Engineering Optimization: Theory and Practice, Fourth Edition

16 Introduction to Optimization

Here the design variables are functions of the length parametert. This type of problem,

where each design variable is a function of one or more parameters, is known as a

trajectoryordynamic optimization problem[1.55].

1.5.3 Classification Based on the Physical Structure of the Problem

Depending on the physical structure of the problem, optimization problems can be

classified as optimal control and nonoptimal control problems.

Optimal Control Problem. Anoptimal control(OC)problemis a mathematical pro-

gramming problem involving a number of stages, where each stage evolves from the

preceding stage in a prescribed manner. It is usually described by two types of vari-

ables: the control (design) and the state variables. Thecontrol variables define the

system and govern the evolution of the system from one stage to the next, and thestate

variablesdescribe the behavior or status of the system in any stage. The problem is

to find a set of control or design variables such that the total objective function (also

known as theperformance index, PI) over all the stages is minimized subject to a

set of constraints on the control and state variables. An OC problem can be stated as

follows [1.55]:

FindXwhich minimizesf (X)=

∑l

i= 1

fi(xi, yi) (1.6)

subject to the constraints

qi(xi, yi)+yi=yi+ 1 , i= 1 , 2 ,... , l

gj(xj) ≤ 0 , j= 1 , 2 ,... , l

hk(yk) ≤ 0 , k= 1 , 2 ,... , l

wherexiis the ith control variable,yithe ith state variable, andfithe contribution

of theith stage to the total objective function;gj,hk, andqi are functions ofxj, yk,

andxiandyi, respectively, andlis the total number of stages. The control and state

variablesxi andyican be vectors in some cases. The following example serves to

illustrate the nature of an optimal control problem.

Example 1.2 A rocket is designed to travel a distance of 12sin a vertically upward

direction [1.39]. The thrust of the rocket can be changed only at the discrete points

located at distances of 0, s, 2 s, 3 s,... , 12 s. If the maximum thrust that can be devel-

oped at pointieither in the positive or negative direction is restricted to a value of

Fi, formulate the problem of minimizing the total time of travel under the following

assumptions:

1.The rocket travels against the gravitational force.

2.The mass of the rocket reduces in proportion to the distance traveled.

3.The air resistance is proportional to the velocity of the rocket.