guidance, flight mechanics and trajectory optimization

The combined solution of (2.4.24) and (2.4.25) yieldsR(%,$) which is the
minimum value of the integral starting at the point (y,$. Evaluating R
at the point(z&,y,) provides the solution to the problem.
Two questions arise at this point. First, how are Eqs. (2.2.211) and
(2.4.25) solved; and secondly, once the function/?{%, ) is known, how is
the optimal curvev(x) determined? Both questions ar If interrelated and
can be answered by putting the partial differential equation in (2.lc.24)
in a more usable form.
Note that the minimizationjn Eq. (2.4.24) is a problem in maxima -
minima theory; that is, the slope r/'(,y)is to be selected so that the quantity
![z,y,y'ltj$+fly'
noting that d
is a minimum. Assuming that $
does not depend on 8' , it follows
is differentiable
that
and
or
af aR 0
-jj-/+ ay=
Thus, Eq. (2.4.24) is equivalent to the two equations
(2.4.26)
(2.4.27)
which, when combined, lead to a classical-partial differential equation in
the independent variables % and Y {f/' is eliminated by Eq. (2.4.26) ] and
the dependent variable R(r,y, . This equation can be solved either ana-
lytically or numerically, and then Eq. (2.4.26) used to determj.ne the
optimal decision sequence Y'(Z) for (X0'-z"X/).
2.4.3 An Example Problem
The problem of minimizing the integral
has been shown to be equivalent to solving the partial differential equations
67
(2.4.28)