guidance, flight mechanics and trajectory optimization

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Performing the minimization indicated in (2.5.19)
derivative with respect to u to zero) provides
which can be rewritten as
Substituting this expression in (2.5.19) yields
It can be shown that (2.5.22) has a solution of the form
i.e., setting the
(2.5.21)
(2.5.22)
(2.5.23)
where S(t) is an fixn time dependent symmetric matrix and ,&ft) is
a time varying scalar. This expression will satisfy the boundary condition
of Eq. (2.5.20) provided
set,) =A
pczy = 0
Also, by substituting Eq. (2.5.23).into (2.5.22), it follows that the
proposed R function will satisfy Eq. (2.5.22) if
(2.5.24)
.i? +Q,-SGQ;IGTS+SA +A% = 0 (2.5.25)
p’ + h(C$) =o (2.5.26)
Collecting results, the solution is achieved by integrating Eqs.
(2.5.25) and (2.5.26) backwards from f, to to and using the boundary
conditions in (2.5.24). From (2.5.21) and (2.5.23), the optimal control
action is then determined from.
Ll= - Q2-‘GTSx (2.5.27)
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