Chemistry Reference
In-Depth Information
p
Im E
|
The vector
p
is orthogonal to Im
E
.
Figure 3.
Proof
Since Im
E
n
, there exists a nonzero vector
p
such that
= R
/
T
ϕ
(
T
)
ϕ
−
1
(
s
)
B
(
s
)
δu
(
s
)
ds
p
·
=
0
,
∀
δu
0
The vector
p
can be associated with the forbidden direction of the dynamics when
varying
u
by a small amount
δu
. This remark is illustrated by Fig. 3 and
this
equation is equivalent to
pϕ
(
T
)
ϕ
−
1
(
s
)
B
(
s
)
=
0
pϕ
(
T
)
ϕ
−
1
(
s
), one obtains
Introducing the adjoint vector
p
such that
p
(
s
)
=
p
∂F
p
(
t
)
=−
∂x
(
x, u
)
with
p
(
T
)
=
p
. This can be rewritten as
p
=−
(
∂H/∂x
), where
H
=
p
·
F
(
x, u
).
The relation
x
=
F
(
x, u
) is also given by
x
=
∂H/∂p
. Since
p
(
t
)
B
(
t
)
=
0 almost
everywhere, one deduces that
∂H/∂u
=
p
(
∂F /∂u
)
=
0.
Note that for a singular control
u
on the interval [0
,T
], we have
p
(
t
)
⊥
Im
E
u
|
[0
,t
]
since
u
(
t
) is singular and
p
(
t
)
=
p
(
t
) (see Fig. 4 for an illustration).
Im E
|
p
(
t
)
Figure 4.
Plot of a trajectory for a
given control parameter. The set Im
E
(
x
0
,t
)
is
schematically represented as a vertical line.
The vector
p
(
t
) is orthogonal to this set.
