guidance, flight mechanics and trajectory optimization

2.2.2 Applications to the Calculus of Variations
So far, the use of Dynamic Programmin g has been applied to multi-stage
decision processes. The same concepts can, however, be applied to the
solution of continuous variational problems providing the problem is
formulated properly. As might be expected, the formulation involves a
discretizing process. The Dynamic Programming solution will be a discretized
version of the continuous solution. Providing there are no irregularities,
the discretized solution converges to the continuous solution in the limit
as the increment is reduced in size. It is interesting to note that 'the
formal mathematical statement of the concepts already introduced can be
shown to be equivalent to the Euler-Lagrange equation in the Calculus of
Variations in the limit (see Section 2.4). The two classes of problems
that are considered in this section are the problem of Lagrange and the
and the problem of Mayer. The general computational procedure for the
application of Dynamic Programming to each of these problem classes will
be discussed in the following paragraphs. Some illustrative examples are
included in Sections 2.2.2.1, 2.2.2.2, and 2.2.2.3 so that the specific
applications can be seen.
The problem of Lagrange can be stated as finding that function y(x)
such that the functional
is a minimum. That is, of all the functions passing through the points
(x0, Y. ) and h , y ), find that particular one that minimizes J. The
classical trea e men E of this problem is discussed in Reference (2.1).
The approach taken here is to discretize this space in the region of
interest. The following sketch indicates how the space could be divided.
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(2.2.1)