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 179
φ
′
(t)). Eq uation (6.5.9) is the Euler equation. If we assume that φ
′
is piecewise
continuous, then (6.5.9) holds between corners of φ.
We next discuss the transversality condition. If in (6.5.4) we take λ
0
= −1
and use (6.5.1) and (6.5 .6), then (6.5.4) beco mes
(−f
0
∗
(t
0
) + hf
0
∗
z
(t
0
), φ
′
(t
0
)i, −f
0
∗
z
(t
0
), f
0
∗
(t
1
) − hf
0
∗
z
(t
1
), φ
′
(t
1
)i, f
0
∗
z
(t
1
)),
(6.5.10)
where f
0
∗
(t) = f
0
(t, φ(t), φ
′
(t)) and f
0
∗
z
is given by (6.5 .8). The transversality
condition now states that (6.5.10) is orthogonal to B at e(φ).
If we take λ
0
= −1 and use (6.5.1) and (6.5.6), then the statement that
z = u(t) maximize s (6.5.3) over U becomes the following statement. For almost
all t in [t
0
, t
1
] and all z in U
− f
0
(t, φ(t), φ
′
(t))
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