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 {\ q} belongs to a finite-dimensional manifold \ M and \ u belongs to a control set \ U. Consider the family and assume that every vector field in is complete. For every and every real \ t, denote by \ e^{t f} the flow of \ f at time \ t. The orbit of the control system through a point q_0\in M is the subset of \ M 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. In particular, if the family is symmetric (i.e., if and only if ), 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 \ M. The tangent space to the orbit 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 and \ P is a diffeomorphism of \ M of the form with and. If all the vector fields of the family are analytic, then where is the evaluation at \ q 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 \ q\in M and if \ M is connected, then each orbit is equal to the whole manifold \ M.