Orbit (control theory)


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.

Definition

Let
be a control system, where
belongs to a finite-dimensional manifold and belongs to a control set. Consider the family
and assume that every vector field in is complete.
For every and every real, denote by the flow of at time.
The orbit of the control system through a point is the subset of defined by
;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 is symmetric, 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.

Orbit theorem (Nagano–Sussmann)

Each orbit is an immersed submanifold of.
The tangent space to the orbit
at a point is the linear subspace of spanned by
the vectors where denotes the pushforward of by , belongs to and is a diffeomorphism of of the form with and.
If all the vector fields of the family are analytic, then where is the evaluation at of the Lie algebra generated by with respect to the Lie bracket of vector fields.
Otherwise, the inclusion holds true.

Corollary (Rashevsky–Chow theorem)

If for every and if is connected, then each orbit is equal to the whole manifold.