guidance, flight mechanics and trajectory optimization

or, as
. -... .-. . .._-_---
to indicate that the minimization is performed through the selection of the
control u and that this control must lie in the set U .
To generalize Eq. (2.4.106), let&t,xti)) denote the minimum value of
the performance index for the solution which starts at the point (t, x(t))
and satisfies the constraint conditions of Eqs. (2.4.103) and (2.4.104);
that is,
Similarly,
(2.4.106)
(2.4.107)
where the solution starts at the point ($t~t , XC($+&)) and satisfies
constraints (2.4.103) and (2.4.104). Now, the Principle of Optimality states
that if a solution which starts at the point (t, x(t)) is at the point
(ttdt,xlttAL)) after the first decision or the first set of decisions
1,
L
all the remaining decisions must be
op in-al decisions if the solution itself
is to be optimal. Putting this statement into mathematical form, leads to
the equation
Q(f) xtt)) = M/N
U(T)C U
(f L ‘t* ttat)
I
R (8 tot, pAttAt))+ .-b,u)nt (2.4.109)
Note the similarity between this equation and Eq. (2.4.21) developed for the
problem of Lagrange. Again, it is to be emphasized that Eq. (2.4.109) is
simply a mathematical statement of how the search procedure for the decision
sequence is to be conducted.
88