important role in variational calculus is played by smooth functionals in which, in a given parameter range, $ {\mathcal E} ( \cdot ,\ \dot{x} ,\ x ^ \prime ) \geq 0 $ for all $ \dot{x} ,\ x ^ \prime $, or, stronger, if $ {\mathcal E} ( \cdot ,\ \dot{x} ,\ x ^ \prime ) >0 $ for all $ \dot{x} \neq x ^ \prime $. They are known as quasi-regular and regular or elliptic, respectively. For such functionals the Legendre condition and the necessary Weierstrass conditions (for a variational extremum) are invariably valid, as are the theorems of existence and regularity [7].

References

[1]	K. Weierstrass, "Vorlesungen über Variationsrechnung" , Math. Werke , 7 , Akademie Verlag (1927)
[2]	C. Carathéodory, "Calculus of variations and partial differential equations of the first order" , 1–2 , Holden-Day (1965–1967) (Translated from German)
[3]	O. Bolza, "Lectures on the calculus of variations" , Chelsea, reprint (1960) (Translated from German)
[4]	N.I. Akhiezer, "The calculus of variations" , Blaisdell (1962) (Translated from Russian)
[5]	L.S. Pontryagin, V.G. Boltayanskii, R.V. Gamkrelidze, E.F. Mishchenko, "The mathematical theory of optimal processes" , Wiley (1962) (Translated from Russian)
[6]	M.R. Hestenes, "Calculus of variations and optimal control theory" , Wiley (1966)
[7]	"Hilbert problems" Bull. Amer. Math. Soc. , 8 (1902) pp. 437–479 (Translated from German)
[8]	G.A. Bliss, "Lectures on the calculus of variations" , Chicago Univ. Press (1947)

References

[a1]	L. Cesari, "Optimization - Theory and applications" , Springer (1983)
[a2]	G.M. Ewing, "Calculus of variations with applications" , Dover, reprint (1985)
[a3]	E.B. Lee, L. Marcus, "Foundations of optimal control theory" , Wiley (1967)