Engineering Optimization: Theory and Practice, Fourth Edition

678 Optimal Control and Optimality Criteria Methods

The value of the unknown constantλcan be found by using Eq. (E 7 ) as

m= 2 ρ

∫L

0

y(x) dx= 2 ρ

(

c 1 L+c 2

L^2

2

+c 3

L^3

3

)

that is,

m

2 ρL

=c 1 +c 2

L

2

+c 3

L^2

3

=

hL

2 (kρλ)^1 /^2

−

1

3

hL^2

k

(E 20 )

Equation (E 20 ) gives

λ^1 /^2 =

hL

(kρ)^1 /^2

1

(m/ρL)+^23 (hL^2 /k)

(E 21 )

Hence the desired solution can be obtained by substituting Eq. (E 21 ) in Eq. (E 16 ).

12.2.4 Generalization

The concept of including constraints can be generalized as follows. Let the problem be

to find the functionsu 1 (x, y, z), u 2 (x, y, z),... , un(x, y, z) that make the functional

∫

V

f

(

x, y, z, u 1 , u 2 ,... , un,

∂u 1

∂x

,...

)

dV (12.18)

stationary subject to themconstraints

g 1

(

x, y, z, u 1 , u 2 ,... , un,

∂u 1

∂x

,...

)

= 0

gm

(

x, y, z, u 1 , u 2 ,... , un,

∂u 1

∂x

,...

)

= 0

(12.19)

The Lagrange multiplier method consists in taking variations in the functional

A=

∫

V

(f+λ 1 g 1 +λ 2 g 2 + · · · +λmgm) dV (12.20)

whereλiare now functions of position. In the special case where one or more of the

giare integral conditions, the associatedλiare constants.

12.3 Optimal Control Theory

The basic optimal control problem can be stated as follows:

Find the control vectoru =











u 1

u 2

..

.

um









