Chemistry Reference
In-Depth Information
1
0.5
0
−0.5
−1
−1
−0.5
0
0.5
1
y
Figure 17.
2
Division of
R
by the sets
C
(in solid line) and
S
(in dashed and dot-dashed lines)
for
γ
−
=
0. The dashed and dot-dashed lines represent, respectively, antiturnpike and turnpike arcs
(see text). The exterior circle in the solid line corresponds to the limit of the Bloch ball in the plane
(
y, z
). The small open circle indicates the position of the fixed-point of the free dynamics. Numerical
values are taken to be
/
=
1,
γ
−
=−
0
.
2 and
γ
+
=
0
.
4.
the field is on, which are defined by
F
0. Since
F
and
G
are parallel, the
fixed points belong to
C
. Inversely, one can ask if every point of
C
corresponds to
a limit point of the dynamics. The answer is positive for a real nonbounded control
u
since the limits can be written
y
+
uG
=
uγ
−
γ
+
+
−
=
u
2
γ
z
=
−
u
2
/
γ
+
+
The set
C
and
S
are represented in Fig. 17 for particular values of the dissipative
parameters.
On the complementary of
C
, we can define the clock form that will be used in
the construction of the optimal synthesis.
The Clock Form
In this section, we derive the expression of the clock form is denoted
α
[8]. By
definition, the clock form is a 1-form that fulfills the following conditions:
α
(
F
)
=
1
α
(
G
)
=
0
