guidance, flight mechanics and trajectory optimization

#R
where tr denotes the trace of the matrix C(f) ~7 . This last term is
derived from the expected value of the'quantity
E
f(T)
f'T'f+4t
f
f'rLt*At
(2.5.18)
The Dirac delta appearing in the variance expression for F in Eq. (2.5.10)
causes this term to reduce to first order in dt . Substitution of
Eq. (2.5.18) into (2.5.16) and taking the ,limit as At goes to zero
provides the final result
The boundary condition on Rtz,t) is easily developed from the
definition of R given in (2.5.13). Thus,
or alternately
Rx,+) =;rTA;r (2.5.20‘)
Eq. (2.5.19) is similar to that developed in the deterministic case
[ see Equation (2.4.113)] , the only difference being the appearance of
the term $(r;$) . This, however, is a major difference.
Wh;Lle the Bellman equation is a first order partial differential equation
and can be solved in a straightforward manner using the method of
characteristics, this equation is a second order equation of the diffusion
type. As a general rule, diffusion processes are rather difficult to
solve. Fortunately, Eq. (2.5.19) solves rather easily.
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