Feedback linearization and driftless systems

The problem of dynamic feedback linearization is recast using the notion of dynamic immersion. We investigate here a “generic” property which holds at every point of a dense open subset, but may fail at some points of interest, such as equilibrium points. Linearizable systems are then systems that can be immersed into linear controllable ones. This setting is used to study the linearization of driftless systems : a geometric sufficient condition in terms of Lie brackets is given ; this condition is shown to be also necessary when the number of inputs equals two. Though non invertible feedbacks are not a priori excluded, it turns out that linearizable driftless systems with two inputs can be linearized using only invertible feedbacks, and can also be put into chained form by (invertible) static feedback. Most of the developments are done within the framework of differential forms and Pfaffian systems.