August 2012
Intermediate to advanced
392 pages
10h 50m
English
Content preview from Nonlinear Optimal Control Theory
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The Maximum Principle and Some of Its Applications 169
Remark 6.3.25. If (dt
0
, dx
0
, dt
1
, dx
1
) denotes an arbitrary tangent vector to
B at e(ψ), then the transversality condition says that the vector in (6.3.24) is
orthogo nal to (dt
0
, dx
0
, dt
1
, dx
1
). Hence the inner product of (6.3.24) with an
arbitrar y tangent vector (dt
0
, dx
0
, dt
1
, dx
1
) is zero. Thus,
[ − λ
0
g
t
0
+ H
r
(π(t
0
)]dt
0
+ h−λ
0
g
x
0
− λ(t
0
), dx
0
i (6.3.26)
+ [−λ
0
g
t
1
− H
r
(π(t
1
)]dt
1
+ h−λ
0
g
x
1
+ λ(t
1
), dx
1
i = 0,
for all tange nt vectors (dt
0
, dx
0
, dt
1
, dx
1
), where the partials of g are evaluated
at e(ψ).
Since B has dimension r, the vector space of tangent vectors to B at e(ψ)
is r-dimensional. We therefore ...
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