576 Dynamic Programming
In Eqs. (9.37) to (9.39),θoryiis a continuous variable. However, for simplicity,
we treatθ oryias a discrete variable. Hence for each value ofi,we find a set of
discrete values thatθoryican assume and find the value offi∗(θ ) or each discretef
value ofθoryi. Thusfi∗(θ ) willbe tabulated for only those discrete values thatθcan
take. At the final stage, we find the values off 0 ∗(α) andy 1 ∗. Oncey 1 ∗is known, the
optimal values ofy 2 , y 3 ,... , yncan easily be found without any difficulty, as outlined
in the previous sections.
It can be seen that the solution of a continuous decision problem by dynamic
programming involves the determination of a whole family of extremal trajectories as
we move frombtowarda. In the last step we find the particular extremal trajectory that
passes through both points (a,α) and (b,β). This process is illustrated in Fig. 9.16. In
this figure,fi∗(θ ) s found by knowing which of the extremal trajectories that terminatei
atxi+ 1 pass through the point (xi,. If this procedure is followed, the solution of aθ)
continuous decision problem poses no additional difficulties. Although the simplest type
of continuous decision problem is considered in this section, the same procedure can be
adopted to solve any general continuous decision problem involving the determination
of several functions,y 1 (x), y 2 (x),... , yN(x) subjecttomconstraints (m < N) in the
form of differential equations [9.3].9.10 Additional Applications
Dynamic programming has been applied to solve several types of engineering problems.
Some representative applications are given in this section.9.10.1 Design of Continuous Beams
Consider a continuous beam that rests onnrigid supports and carries a set of pre-
scribed loadsP 1 , P 2 ,... , Pnas shown in Fig. 9.17 [9.11]. The locations of the supports
are assumed to be known and the simple plastic theory of beams is assumed toFigure 9.16 Solution of a continuous dynamic programming problem.