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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.

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Definition

Let {\ }\dot q=f(q,u) be a \ {\mathcal C}^\infty control system, where {\ q} belongs to a finite-dimensional manifold \ M and \ u belongs to a control set \ U. Consider the family {\mathcal F}=\{f(\cdot,u)\mid u\in U\} and assume that every vector field in {\mathcal F} is complete. For every f\in {\mathcal F} and every real \ t, denote by \ e^{t f} the flow of \ f at time \ t.

The orbit of the control system {\ }\dot q=f(q,u) through a point q_0\in M is the subset {\mathcal O}_{q_0} of \ M defined by

{\mathcal O}_{q_0}=\{e^{t_k f_k}\circ e^{t_{k-1} f_{k-1}}\circ\cdots\circ e^{t_1 f_1}(q_0)\mid k\in\mathbb{N},\ t_1,\dots,t_k\in\mathbb{R},\ f_1,\dots,f_k\in{\mathcal 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 {\mathcal F} is symmetric (i.e., f\in {\mathcal F} if and only if -f\in {\mathcal F}), then orbits and attainable sets coincide.

The hypothesis that every vector field of {\mathcal 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.

Orbit theorem (Nagano-Sussmann)

Each orbit {\mathcal O}_{q_0} is an immersed submanifold of \ M.

The tangent space to the orbit {\mathcal 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 {\mathcal F} and \ P is a diffeomorphism of \ M of the form e^{t_k f_k}\circ \cdots\circ e^{t_1 f_1} with  k\in\mathbb{N},\ t_1,\dots,t_k\in\mathbb{R} and f_1,\dots,f_k\in{\mathcal F}.

If all the vector fields of the family {\mathcal F} are analytic, then \ T_q{\mathcal O}_{q_0}=\mathrm{Lie}_q\,\mathcal{F} where \mathrm{Lie}_q\,\mathcal{F} is the evaluation at \ q of the Lie algebra generated by {\mathcal F} with respect to the Lie bracket of vector fields. Otherwise, the inclusion \mathrm{Lie}_q\,\mathcal{F}\subset T_q{\mathcal O}_{q_0} holds true.

Corollary (Rashevsky-Chow theorem)

If \mathrm{Lie}_q\,\mathcal{F}= T_q M for every \ q\in M then each orbit is equal to the whole manifold \ M.

References

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