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CALCULUS OF VARIATIONS
- Introduction
- Euler Equation
- Functions, Functionals and Neighborhoods
- More Complex Problems
- Constrained Variational Problems
- Closure
- References
- Problems
| Introduction | top |
The calculus of variations and its extensions are devoted to finding the optimum function that gives the best value of the economic model and satisfies the constraints of a system. The need for an optimum function, rather than an optimal point, arises in numerous problems from a wide range of fields in engineering and physics, which include optimal control, transport phenomena, optics, elasticity, vibrations, statics and dynamics of solid bodies and navigation. Two examples are determining the optimal temperatures profile in a catalytic reactor to maximize the conversion and the optimal trajectory for a missile to maximize the satellite payload placed in orbit. The first calculus of variations problem, the Brachistochrone problem, was posed and solved by Johannes Bernoulli in 1696(1). In this problem the optimum curve was determined to minimize the time traveled by a particle sliding without friction between two points.
This chapter is devoted to a relatively brief discussion of some of the key concepts of this topic. These include the Euler equation and the Euler-Poisson equations for the case of several functions and several independent variables with and without constraints. It begins with a derivation of the Euler equation and extends these concepts to more detailed cases. Examples are given to illustrate this theory.
The purpose of this chapter is to develop an appreciation for what is required to determine the optimum function for a variational problem. The extensions and applications to optimal control, Pontryagin's maximum principle and continuous dynamic programming are left to books devoted to those topics.
| Functions, Functionals and Neighborhoods | top |
It will be necessary to discuss briefly functionals and neighborhoods before developing the Euler equation for the solution of the simplest problem in the calculus of variations. In mathematical programming the maximum or minimum of a function was determined to be an optimal point or set of points. In the calculus of variations the maximum or minimum value of a functional is determined to be an optimal function. A functional is a function of a function and depends on the entire path of one or more functions rather than a number of discrete variables.
For the calculus of variations the functional is an integral, and the function that appears in the integrand of the integral is to be selected to maximize or minimize the value of the integral. The texts by Forray(1), Ewing(2), Weinstock(3), Schechter(4), and Sagan(6) elaborate on this concept. However, at this point let us examine an example of the functional given by equation (8-1). The minimum of this functional is a function y(x) that gives the shortest distance between two points [x0,y(x0)] and [x1,y(x1)].
In this equation y' is the first derivative of y with respect to x. The function that minimizes this integral, a straight line, will be obtained as an illustration of the use of the Euler equation in the next section.
The concept of
a neighborhood is used in the derivation of the Euler equation to convert
the problem into one of finding the stationary point of a function of
a single variable. A function is said to be in the neighborhood of a function
y if | 
| < h (5). This is illustrated in Figure
8-1(a). The concept can be extended for more restrictive conditions
such as that shown in Figure 8-1(b)
when | |
< h and | 
|
< h. For this case 
is said to be in the neighborhood of first order to y. Consequently, the
higher the order of the neighborhood the more nearly the functions will
coincide. Extensions of these definitions will lead to what are referred
to as strong and weak variations(6).
| Euler Equation | top |
The simplest form of the integral to be optimized by the calculus of variations is the following one.
In addition, the values of y(x0) and y(x1) are known; and an example of the function F(x, y, y') was given in equation (8-1) as:
F(x, y, y') = [1 + (y')2]1/2 (8-3)
To obtain the optimal function that minimizes the equation (8-2), it is necessary to solve the Euler equation, which is the following second order ordinary differential equation.
It is not obvious that equation (8-4) is a second order ordinary differential equation. Also, it probably appears unusual to be partially differentiating the function F with respect to y and y'. In addition, although the term minimize will be used, stationary points are being located, and their character will have to be determined using sufficient conditions. Consequently, it should be beneficial to outline the derivation of the Euler equation.
First, y(x) is specified as the function
that minimizes the functional I[y(x)], equation (8-2). (however, the form
of y(x) has to be determined.) Then a function (x)
is constructed to be in the neighborhood of y(x) as follows:
where a is a parameter that can be made arbitrarily small. Also n(x) is a continuously differentiable function defined on the interval x0 < x < x1 with n(x0) = n(x1) = 0 but is arbitrary elsewhere. The results from the derivation using equation (8-5) are described mathematically as weak variations (1), for an(x) and an'(x) are small.
Now equation (8-2)
is written in terms of the function (x)
as:
The above equation can be put in terms of the optimal function y(x) and the arbitrary function n(x) using equation (8-5).
