Chemistry Reference
In-Depth Information
Using the relations
⎧
⎨
sin
ϕ
sin
θ
∂
∂y
1
cos
θ
cos
ϕ
∂θ
−
∂
∂ϕ
=
∂
∂y
2
sin
θ
∂θ
=−
⎩
∂
∂y
3
cos
θ
sin
ϕ
∂θ
+
cos
ϕ
sin
θ
∂
∂ϕ
=
one deduces that
sin
ϕ
cos
ϕ
cot
θ
cos
ϕ
sin
ϕ
cot
θ
−
F
1
=
F
2
=
In the coordinates (
θ, ϕ
), the system (17) reads
θ
ϕ
sin
ϕ
cos
ϕ
cot
θ
cos
ϕ
sin
ϕ
cot
θ
−
=
u
1
+
u
2
The following rotation on the control:
v
1
=−
cos
ϕu
1
+
sin
ϕu
2
v
2
=
sin
ϕu
1
+
cos
ϕu
2
which does not modify the cost, leads to
θ
ϕ
1
0
0
cot
θ
=
v
1
+
v
2
(18)
The pseudo-Hamiltonians
H
P
associated to this system are, respectively, given by
1
2
(
v
1
+
v
2
)
H
P
=
v
1
p
θ
+
v
2
p
ϕ
cot
θ
−
for the energy minimization problem and by
H
P
=
v
1
p
θ
+
v
2
p
ϕ
cot
θ
for the time-optimal control. In the first case, the constant
p
0
has been normalized
to
1
/
2 and in the second case, this constant has been substracted in the definition
of
H
P
. The application of the PMP gives the following extremal controls
v
1
=
−
p
θ
R
(19)
p
ϕ
cot
θ
R
v
2
=
p
θ
+
p
ϕ
cot
2
θ
for the time-minimum
problem. The extremal trajectories correspond to the flows of the Hamiltonians
where
R
=
1 for the energy and
R
=
