guidance, flight mechanics and trajectory optimization

But from (2.4.155B) it follows that the quantity~(x,L()+C~~~(;t,u)c-
has a minimum of u=uopt and that this minimum value is‘lero. IFU
is held fixed at its optimum value which. corresponds to some point(z,2)
{ia +t = u,,t (*,,!I] then this bracketed quantity, considered as a function
of x,t, will have a minimum at the point f, z . Hence,
Z=x^
=0 f=/,n
and substituting this expression into (2.4.157), yields the desired result;
namely,
d'=-
df
The fourth condition follows directly from the original assumption on the
R function needed to develop the Bellman equation. This assumption [see
(2.4.W)] re q uired that the second derivatives of R be bounded; hence,
the first derivatives must be continuous. Thus,
and condition (4) is satisfied. As discussed at the start of this section,
this requirement in the second derivative s is not always satisfied, a point
which will be treated later on.
The conditions of the Maximum Principle as developed from the Bellman
equation and represented in (2.4.152) to (2.4.155) will now be used to
solve the first example problem in Section (2.4.8).
The guidance problem of Section (2.4.8) is represented by the equations
101
ae
(2.4.160)