Chemistry Reference
In-Depth Information
G
(z
2
)
G
(z
1
)
Figure 23.
Possible optimal synthesis around the origin.
form given by Fig. 23. To answer this question, we use the switching function
.
For
x
(
t
)
0, that is, the vectors
p
(
t
) and
G
(
x
(
t
)) are orthogonal.
Since the direction of
G
is known (
G
is orthoradial), one can deduce the direction
of
p
(
t
). Let
z
1
and
z
2
be two points belonging, respectively, to
C
S
and
S
. The
vectors
G
(
z
) associated to these points are schematically represented in Fig. 23.
Now, we let the states
z
1
and
z
2
go to (0
,
0) and we determine the directions of
the different adjoint states. We recall that the PMP states that
p
is a continuous
function that does not vanish. When
z
1
goes to (0
,
0), one deduces by a continuity
argument that
p
1
is vertical in O. When
z
2
goes to (0
,
0), the limit direction of
p
2
is given by the switch curve
C
S
. To respect the continuity of
p
, one sees that
C
S
has to be tangent to the line
y
∈
C
S
∪
S
,
(
t
)
=
0 in O. Due to the complexity of analytical
calculations, we have checked numerically that this is not the case. The singular
line for
z<
0 is therefore not optimal.
=
trajectories cross
C
, the angle between
the vectors
F
(
x
) and
G
(
x
) changes its sign. New optimal trajectories originate
from this point of intersection and correspond to two new regions of the reachable
set. The optimal synthesis is represented in Fig. 22.
In addition, when the initial
X
and
Y
−
−
Physical Interpretation
From the results obtained in the preceding paragraphs, some qualitative con-
clusions can be made with respect to the dissipation effect on the time optimal
