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 formula_1 be a formula_2 control system, where formula_3 belongs to a finite-dimensional manifold formula_4 and formula_5 belongs to a control set formula_6. Consider the family formula_7 and assume that every vector field in formula_8 is complete. For every formula_9 and every real formula_10, denote by formula_11 the flow of formula_12 at time formula_10. The orbit of the control system formula_1 through a point formula_15 is the subset formula_16 of  formula_4 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 formula_8 is symmetric (i.e., formula_9 if and only if formula_21), then orbits and attainable sets coincide. The hypothesis that every vector field of formula_8 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 formula_16 is an immersed submanifold of formula_4. The  tangent space to the orbit formula_16 at a point formula_26 is the linear subspace of formula_27 spanned by the vectors formula_28 where formula_29 denotes the pushforward of formula_12 by formula_31, formula_12 belongs to formula_8 and formula_31 is a diffeomorphism of formula_4 of the form formula_36 with formula_37 and formula_38. If all the vector fields of the family formula_8 are analytic, then formula_40 where formula_41 is the evaluation at formula_26 of the Lie algebra generated by formula_8 with respect to the Lie bracket of vector fields. Otherwise, the inclusion formula_44 holds true. Corollary (Rashevsky–Chow theorem). If formula_45 for every formula_46  and if formula_4 is connected, then each orbit is equal to the whole manifold formula_4.