guidance, flight mechanics and trajectory optimization

2.5
DYNAMIC PROGRAMMING AND THE OPTIMIZATION OF STOCR$STIC~S$3Tl34S
2.5.1 Introduction
The previous sections of this report have dealt exclusively with the
optimization of deterministic systems. In this section, some optimization
problems are considered in which the equations describing the system
contain stochastic or random elements. This extension is considered
desirable, if not necessary, since all phenomena occurring in nature are
stochastic. That is, every physical process contains some parameters or
elements which are not known exactly but which are known in some statistical
sense. Fortunately, in many systems, the total effect of these random
parameters on system behavior is negligible and the system can be approxi-
mated by a deterministic model and analyzed using standard procedures. In
other cases, however, the random elements are not negligible and may
dominate those elements which are known precisely. The midcourse correction
problem encountered in lunar and planetary transfer maneuvers is a case in
point.
Due to injection errors at the end of the boost phase of a planetary
transfer, the vehicle's trajectory will differ slightly from the desired
nominal condition, and hence, some correction maneuver will be required.
To make such a maneuver, the trajectory error must be known;
and so radar and optical measurement data are collected. This data will
lead to a precise determination of the trajectory error only if the data
itself are precise. Unfortunately, the measurements and measuring devices
are not perfect. Hence, the midcourse maneuver which is made will not
null the trajectory error. Rather, it will null some estimate of the error,
for example, the most probable value of the error. The determination of when
and how to make these corrections SO that the fuel consumed is a minimum is
a problem of current interest in stochastic optimization theory. Note that
if a deterministic model of the planetary transfer problem were used, the
problem itself would cease to exist.
At the present time, the area of optimal stochastic control is just
beginning to be examined. Thus, there are no standard equations or
standard approaches which can be applied to such systems. In fact, the
literature on the subject contains very few problems which have been solved.
One reason for this limited amount of literature is that the fundamental
equations which are encountered are of the diffusion type; that is, they
are second order partial differential equations, Hence, the method of
characteristics, which is used in the deterministic case and which reduces
the Bellman equation to a set of ordinary differential equation, can not be
applied; rather, the partial differential equations must be utilized
directly.
A second factor contributing to the difficulty in handling stochastic
problems is that the type of feedback being considered must be explicitly
accounted for. This situation is just the opposite of that encountered
in the deterministic case. If the initial state is known along with the
control to be applied in a deterministic system, then all subsequent states
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