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.
Read more about this topic: Orbit (control Theory)
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