guidance, flight mechanics and trajectory optimization
guidance, flight mechanics and trajectory optimization
guidance, flight mechanics and trajectory optimization
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To reduce (2.4.109) to a partial differential equation, one must assume<br />
that all second derivatives of R with respect to t <strong>and</strong> x are bounded; that<br />
is,<br />
Under this assumption,R(z!+df,t(~ tAf))has the series expansion<br />
where T denotes transpose<br />
dx<br />
dt<br />
Substituting (2.4.111) into (2.4.109) along with the values for k from<br />
(2.4.108), provides<br />
In the limit asdt + 0 this expression becomes<br />
(2.4.110)<br />
(2.4.113)<br />
which is a first-order, partial differential equation <strong>and</strong> will be referred to<br />
as the Bellman equation for the Problem of Rolza. The boundary condition<br />
which R(t, x(t)) must satisfy, will be considered next.<br />
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