Engineering Optimization: Theory and Practice, Fourth Edition

672 Optimal Control and Optimality Criteria Methods

Figure 12.2 Curve of minimum time of

descent.

Since potential energy is converted to kinetic energy as the particle moves down the

path, we can write

1

2 mν

(^2) = mgx
Hence
dt=

[

1 +(y′)^2

2 gx

] 1 / 2

dx (E 1 )

and the integral to be stationary is

t=

∫xB

0

[

1 +(y′)^2

2 gx

] 1 / 2

dx (E 2 )

The integrand is a function ofxandy′and so is a special case of Eq. (12.1). Using

the Euler–Lagrange equation,

d

dx

(

∂F

∂y′

)

−

∂F

∂y

=0 withF=

[

1 +(y′)^2

2 gx

] 1 / 2

we obtain

d

dx

(

y′

{x[1+(y′)^2 ]}^1 /^2

)

= 0

Integratingyields

y′=

dy

dx

=

(

C 1 x

1 −C 1 x

) 1 / 2

(E 3 )

whereC 1 is a constant of integration. The ordinary differential equation (E 3 ) yields on

integration the solution to the problem as

y(x)=C 1 sin−^1 (x/C 1 )−( 2 C 1 x−x^2 )^1 /^2 +C 2 (E 4 )

Example 12.2 Design of a Solid Body of Revolution for Minimum Drag Next we

consider the problem of determining the shape of a solid body of revolution for mini-

mum drag. In the general case, the forces exerted on a solid body translating in a fluid