guidance, flight mechanics and trajectory optimization

Since V is positive definite for t p$, and &' is positive, one half of
the inequality in (2.5.78) is established. To establish the other half,
note that from Eqs, (2.5.33B) and (2i5.77)
Note, also, that the variance functions, v , are different in the two
cases. However, making use of Eq, (2.5.55) for the inobservable case and
Eq. (2.5.80) for the partially observable case reduces (2,.5.81) to
Now, since V in' the partially observable case is less than r/ in the
perfectly inobservable case (i.e., the observations y reduce the variance
in the estimate of ,X ) the inequality
is established.
2.5.2.4 Discussion
In all three cases, perfectly observable, perfectly inobservable and
partially observable, tie form of the optimal control action is the same.
Specifically, the optimal control is a linear function of either the state,
or the expected value of the state, with the proportionally factor being
the same for each case. This is a rather striking similarity, but one
which appears to hold only for the linear - quadratic cost problem.
Note that the performance index, which is to be minimized, decreases
as a quality of the observational data increases. The two limiting cases,
the perfectly observable and perfectly inobservable systems, provide lower
and upper bounds, respectively, for the performance index value which can
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