guidance, flight mechanics and trajectory optimization

2.2.2.3 Simple Guidance Problem
As an example of the application of Dynamic Programming to a problem of
the Mayer type, a simple guidance problem will be examined. Consider a
throttlable vehicle with some initial condition state vector X(0) where
X(0, = (2.2.7)
and some initial mass 111 . It is desired to guide the vehicle to some terminal
point x(f), Y(f) subjece to the constraint that its terminal velocity vector
is a certain magnitude, i.e.
where t f is not explicitly specified.
(2.2.8)
Further, it is desired to minimize the amount of propellant that is used
in order to acquire these terminal conditions (this problem is equivalent
to maximizing the burnout mass). In order to simplify the problem, a flat
earth will be assumed and the vehicle being considered will be restricted
to two control variables U and U2. Ul is a throttle setting whose range is
0 5 u15 1. This variab I e applies a thrust to the vehicle equal to
(2.2.9)
where T is the maximum thrust available. U2 is the control variable that
governs m%e direction of thrust. This variable is defined as the angle between
the thrust vector and the horizontal. The following sketch shows the geometry
of these parameters.
27