9.3 Concept of Suboptimization and Principle of Optimality 549
Figure 9.5 Types of multistage problems: (a)initial value problem; (b)final value problem;
(c)boundary value problem.
3.Boundary value problem.If the values of both the input and output variables
are specified, the problem is called aboundary value problem. The three types
of problems are shown schematically in Fig. 9.5, where the symbol|→is used
to indicate a prescribed state variable.
9.3 Concept of Suboptimization and Principle of Optimality
A dynamic programming problem can be stated as follows.†Findx 1 , x 2 ,... , xn, which
optimizes
f (x 1 , x 2 ,... , xn)=
∑n
i= 1
Ri=
∑n
i= 1
ri(si+ 1 , xi)
andsatisfies the design equations
si=ti(si+ 1 , xi), i= 1 , 2 ,... , n
The dynamic programming makes use of the concept of suboptimization and the prin-
ciple of optimality in solving this problem. The concept of suboptimization and the
principle of optimality will be explained through the following example of an initial
value problem.
†In the subsequent discussion, the design variablesxiand state variablessiare denoted as scalars for
simplicity, although the theory is equally applicable even if they are vectors.
