August 2012
Intermediate to advanced
392 pages
10h 50m
English
Content preview from Nonlinear Optimal Control Theory
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Systems Governed by Integrodifferential S ystems 293
a row vector. Now the las t inequality c an be rewritten as
λ
0
g
′
(φ
0
(t
1
))q(t
1
) + λW
′
(φ
0
(t
1
))q(t
1
)
+ λ
Z
t
1
0
M(φ
0
(t), (ν − ν
0
)
t
, t)dt +
Z
t
1
0
ψ(t)q(t)dt ≥ 0. (9.7.10)
Recalling the definition of q(t) (9.7.9) we can rerite (9.7.10) as
Z
t
1
0
[λ
0
g
′
(φ
0
(t
1
)) + λW
′
(φ
0
(t
1
))]L(t
1
, φ
0
(t), ν
t
, t)dt
+
Z
t
1
0
Z
t
1
t
ψ(s)L(s, φ
0
(t), ν
t
, t)ds
dt + λ
Z
t
1
0
M(φ
0
(t), ν
t
, t)dt
≥
Z
t
1
0
λ
0
g
′
(φ
0
(t
1
)) + λW
′
(φ
0
(t
1
))
L(t
1
, φ
0
(t), ν
0t
, t)dt
+
Z
t
1
0
Z
t
1
t
ψ(s)L(s, φ
0
(t), ν
0t
, t)ds
dt + λ
Z
t
1
0
M(φ
0
(t), ν
0t
, t)dt.
(9.7.11)
We use (9.7.11) to state our theore m next.
Theorem 9.7.2. Under Assumption 9.7.1 let (φ
0
, ν
0
) be optimal for the prob-
lem of minimizing the cost (9.7.4) under the conditions (9.7.1) to (9.7.3). Then
the following conditions are met:
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