Chemistry Reference
In-Depth Information
Expanding this equation up to the second variation order leads to
x
+
δ x
=
F
(
x, u
)
+
F
x
(
x, u
)
δx
+
F
u
(
x, u
)
·
δu
1
2
F
xx
(
δx, δx
)
1
2
F
uu
(
δu, δu
)
+
+
F
xu
(
δx, δu
)
+
where
F
x
and
F
u
denote, respectively, the derivatives of
F
with respect to
x
and
u
. We decompose
δx
into its linear
δ
1
x
and quadratic
δ
2
x
variations with respect
to
u
:
δx
=
δ
1
x
+
δ
2
x
It is then straightforward to show that
δ
1
x
=
A
(
t
)
δ
1
x
+
B
(
t
)
δu
with
δ
1
x
(0)
=
0,
A
(
t
)
=
F
x
(
x, u
),
B
(
t
)
=
F
u
(
x, u
), and
1
2
F
xx
(
δ
1
x, δ
1
x
)
1
2
F
uu
(
δu, δu
)
δ
2
x
=
A
(
t
)
δ
2
x
+
+
F
uu
(
δ
1
x, δu
)
+
with
δ
2
x
(0)
0. From the equation satisfied by
δ
1
x
, one deduces by a direct
integration of the first-order inhomogeneous equation that
=
ϕ
(
T
)
T
0
E
(
x
0
,T
)
ϕ
−
1
(
s
)
B
(
s
)
δu
(
s
)
ds
=
(1)
where
ϕ
is the matrix solution of
ϕ
Id. Note that, following
the same idea as for
E
, a direct and explicit computation of each derivative can
be made.
=
Aϕ
with
ϕ
(0)
=
The parameter
u
is said to be regular on
[0
,T
]
if the image of
E
Definition 2
n
and singular otherwise.
is
R
Geometric Interpretation
An equivalent statement of the fact that the image of the map
E
n
is the
following. A regular control means that, by varying
u
by a small amount
δu
, one
can reach any point in a neighborhood of the final point
x
(
T, x
0
,u
). This notion
can be more precisely defined as follows. We introduce the set
A
of accessible
states in time
T
from
x
0
:
is
R
A
(
x
0
,T
)
=
x
(
T, x
0
,u
)
u
∈
U
