Chapter 2 Quadratic Variational Problems

Chapter 2

Q u a d r a t i c Var i a t i on a I Pro b le m s

1. Introduction In this chapter we wish to review some of the more important classical methods for treating linear differential equations and quadratic variational problems. An understanding of what is required will motivate o u r subsequent application of a different approach, dynamic programming to the same problems. At the outset however, it should be explicitly pointed out that all methods race dilficulties in treating problems of high dimension. These difficulties are different for different methods. We shall first discuss the minimization o f the scalar quadratic functional

J(u)

(u”

=

+ q(t)U2)dt,

JO’

subject to u ( 0 ) = c , , u ( T ) = cz and then the N-dimensional counterpart, the problem of minimizing

J(x) = 6

I’

[(x’, x’)

+ (x, A ( t )x)] dt ,

(2)

7

2. Variational Approach

subject to x(O)=c, x ( T ) = d . Our tool will be the associated EulerLagrange equation. Following this, we shall briefly sketch the Rayleigh-Ritz and BubnovGalerkin methods. All three procedures ultimately require the solution of linear algebraic equations, and the obstacles stem from this.

2. Variational Approach Let us suppose that there is a function u such that u and u' belong to Lz (0, T ) , u satisfies the boundary conditions, and u furnishes the absolute minimum of J(u). Our aim is to obtain a necessary condition that u must satisfy. This is the Euler-Lagrange equation which in this case turns out to be sufficient. We proceed as follows. Let u be another function such that u and u' belong to L'(0, T ) with v(0) = u(T) = 0. Then for a n y real E , the function u+ EU satisfies the original boundary conditions and is such that it and its derivative are in L2(0,T ) . Consider next the expression J ( U + E U )= J ( u )

+ c2J(u) + 2~

[u'u'+q(r)u~]dt,

which by assumption possesses an absolute minimum at yields the variational condition

E = 0.

(1)

This fact

for all u of the foregoing nature. Integrating by parts, this becomes

+

u l q ( t l ) u ( t , ) d t l ]0 T / T 0 [ u ' u ' - ~ r/ f0q ( t , ) u ( t , ) d t ,

The integrated term drops out since v ( 0 ) = u ( T ) = 0. Hence, for any constant c j , c u t [c3

+ u' - ~ q ( r l ) u ( r l ) d r l dr, ] = 0.

(4)

Choose q(tl)u(r,) dt,

v(0) = 0 ,

(5)

8

2 Quadratic Variational Problems

where c j is determined by the condition v ( T ) = O . Then (4) yields the relation c3

+ u’ -

J6’4(f)’

df, = 0

u(f1)

almost everywhere. The almost everywhere may be replaced by everywhere. Thus (6) yields the Euler-Lagrange equation u” - q ( t ) u = 0 ,

u(T) =

u(0) = c , ,

c2.

(7)

3. Positive Definiteness, Existence and Uniqueness of Solution It is not difficult to show that if J ( u ) is positive definite, then (2.7) possesses a unique solution. We proceed as follows. If there are two solutions, there is a solution u such that ~ ( 0=) v(T)= 0. Consider then the expression T

0

=

J’,

T

V(V”

L

T

- q ( f ) v ) d f= v z ~ ’ ]-~

L

T

=

0-

[v’2

[ d 2+ q ( t ) v 2 ] d f

+ q ( t )V 2 ] df,

(1)

a contradiction to the assumed positive-definite character of J ( v ) if u is not identically zero. That (2.7) possesses a solution follows from the discussion below.

4. Computational Aspects Let u1 and u2 be the principal solutions of (2.7), i.e., U ’ ( 0 ) = 1,

u,(O) = 0 ,

u; (0) = 0 ,

ui(0) = 1 .

(1)

Write u = a, u ,

+ a,u,,

(2)

where a , and a , are to be determined by the conditions (‘1

=

a,,

c2

=

a , UI ( T )

+ a, u,(T).

(3)

The positive-definite nature of J ( u ) guarantees that u 2 ( T ) # 0, as indicated in Section 3, whence a , and a, are uniquely determined. The values of the principal solutions are determined by numerical integration of the differential equation (2.7) with the appropriate initial conditions.

