The notion of orbit of a control system used in mathematical control theory is a particular case of the notion of orbit in group theory.
Let
q
˙
=
f
(
q
,
u
)
be a
C
∞
control system, where
q
belongs to a finite-dimensional manifold
M
and
u
belongs to a control set
U
. Consider the family
F
=
{
f
(
⋅
,
u
)
∣
u
∈
U
}
and assume that every vector field in
F
is complete. For every
f
∈
F
and every real
t
, denote by
e
t
f
the flow of
f
at time
t
.
The orbit of the control system
q
˙
=
f
(
q
,
u
)
through a point
q
0
∈
M
is the subset
O
q
0
of
M
defined by
O
q
0
=
{
e
t
k
f
k
∘
e
t
k
−
1
f
k
−
1
∘
⋯
∘
e
t
1
f
1
(
q
0
)
∣
k
∈
N
,
t
1
,
…
,
t
k
∈
R
,
f
1
,
…
,
f
k
∈
F
}
.
Remarks
The difference between orbits and attainable sets is that, whereas for attainable sets only forward-in-time motions are allowed, both forward and backward motions are permitted for orbits. In particular, if the family
F
is symmetric (i.e.,
f
∈
F
if and only if
−
f
∈
F
), then orbits and attainable sets coincide.
The hypothesis that every vector field of
F
is complete simplifies the notations but can be dropped. In this case one has to replace flows of vector fields by local versions of them.
Each orbit
O
q
0
is an immersed submanifold of
M
.
The tangent space to the orbit
O
q
0
at a point
q
is the linear subspace of
T
q
M
spanned by the vectors
P
∗
f
(
q
)
where
P
∗
f
denotes the pushforward of
f
by
P
,
f
belongs to
F
and
P
is a diffeomorphism of
M
of the form
e
t
k
f
k
∘
⋯
∘
e
t
1
f
1
with
k
∈
N
,
t
1
,
…
,
t
k
∈
R
and
f
1
,
…
,
f
k
∈
F
.
If all the vector fields of the family
F
are analytic, then
T
q
O
q
0
=
L
i
e
q
F
where
L
i
e
q
F
is the evaluation at
q
of the Lie algebra generated by
F
with respect to the Lie bracket of vector fields. Otherwise, the inclusion
L
i
e
q
F
⊂
T
q
O
q
0
holds true.
If
L
i
e
q
F
=
T
q
M
for every
q
∈
M
and if
M
is connected, then each orbit is equal to the whole manifold
M
.
