guidance, flight mechanics and trajectory optimization

Note that since there is no feedback loop, the optimal control can be
computed only as a function of time and whatever knowledge is available
concerning the initial state x0 .
Again the state equations are
with F a Gaussian white noise with
2 =Ax+Gu+f (2.5,291
El)-, = 0
&E{pw fw] = CW m-Y)
(2.5.30)
The initial state ,z, is assumed to be a Gaussian random variable with
mean c6 and covariance v, , that is
The performance index is again
(2.5.31.)
(2.5.32)
There are two means available for evaluating the expected value of
the functional J . First, the state equation can be solved to develop
the function relationship between z and the random variables 5 and x6 .
Following this development, the expected value of J can be computed by
using the appropriate density function for f and x0 . A second approach
is to develop the probability density function for X , p(x,t) , given
the densities of X0 and p . This approach is more direct and will be
used here since it leads to the rather simple relationship
from which the .optimal control can be readily determined.
Since the state equation is linear and since 5 and X, are Gaussian,
it follows that the random process Z(C) is also Gaussian*. The mean and
* See Reference (2.5.3) for the demonstration that linear transformation
on Gaussian random processes lead to Gaussian random processes.
119