The mathematical argument(3) is made that all of the possible functions lie in an arbitrarily small neighborhood of y because a can be made arbitrarily small. As such, the integral of equation (8-7) may be regarded as an ordinary function of a, F(a), because would specify the value of the integral knowing F(a = 0) at the minimum from y(x).
The minimum of F(a) is obtained by sitting the first derivative of F with respect to a equal to zero. The differentiation is indicated as:
Leibnitz' rule, equation (8-10), is required to differentiate the integral given in equation (8-9).
where x0 and x1 correspond to a1 and a2, and a corresponds to t.
The upper and lower limits, x0 and x1, are constants; and the following derivatives in the second and third terms on the right hand side of equation (8-10) are zero for this case.
Consequently, the order of integration and differentiation is interchanged, and equation (8-9) can be written as:
The integrand can be expanded as follows:
where dx/da = 0, because x is treated as a constant in the mathematical argument of considering changes from curve to curve at constant x.
Substituting equation (8-13) into equation (8-12) gives:
The following results are needed:
Using equation (8-15), we can write equation (8-14) as:
An integration-by-parts will give a more convenient form for the term involving n' i.e.:
The first term on the right hand side is zero for n(xo) = n(x1) = 0. Combining the results from equation (8-17) with equation (8-16) gives:
At the optimum dF/da
= 0, and letting a
® 0 has
®
y and '®
y'. Therefore, the above equation becomes:
To obtain the Euler equation the fundamental lemma of the calculus of variation is used. This lemma can be stated after Weinstock (3) as:
"If x0 and x1 (>x0) are fixed constants and G(x) is a particular continuous function in the interval x0 < x < x1 and if:
for every choice of the continuously differentiable function n(x) for which n(x0) = n(x1) = 0 then G(x) = 0 identically in the interval x0 < x < x1."
The proof of this lemma is by contradiction, and is given by Weinstock (3).
Applying this lemma to equation (8-18) gives the Euler equation.
This equation is a second order ordinary differential equation and has boundary conditions y(xo) and y(x1). The solution of this differential equation y(x) optimizes the integral I[y(x)].
A more convenient form of the Euler equation can be obtained by applying the chain rule to ¶F(x, y, y')/¶y'.
Substituting (equation 8-20) into equation (8-4) and rearranging gives a more familiar form for a second order ordinary differential equation.
A more convenient way to write this equation is:
The coefficients for the differential equation come from partially differentiating F.
A special case that sometimes occurs is to have F(y, y'), i.e., F is not a function of x. For this situation it can be shown that:
This equation may be integrated once to obtain a form of the Euler equation given below which can be a more convenient starting point for problem solving.
where the constant is evaluated using one of the boundary conditions.
At this point it should be noted that the necessary conditions of the classical theory of maxima and minima have been used to locate a stationary point. This point may be a minimum, maximum or saddle point. To determine its character, sufficient conditions must be used; and these will be discussed subsequently. However, before that the following example is used to illustrate an application of the Euler equation.
Example 8-1
Determine the function that gives the shortest distance between two given points. Referring to Figure 8-2, we can state the problem as:
and from the figure it follows that:
ds = [(dx)2 + (dy)2]½ = [1+ (y')2]½ dx
Substituting for ds in the integral gives:
Evaluating the partial derivatives for the Euler equation.
Fy = 0 Fy 'x = 0 Fy 'y = 0 Fy 'y ' = 1 / [ 1 + (y')2]3/2
Substituting, equation 8-22 becomes:
simplifying gives
Integrating the above equation twice gives:
y = c1 x + c2
This is the equation of a straight line; and the constants, c1 and c2, are evaluated from the boundary conditions.
Another classic problem of the calculus of variation as mentioned earlier is the Brachistochrone problem(10). The shape of the curve between two points is to be determined to minimize the time of a particle sliding along a wire without frictional resistance. The particle is acted upon only by gravitational forces as it travels between the two points. The approach to the solution is the same as for Example 8-1, and the integral for the time of travel, T, is given by Weinstock(3) as:
and the solution is in terms of the following parametric equations.
x = x0 + a[q - sin(q)] y = y0 + a[1 - cos(q)] (8-26)
The details of the solution are given by Weinstock(3). The solution is the equations for a cycloid.
The method of obtaining the Euler equation is used almost directly to obtain the results for more detailed forms of the integrand of equation (8-2). The next section extends the results for more complex problems.