guidance, flight mechanics and trajectory optimization

Equation (2.4.131) governing the evolution of the matrix S is nonlinear
and, hence, difficult to solve. However, the matrix S need not be explicitly
evaluated to determine the optimal solution which from (2.4.130) to (2.4,135)
depends only on the terms SX and z. It will be shown next that these terms
satisfy a linear equation and can be evaluated rather easily.
Let P be the n-dimensional vector
Substitution of this variable into (2.4.130) to (2.4.134) and using the
state equation for x provides
with the boundary conditions
u=
Q,-’ G ‘P
2
p’ = -A; +ZQ,/u
i
G Q;'G;
-Ax- 2
(2.4.135)
(2.4.136)
(2.4.13’7)
(2.4.138)
x = %o AT t=c$ (2.4.139)
Note that the new equations in p and x are linear and that the two-point
boundary problem as represented in Eqs. (2.4.137) to (2.4.140) can be solved
directly (i.e., without iteration). The optimal control is then evaluated
using EYQ. (2.4.136). The method will be illustrated next on the simple
attitude control problem of Section (2.4.8).
95
(2.4.140)