guidance, flight mechanics and trajectory optimization

For a measure of the fuel consumption (assumed to be a small fraction
of the total mass of the spacecraft), the total velocity increment added
will be used as the seventh variable ~7. This differential equation is
The integral is expressed simply as
ur; =b with NJ7 (0) = 0. 4.13
Lu, =
f 0
‘bo+
Many presentations of the Pontryagin Maximum Principle such as that
given by Kopp (Ref. 4.13) require the introduction of an additional variable
when the payoff function involves the time; here, the payoff function is the
final time itself. The treatment by McIntyre (Ref. 4.12) on the other hand
allows J to contain Qf; this procedure will be followed, though it'is
noted that the adjoint variables and the Hamiltonian can take the same form
for both problems. Thus, seven adjoint variables corresponding to the seven
state variables w, w7 are introduced as p, p7 and the Hamiltonian is
H = pT (Aw tf)+p,b
where the forcing vector is fr= (k”, $5, $,iU, , 0, UJ )
and where the differential equations for the seven variables p, p7 are:
4-u
4.15
P ’ 5 -AT/o 4.16
4 =0
4.17
Since a fundamental motion for -AT is FT (-Q), the solutions can be written
as
where Q. = 0 and F (0) = I
p=FT (4) F (40) P, = FT(-Q)p
p7 = P7 , a constant,
The control variables in this problem are to be chosen so as to
maximize the Hamiltonian along the path. In order to determine these control
conditions, note that H can be written as:
63
0
4.18
4.19