Chemistry Reference
In-Depth Information
considering a single-input problem in a plane of the form:
y
z
u
−
y
−
z
y
=
+
γ
−
γz
where the subscript
x
has been omitted for the control parameter. We can then apply,
the theoretical description of the previous paragraph where
F
=
(
−
y, γ
−
γz
)
and
G
z, y
).
We introduce the switching function
=
(
−
=−
p
y
z
+
p
z
y
[9]. In this case, the set
S corresponds to the union of the vertical line
y
=
0 and of the horizontal one with
z
given by
γ
T
2
2(
T
1
−
z
0
=−
γ
)
=−
2(
−
T
2
)
if
/
=
γ
(or equivalently if
T
1
=
/
T
2
). The corresponding singular control
u
s
is
given by
2
yz
0
(
γ
2
2
)
=
−
yγ
(
−
2
γ
)
−
−
u
s
(
y, z
)
(47)
z
0
)
γ
)(
y
2
2(
−
−
−
γz
0
One deduces that the singular control vanishes on the vertical singular line and
that it is admissible, that is,
|
u
s
|≤
2
π
, on the horizontal one if
|
y
|≥|
γ
(
γ
−
2
)
2
γ
)]. For smaller values of
y
, the system cannot follow the hori-
zontal singular arc and a switching curve appears from the point where the admis-
sibility is lost [9]. The optimality of the singular trajectories can be determined
geometrically by using the clock form. It can be shown that the horizontal singular
line is locally optimal and that the vertical one is optimal if
z>z
0
.
We consider the control problems defined by the relaxation parameters
γ
−
1
and
−
1
(expressed in the normalized time unit defined above) of 23.9 and 1.94,
respectively, and
M
0
≈
|
/
[2
π
(2
−
10
−
5
. We compare the optimal control law with
an intuitive one used in NMR. The intuitive solution is composed of a bang pulse
to reach the opposite point of the initial state along the
z
- axis followed by a zero
control, where we let the dissipation act up to the center of the Bloch ball. The
optimal and the intuitive solutions are plotted in Fig. 24. Geometric tools allow
us to show that the optimal control is the concatenation of a bang pulse, followed
successively by a singular control along the horizontal singular line, another bang
pulse and a zero singular control along the vertical singular line. Figure 24 also
displays the switching curve that has been determined numerically by considering
a series of trajectories with
u
2
.
15
×
=+
2
π
originating from the horizontal singular set,
where
ϕ
=
0. The points of the switching curve correspond to the first point of each
