In mathematics, Krener's theorem is a result attributed to Arthur J. Krener in geometric control theory about the topological properties of attainable sets of finite-dimensional control systems. It states that any attainable set of a bracket-generating system has nonempty interior or, equivalently, that any attainable set has nonempty interior in the topology of the corresponding orbit. Heuristically, Krener's theorem prohibits attainable sets from being hairy.
Theorem.
Let 
formula_1 
be a smooth control system, where 
formula_2 
belongs to a finite-dimensional manifold formula_3 and formula_4 belongs to a control set formula_5. Consider the family of vector fields formula_6.
Let formula_7 be the Lie algebra generated by formula_8 with respect to the Lie bracket of vector fields. 
Given formula_9, if the vector space formula_10 is equal to formula_11,
then formula_12 belongs to the closure of the interior of the attainable set from formula_12.
Remarks and consequences.
Even if formula_14 is different from formula_11,
the attainable set from formula_12 has nonempty interior in the orbit topology,
as it follows from Krener's theorem applied to the control system restricted to the orbit through formula_12.
When all the vector fields in formula_18 are analytic, formula_19 if and only if formula_12 belongs to the closure of the interior of the attainable set from formula_12. This is a consequence of Krener's theorem and of the orbit theorem.
As a corollary of Krener's theorem one can prove that if the system is bracket-generating and if the attainable set from formula_9 is dense in formula_3, then the attainable set from formula_12
is actually equal to formula_3.