guidance, flight mechanics and trajectory optimization

2.2.2.2 Variational Problem with Movable Boundary
Dynamic Progr amming will now be applied to the solution of a variational
problem with a movable boundary. Consider the minimization of the functional
subject to the constraint that
L/co, = 0
6
= q-5
(2.2.6)
This problem appears as an example in Ref. 2.2.2 to illustrate the classical
solution of a problem with a movable boundary. Note that this problem dif-
fers in concept from the preceding problem in that the upper limit of
integration is not explicitly specified. However, as would be suspected from
previous problems, the Dynamic Programming approach still involves the divi-
sion of the space into segments and the calculation of the cost of each
transition. The set of end points is located on the line vr = )c~ -5. As
mentioned earlier, there are two ways to perform the Dynamic Programming
calculations in most problems. One method initiates the computation at the
first stage and progresses to the last stage; the second method begins the
computation at the end of the process and progresses to the first stage.
Both methods are equivalent and yield the same answers as shown in an earlier
example. The following example will be partially solved by using the second
method. (The number of computations prohibits the complete manual solution.)
The other problems in this section use the first method.
To begin the Dynamic Programming solution, the space is divided as shown
in the following sketch,
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