guidance, flight mechanics and trajectory optimization

performed using the Pontryagin Maximum Principle as in the last section.
The notation used here is the same as in the previous monograph on the
Pontryagin Maximum Principle (Reference 1.8). In order to make use of the
maximum principle in the form given in the reference, Equation (4.36) must
be reduced to a system of first order differential equations. This is
accomplished by making the change of variables.
The new equations of motion are:
.
or & = A & + Ba,. (4.37)
The target set in which the terminal state must lie is given by:
and, the function to be minimized is:
The differential constraints are given by Equation (4.37); (in component
form) they are:
where Ai. J and Bj are elements of A and B, respectively,
Notice that
X,($~=O
the target set 3v XF, tF 1 does not contain the condition
. This condition is unnecessary since Xl (tf) can be nulled
by advancing or delaying the time to start the rendezvous maneuver. If x1
(t ) were constrained to be zero, then the problem would have to be refo$mulated
to allow for coast periods, i.e., the acceleration should be
allowed two values cl,&- , and 0 instead of just a, . In addition, the final
time of the maneuver would no longer be proportional to the fuel used.
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