Section: New Results
Local topological linearization of control systems
The article  states the following result: if a system is locally equivalent to a controllable linear system via a bi-continuous transformation –a local homeomorphism in the state-control space– it is also equivalent to this same controllable linear system via a transformation that bears a triangular structure, inducing a diffeomophism on state spaces that is as smooth as the system itself (real analytic for a real analytic system), and a transformation on inputs that bears this smoothness but whose inverse may not, at some singularities. This basically says that topological linearization is almost as stringent as smooth linearization.
This negative answer to the question raised in the last paragraph of section 3.2.3 calls for the following question, which is important for modeling control systems: are there local “qualitative” differences between the behavior of a non-linear system and that of its linear approximation when the latter is controllable? It would also be interesting to know whether, for equivalence between arbitrary systems (not assuming that one of them is linear controllable), the gap between topological and smooth equivalence is still negligible.