guidance, flight mechanics and trajectory optimization

The minimum value of the performance index is given by
(2.5.28)
Two observations concerning the control law of Eq. (2.5.27) can be
made. First the control law in the stochastic case in identical to the
control law for the deterministic case in which the random variable 5 in
Eq. (2.5-g) is set to zero and the criterion of minimizing the expected
value of J is replaced by minimizing J itself. Dreyfus in Reference (2.4.1)
refers to this property as "certainty equivalence" and points out that
it occurs infrequently in stochastic problems. However, a non-linear
example of certainty equivalence is given in Reference (2.5.1). A second
observation is that the.control law is an explicit function of the state,
the actual system output. To implement this law, the state must be
observed at each instant of time, a requirement that can be met only in the
perfectly observable case; that is, the control law could not be
implemented if something less than perfect knowledge of the system output
were available. This point clearly demonstrates that the optimal control
law in a stochastic problem is very much a function of the type of
observational data being collected.
For the treatment of additional stochastic problems in which perfect
observability is assumed, the reader is referred to References (2.l.3),
(2.4.1), (2.5-l) and (2.5.2).
2.5.2.2 Perfectly Inobservable Case
In this case, it is assumed that no knowledge of the output of the
system is available for f p t, . A diagram of this type of controller
is given in Sketch (2.5.2) below.
,f (Disturbing forces)
Sketch (2.5.2)
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