guidance, flight mechanics and trajectory optimization
guidance, flight mechanics and trajectory optimization
guidance, flight mechanics and trajectory optimization
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2.4.10 Ljnear Problem with Quadratic Cost<br />
To illustrate the method of solution by means of the Bellman partial<br />
differential equation, consider the following linear problem. Let the system<br />
state be governed by<br />
or in the vector notation<br />
i = A(f) 3: +G(t)u<br />
where A is an n x n matrix <strong>and</strong> G is an n x r matrix. The initial state is<br />
specified, while the terminal state must satisfy the m constraint conditions<br />
i=ty<br />
which can also be written as<br />
?Cij ,Z (1 )-a!=0<br />
; =I .I f i i ( . = /, m<br />
(2.4.1216)<br />
(2.4.121B)<br />
(2.4.122A)<br />
Cr, -d =O at f= tr (2.4.122B)<br />
where C is an m x n constant matrix <strong>and</strong> d is an m-dimensional constant vector.<br />
The problem is to select the control u so that the integral<br />
(2.4.123)<br />
with Q a n x n synn$,ric matrix with elements 9;" <strong>and</strong> Q a r x r symmetric<br />
matrix g th element fir9 . It is required that Q 'se a posl -2. lve, definite<br />
matrix (i.e.,urQIU 1s always greater than zero 1 or any control u not equal<br />
to zero). Furthermore, the admissible control set U is the entire r-dimensional<br />
control space; or in other words, no constraints are imposed on the components<br />
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