9

5. Vector-Matrix Case

5. Vector-Matrix Case Let us consider in similar fashion the task of minimizing

J(x)

=

/)(xf,x’)

+ (x,=4(t)x>3d t ,

where A ( t ) is positive definite. We suppose that x(0) = c, x(T)= d and x’~L’(0,T ) . Then proceeding as before the variational equation is X”

x ( 0 ) = c , x(T)= d .

-A(t)x = 0,

(2)

Analogous arguments to those given in Section 3 show that this equation possesses a unique solution. The computational aspects, however, require a careful examination. As a straightforward extension of the method used in the scalar case, write

where X , and X 2 are the principal matrix solutions of

X ” - A ( t ) X = 0,

(4)

that is,

X,(O)

=

I,

Xi(0)

X,(O)

=

0,

Xi(0) = I ,

=

0,

and a and b are constant vectors to be determined by the boundary conditions given in (2). These lead to the equations c =

a,

d

=

X,(T)a+ X2(T)b.

(6)

It is not difficult to extend the method sketched in Section 2 to the present case to show that X , ( T ) is not singular. Hence a and 6 , and thus x ( t ) , are uniquely determined by (2). This determination of b, however, requires a solution of a system of linear algebraic equations, always a ticklish matter. It is particularly so when the dimension of a is large and Tis large. The fact that T is large means that X 2 ( T )is close to a singular matrix. This combined with the fact that dim(X2) is large means that numerical accuracy is not readily ensured. A detailed discussion of these matters will be found in the reference cited in the bibliography at the end of the chapter. I t is the block to a straightforward solution of linear differential equations of high dimension that motivates the constant search for new approaches to linear partial differential equations and quadratic variational problems.

10

2 Quadratic Variational Problems

6. Rayleigh-Ritz Method

Let us briefly sketch two of the most powerful approaches which avoid the route indicated i n Section 5. The first is the Rayleigh-Ritz method. Consider the functional J(x) =

lr

[(x', x')

+ (x,A ( t ) x)] d t ,

with x(0) = c, x ( T ) = d, and use the trial function

We have a choice of letting the 4k be scalars and the 6, vectors or conversely. In any case, let us choose the & as known functions and the b, as unknowns, subject only to the boundary conditions. Then J ( x ) = J @ , , b,, ..', bM)

(3)

and the minimization is now over the 6,. This leads to a set of linear algebraic equations, of degree M if the bi are scalars. If M N , this is a much more tractable problem than the original. If we do not wish to use the boundary conditions, we can form the new functional

J(x, I . 1 , & )

=

J(x) + AI(x(0)- c,x(O)

- C)

+ A ~ ( x ( T-) d , x ( T ) - d ) ,

(4) where I., and A2 are Courant parameters and A, and A2 % 1 . The choice of the sequence { 4 , ( f ) }is critical and is determined in any particular case by some combination of mathematical reasoning, physical intuition, and experience.

7. Bubnov-Galerkin Method I n place of solving the Euler-Lagrange equation x"- A ( t ) x =

0,

x(0) =

C,

x(T) = d ,

consider the problem of minimizing PT

J,(x) =

J -(x"-A(t)x,x"0

A(t)x)dt

11

Bibliography and Comment

over x ( t ) such that the boundary conditions are fulfilled and the integral exists. Now use the Rayleigh-Ritz method to carry out this minimization. It is sometimes convenient t o introduce the mixed expression J , (x) = J(x)

+ AJl (x)

(3)

in place of J(x) alone, where I again is a Courant parameter. BIBLIOGRAPHY A N D COMMENT Sections 1-5. A detailed discussion of these problems plus a number of additional references may be found in

R. Bellman, Introduction

to the Mathernutical Theory of Control Processq I : Linear Equations and Quadratic Critwia, Academic Press, New York, 1967.

For an introduction t o the regularization techniques of Tychonov, see

R. Bellman, R. Kalaba, and J . Lockett, Nunwrical Inriersion Amer. Elsevier, New York, 1966. Sections 6-7.

of the Laplace Transfimn.

See

R. Bellman, Methods of Nonlinear .4nalvsis, Vol. I , Academic Press, New York, 1970.