guidance, flight mechanics and trajectory optimization

of the p and $ equations (i.e., Eqs. (2.5.106A) and (2.5.111) with the
initial condition of Eq. (2.5.112B), the terminal condition of Eq. (2.5.106~)
and with A selected so that Ptt,) satisfies the terminal variance
constraint which is imposed.
In.most cases, the solution will have to be achieved iteratively.
Thus, the process might proceed as follows:
(1) Guess the diagonal matrix A . As has been noted, the number of
independent diagonal elements (i.e., the number of different
quantities that can be guessed) is equal to the number of terminal
constraints imposed. For example, if Eq. (2.5.87A) is used,
then only one constraint is imposed and all the diagonal elements
of -A are equal to some number, say A-. This number would be
guessed to start the iteration.
(2) Integrate the equation for Zi backwards in time with s (&+,) = n
C i.e., integrate Eq. (2.5.106A) 1 .
(3) Set p&J = ;,;zand integrate the P equation forward from
*, to t, [i.e., Eq. (2.5.111) I.
(4) Test P 14)to see if the specified terminal constraints are
satisfied.
(5) ze;hT2;onstraints are not satisfied, adjust h and go back to
.
Since the terminal constraints are inequality constraints [see
Eq. (2.5.87)] , this iteration scheme will not lead to a unique solution.
However, it can be shown, using standard methods from the Calculus of
Variations, that A must be a negative semi-definite matrix, with the
diagonal elements all less than or equal to zero. This condition
suggests that the iteration loop above should start with the condition
nzo; furthermore, it generally allows for a unique solution to the
iteration problem.
Summarizing the results for the perfectly observable case, the optimal
feedback control is given by Eq, (2.5.107) where the matrix s is determined
from Eq. (2.5.106A). The Lagrange multiplier matrix A is selected so that
the simultaneous solution of Eq. (2.5.106A) and (2.5.111) lead to a control
which satisfies the specified terminal constraints.
2.5.3.2 Perfectly Inobservable Case
The treatment of the, perfectly inobservable case parallels that given
in Section (2.5.2.2) where no terminal conditions were imposed. Again,
the problem is to minimize the performance index
140
(2.5.114)