guidance, flight mechanics and trajectory optimization

This process continues in the same manner until an end point is reached.
In this case, the end point is a set of points all of which have the same
coordinates for x and y but have many combinations of x and y subject to
the constraint that
After the optimum mass is calculated for all possible terminal points, the
best one is selected. The optimum path is then traced to the initial point
just as was done in previous problems.
Formulation
The equations to be used to calculate the cost of each transition can
be developed from Equations 2.2.10, 2.2.11 and 2.2.12. Since the transitions
from one point to another point in the state space are assumed to be short
in duration, it is assumed that the vehicle's mass is constant during the
transition and that the acceleration in the x and y direction is constant.
This is a reasonable assumption since the state space is divided into many
smaller parts and the mass change is not very significant during a typical
transition from one point to another. Thus, since the mass is practically
constant and the control by the nature of its computation is constant during
a transition, a constant acceleration is a reasonable assumption for a
typical transition.
The laws of constant acceleration motion can now be used for each short
transition. The acceleration thaf. is required in order to force a particle
to position with the.velocity x2 at that position from a position xl with
an initial ve 3 ocity of xl is
Similarly, in the y direction
;r=
2; - if
2(x, 3,)
Now, recalling that the x and y components of thrust are
30
(2.2.13)
(2.2.14)