Chemistry Reference
In-Depth Information
x
(
T,x
0
, u
1
)
x
(
T,x
0
, u
2
)
Figure 2.
Schematic representation
of the accessibility set
A
(
x
0
,T
) in gray.
The control fields
u
1
and
u
2
are respec-
tively singular, and regular.
x
0
Let
x
1
=
x
(
T
) be a point reached in time
T
from
x
0
by a regular trajectory. Then
the accessibility set
A
(
x
0
,T
) is a neighborhood of the point
x
1
, which implies
that the system is locally controllable around
x
1
.If
x
(
T
)
∂A
(
x
0
,T
) (i.e., the
boundary of
A
), then
u
is said to be singular on [0
,T
]. This case is schematically
illustrated in Fig. 2 for a regular and singular control field. The most interesting
case is therefore the singular one and this case can be viewed as a first step to-
ward the maximum principle. Indeed, a key observation for this principle is that
if the trajectory (
x, u
) is optimal, then the point
x
(
T, x
0
,u
) must belong to the
boundary of the accessibility set. This assertion can be understood in the time-
minimal case, where the goal is to reach the target state
x
1
in minimum time.
It is clear that if
x
1
=
∈
x
(
x
0
,T,u
) belongs to the interior of
A
(
x
0
,T
), then one
can find a smaller time
T
<T
and a point
x
(
x
0
,T
,u
)of
∂A
(
x
0
,T
), such that
x
1
=
x
(
x
0
,T
,u
) (one can draw the accessibility set at different times to be con-
vinced of this point). This result finally implies that
u
is not optimal for this control
problem.
Computation of Singular Controls
We assume that Im
E
n
and the corresponding control
u
is singular.
= R
/
n
Theorem 1
If
u
is singular, then there exists
p
(
t
)
∈ R
\{
0
}
, such that if
H
(
x, p, u
)
=
p
·
F
(
x, u
)
, the triplet
(
x, p, u
)
satisfies almost everywhere
⎧
⎨
∂H
∂p
x
=
∂H
∂x
(2)
p
=−
⎩
∂H
∂u
=
0
