1
Extrema of Differentiable Functionals

Motto:

"I wrote them with my nail on the plaster

On a wall of empty cracks,

In the dark, in my solitude,

Unaided by the bull lion vulture

of Luke, Mark and John."

Tudor Arghezi - Flowers of Mildew

Variational calculus aims to generalize the constructions of classical analysis, to solve extrema problems for functionals. In this introductory chapter, we will study the problem of differentiable functionals defined on various classes of functions. The news refers to path integral functionals. For details, see[ 4, 6, 7, 8, 14, 18, 21, 30, 58].

1.1 Differentiable Functionals

Let us consider the point and the function , . If this function has continuous partial derivatives with respect to each of the variables , then increasing the argument with produces

The first term on the right-hand side represents the differential of the function at the point , a linear form of argument growth, the vector ; the second term is the deviation from the linear approximation and is small in relation to , in the sense that

Let be two normed vector spaces. The previous definition can be extended immediately to the case of functions . If is a space of functions, and , then instead of function we use the term functional.

Definition 1.1 The function is called differentiable at a point if there exist a linear and continuous operator and a continuous function such that for any vector to have

The linear continuous operator is called the derivative of the function at given point .

For a given nonzero vector and , the vector

if the limit exists, is called the variation or derivative of the function f at the point x along the direction h.

A differentiable function of real variables has a derivative along any direction. The property is also kept for functions between normed vector spaces. Specifically, the next one takes place:

Proposition 1.1 If the function is differentiable at the point , then for any nonzero vector the function , of real variable , is derivable with respect to , for and

Proof. The derivative sought is obviously and then

Example 1.1: Any linear continuous operator is, obviously, a differentiable function at any point and since

Let be a functional. The simplest examples of functionals are given by formulas: (i) evaluation functional ("application of a function on the value at a point"), , where is a fixed point; (ii) definite integration (functional defined by definite integral); (iii) numerical quadrature defined by definite integration:

where means the set of all polynomials of degree at most ; and (iv) distributions in analysis (as linear functionals defined on spaces of test functions). We will continue to deal with definite integral type functionals, as the reasoning is more favorable to us.

Example 1.2: Let us consider the functional

defined on the space of the continuous functions on the segment , endowed with the norm of uniform convergence. The Lagrangian of the functional (i.e., the function ) is presumed continuous and with continuous partial derivatives of the first order in the domain .

Let us determine the variation of the functional when the argument increases with :

Since the Lagrangian is a differentiable function, we have

where

Therefore

In this way, according to the previous proposition and the derivation formula for integrals with parameters, we find

Example 1.3: Let us consider the functional

defined on the space of functions with a continuous derivative on the segment , endowed with the norm of uniform convergence of derivatives. The Lagrangian of the functional is supposed to have first-order continuous partial derivatives. To write the integral functional, we use in fact the pullback form of the Lagrangian.

Applying the derivation formula of the integrals with parameter we obtain

Analogously, we can extend the result to the functional

defined on the space of functions with continuous derivatives up to the order inclusive, on the segment . In this way we have

Example 1.4: Let us consider a functional which depends on several function variables, for example

defined on the space , whose elements are vector functions , with the norm of uniform convergence of derivatives

( means the derivative with respect to ).

Supposing that the Lagrangian has continuous partial derivatives, denoting , the variation of the functional is

(sum over the index ).

Now, let us consider functionals whose arguments are functions of several real variables.

Example 1.5: Let be a compact domain. As an example we will take, first, the double integral functional

where we noted for abbreviation

The functional is defined on the space of all functions with continuous partial derivatives; the norm of the space is given by

Supposing that the Lagrangian has continuous partial derivatives, the variation of the functional , as the argument grows with , is

Example 1.6: Another example is the curvilinear integral functional

where is a piecewise curve which joins two fixed points in a compact domain .

Suppose the argument grows with . Then the variation of the functional is

1.2 Extrema of Differentiable Functionals

Let us consider the functional , defined on a subset of a normed vector space of functions. By definition, the functional has a (relative) minimum at the point in , if there exists a neighborhood , of point , such that

If the point has a neighborhood on which the opposite inequality takes place

we say that is a point of local maximum for the functional . The minimum and maximum points are called relative extrema points.

In classical analysis, the extrema points of a differentiable function are among the critical points, that is, among the points that cancels first-order derivatives. A similar property occurs in the case of functional ones on normed vector spaces of functions, only in this case the extrema points are found between the extremals (critical points).

Proposition 1.2 If the function is an extremum point for the functional , interior point of the set and if is differentiable at this point, then for any growth .

Proof. Let be a growth of the argument (function); since is an interior point of the set , the function of real variable is defined on hole interval . This function has an extremum point at and is derivable at this point. Then its derivative must vanish at . It follows

Any point at which the variation of the functional is canceled identically with respect to is called either the stationary point or critical point or extremal of the functional.

Hence, for establishing the extrema points of a functional, the variation must be expressed, determine the critical points (those at which the...