August 2012
Intermediate to advanced
392 pages
10h 50m
English
Content preview from Nonlinear Optimal Control Theory
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320 Nonlinear Optimal Control Theory
Thus,
λ
′
= 0.
Since λ(t
−
1
) = 0, it follows that λ ≡ 0. In particular λ(0
+
) = 0 and (i) of
Theorem 11.4.4 is contradicted.
Exercise 11.4.6. Ver ify that λ(0
+
) + |β| + λ
0
6= 0.
11.5 The Bounded State Problem for I ntegrodifferential
Systems
In this section we specialize the problem in Se ction 11.2. Consider the
problem
min
Z
t
1
0
f
0
(φ(t), u(t), t)dt (11.5.1)
subject to
d
dt
φ
i
(t) = f
i
(t, φ(t)) +
Z
t
0
g
i
(t, s, φ(s), u(s))ds, 1 ≤ i ≤ n (11.5.2)
u(t) ∈ Ω, 0 ≤ t ≤ t
1
(11.5.3)
T (φ(0), φ(t
1
)) = 0 (11 .5.4)
G(t, φ(t)) ≤ 0, 0 ≤ t ≤ t
1
(11.5.5)
The assumptions on T and G in (11.5.4) and (1 1.5.5) remain the sa me as
in Section 11.2 and Ω is a
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