Engineering Optimization: Theory and Practice, Fourth Edition

674 Optimal Control and Optimality Criteria Methods

Figure 12.4 Element of surface area acted on by the

pressurep.

Findy(x)which minimizes the dragP given by Eq. (E 5 ) subject to the condition

thaty(x)satisfies the end conditions

y(x= 0 )=0 and y(x=L)=R (E 6 )

By comparing the functionalPof Eq. (E 5 ) with Aof Eq. (12.1), we find that

F (x, y, y′, y′′) = 4 πρν^2 (y′)^3 y (E 7 )

The Euler–Lagrange equation, Eq. (12.10), corresponding to this functional can be

obtained as

(y′)^3 − 3

d

dx

[y(y′)^2 ]= 0 (E 8 )

The boundary conditions, Eqs. (12.11) and (12.12), reduce to

[3y(y′)^2 ] yδ

∣

∣

∣

∣

x 2 =L

x 1 = 0

= 0 (E 9 )

Equation (E 8 ) can be written as

(y′)^3 [− 3 y′(y′)^2 + y( 2 )y′y′′]= 0

or

(y′)^3 + 3 yy′y′′= 0 (E 10 )

This equation, when integrated once, gives

y(y′)^3 =k 13 (E 11 )

wherek^31 is a constant of integration. Integrating Eq. (E 11 ), we obtain

y(x)=(k 1 x+k 2 )^3 /^4 (E 12 )

The application of the boundary conditions, Eqs. (E 6 ), gives the values of the

constants as

k 1 =

R^4 /^3

L

and k 2 = 0