Engineering Optimization: Theory and Practice, Fourth Edition

690 Optimal Control and Optimality Criteria Methods

(g)Functional

(h)Hamiltonian

12.3 Match the following terms and descriptions:

(a)Adjoint variables Linear elastic structures

(b)Optimality criteria methods Lagrange multipliers

(c)Calculus of variations Necessary conditions of optimality

(d)Optimal control theory Optimization of functionals

(e)Governing equations Hamiltonian used

12.4 What are the characteristics of a variational operator?

12.5 What are Euler–Lagrange equations?

12.6 Which method can be used to solve a trajectory optimization problem?

12.7 What is an optimality criteria method?

12.8 What is the basis of optimality criteria methods?

12.9 What are the advantages of using reciprocal approximations in structural optimization?

12.10 What is the difference between free and forced boundary conditions?

12.11 What type of problems require introduction of Lagrange multipliers?

12.12 Where are reciprocal approximations used? Why?

Problems

12.1 Find the curve connecting two pointsA(0, 0) andB(2, 0) such that the length of the

line is a minimum and the area under the curve isπ/2.

12.2 Prove that the shortest distance between two points is a straight line. Show that the

necessary conditions yield a minimum and not a maximum.

12.3 Find the functionx(t )that minimizes the functional

A=

∫T

0

[

x^2 + 2 xt+

(

dx

dt

) 2 ]

dt

with the condition thatx(0)=2.

12.4 Find the closed plane curve of lengthLthat encloses a maximum area.

12.5 The potential energy of an elastic circular annular plate of radiir 1 andr 2 shown in

Fig. 12.7 is given by

π 0 =π D

∫r 2

r 1

[

r

(

d^2 w

dr^2

) 2

+

1

r

(

dw

dr

) 2

+ 2 ν

dw

dr

d^2 w

dr^2

]

dr

− 2 π

∫r 2

r 1

qrw dr+ 2 π

[

rM

dw

dr

−rQw

]

r=r 2