guidance, flight mechanics and trajectory optimization

1.0 STATENI3JT OF THE PROBLEN
This monograph will present both the theoretical and computational
aspects of Dynamic Programming. The development of the subject matter in
the text will be similar to the manner in which Dynamic Programming itself
developed. The first step in the presentation will be an explanation of
the basic concepts of Dynamic Programming and how they apply to simple
multi-stage decision processes. This effort will concentrate on the meaning
of the principle of Optimality, optimal value functions, multistage decision
processes and other basic concepts.
After the basic concepts are firmly in mind, the applications of these
techniques to simple problems will be useful in acquiring the insight that
is necessary in order that the concepts may be applied to more complex
problems. The formulation of problems in such a manner that the techniques
of Dynamic Programming can be applied is not always simple and requires
exposure to many different types of applications if this task is to be
mastered. Further, the straightforward Dynamic Programming formulation
is not sufficient to provide answers in some cases. Thus, many problems
require additional techniques in order to reduce computer core storage
requirements or to guarantee a stable solution. The user is constantly
faced with trade-offs in accuracy, core storage requirements, and computation
time. All of these factors require insight that can only be gained from,the
examination of simple problems that specifically illustrate each of these
problems.
Since Dynamic Progrsmmin g is an optimization technique, it is expected
that it is related to Calculus of Variations and Pontryagin's Maximum
Principle. Such is the case. Indeed,.it is possible to derive the Euler-
Lagrange equation of Calculus of Variations as well as the boundary condition
equations from the basic formulation of the concepts of Dynamic Programming.
The solutions to both the problem of Lagrange and the problem of Mayer can
also be,derived from the Dynamic Programming formulation. In practice,
however, the, theoretical application of the concepts of Dynamic Programming
present a different approach to some problems that are not easily formulated
by conventional techniques, and thus provides a powerful theoretical tool
as well as a computational tool for optimization problems.
The fields of stochastic and adaptive optimization theory have recently
shown a new and challenging area of application for Dynamic Programming.
The recent application of the classical methods to this type of problem has
motivated research to apply the concepts of Dynamic Programming with the hope
that insights and interpretations afforded by these concepts will ultimately
prove useful.
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