August 2012
Intermediate to advanced
392 pages
10h 50m
English
Content preview from Nonlinear Optimal Control Theory
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Proof of t he Maximum Principle 245
where eµ is a relaxed control defined on [0, 1]. The integrand in the transformed
problem is f
0
(t, x, eµ
s
)w and the terminal set
e
B is given by
e
B = {(s
0
, t
0
, x
0
, w
0
, s
1
, t
1
, x
1
, w
1
): s
0
= 0, s
1
= 1 (7.11.3)
(t
0
, x
0
, t
0
, x
1
) ∈ B w
0
= w
1
= t
1
− t
0
}.
As in Section 2.4, it is readily checked that if (
e
ψ, eµ) = (τ, ξ, ω, eµ) denotes
a relaxed admissible pair for the fixed end time problem and (ψ, µ) a relaxed
admissible pair for the variable end time problem, then there is a one- one
correspondence between the pairs (τ, ξ, ω, eµ) and (ψ, µ) defined by
t = τ (s) ψ(t) = ξ(s) µ
t
= eµ
s
ω(s) = t
1
− t
0
, (7.11.4)
where s and t are related by the one to one mapping (7.11.1). More over, if
the pairs (τ, ξ, ω, eµ) and (ψ, µ) are in corresp ...
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