guidance, flight mechanics and trajectory optimization

II
while the optimal control takes the form
u= -Q;’ G’S;
Note that the optimal control is a function of the estimate of the state
which in turn depends on the observations as indicated in Eq. (2.5.62).
Using Eqs. (2.5.74B) and (2.5.70), the minimum value for the performance
index is
The minimum performance index for the partially observable case falls
somewhere between that for the perfectly observable and that for the
perfectly inobservable system; that is
(2.5.78)
This statement can be shown by considering the case where the initial
state is known (i.e., the variance for the initial state is zero, 6 =0 ).
Since the matrix 5 is the same in all three cases, it follows from
Eq. (205*53A), and the definition of ,& in (2.5.75) and (2.5.76) that
But from the definitions of V and s in (2.5.63B) and (2.5.75)
Thus, integrating with V, =O and substituting into (2.5.79) yields
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