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.
control system, where
belongs to a finite-dimensional manifold
belongs to a control set
and assume that every vector field in
is complete.
and every real
, denote by
The orbit of the control system
through a point
is the subset
defined by 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.
), then orbits and attainable sets coincide.
The hypothesis that every vector field of
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.
is an immersed submanifold of
The tangent space to the orbit
at a point
is the linear subspace of
spanned by the vectors
denotes the pushforward of
of the form
If all the vector fields of the family
are analytic, then
of the Lie algebra generated by
with respect to the Lie bracket of vector fields.
Otherwise, the inclusion
holds true.
is connected, then each orbit is equal to the whole manifold