Engineering Optimization: Theory and Practice, Fourth Edition

670 Optimal Control and Optimality Criteria Methods

Figure 12.1 Tentative and exact solutions.

Also, we define the variation of a function of several variables or a functional in a

manner similar to the calculus definition of a total differential:

δF=

∂F

∂u

δu+

∂F

∂u′

δu′+

∂F

∂u′′

δu′′+

∂F

∂x

δx (12.5)

↑

0

(since we are finding variation ofFfor a fixed value ofx, i.e.,δx=0).

Now, let us consider the variation inA(δA) corresponding to variations in the

solution (δu). If we want the condition for the stationariness ofA, we take the nec-

essary condition as the vanishing of first derivative ofA(similar to maximization or

minimization of simple functions in ordinary calculus).

δA=

∫ x 2

x 1

(

∂F

∂x

δu+

∂F

∂u′

δu′+

∂F

∂u′′

δu′′

)

dx=

∫x 2

x 1

δF dx= 0 (12.6)

Integrate the second and third terms by parts to obtain

∫x 2

x 1

∂F

∂u′

δu′dx=

∫x 2

x 1

∂F

∂u′

δ

(

∂u

∂x

)

dx=

∫x 2

x 1

∂F

∂u′

∂

∂x

(δu) dx

=

∂F

∂u′

δu

∣

∣

∣

∣

x 2

x 1

−

∫x 2

x 1

d

dx

(

∂F

∂u′

)

δu dx (12.7)

∫x 2

x 1

∂F

∂u′′

δu′′dx=

∫x 2

x 1

∂F

∂u′′

∂

∂x

(δu′)dx=

∂F

∂u′′

δu′

∣

∣

∣

∣

x 2

x 1

−

∫x 2

x 1

d

dx

(

∂F

∂u′′

)

δu′dx

=

∂F

∂u′′

δu′

∣

∣

∣

∣

x 2

x 1

−

d

dx

(

∂F

∂u′′

)

δu

∣

∣

∣

∣

x 2

x 1

+

∫x 2

x 1

d^2

dx^2

(

∂F

∂u′′

)

δu dx (12.8)