August 2012
Intermediate to advanced
392 pages
10h 50m
English
Content preview from Nonlinear Optimal Control Theory
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254 Nonlinear Optimal Control Theory
From (8.2.13) we get that τ
s
, the time required to go from (x
s
, y
s
) to (0, 0),
is g iven by τ
s
= −y
s
. It then follows from (8 .2.17) that
W (t
0
, x
0
) = y
0
+ 2(y
2
0
/2 + x
0
)
1/2
(x
0
, y
0
) ∈ R
−
. (8.2.18)
For points (x
0
, y
0
) on OA, the right-hand side of (8.2.18) equals −y
0
. It then
follows from (8.2.10) that (8.2.18) is valid for all (x
0
, y
0
) in R
−
∪ OA.
Let R
+
denote t he region below Σ. The n
R
+
= {(x, y) : y < −(signum x)(2|x|)
1/2
, −∞ < x < ∞}.
If (x
0
, y
0
) ∈ R
+
, then the optimal control is u(t) = 1 on a n interval [0, t
s
]
and then u(t) = −1. The optimal trajectory is a segment of the parabola
y
2
− 2x = y
2
0
− 2x
0
. Corresponding to u(t) = 1, until the parabola intersects
OB. The motion proceeds with u(t) = −1 along OB to the
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