Remarks on nonlinear planar control systems which are linearizable by feedback

The possibility that certain nonlinear systems can be linearized by the nonlinear feedback group has recently attracted a great deal of attention in the literature. At least for mechanical systems, the apparent ubiquity of linearizable systems is often accounted for by the claim that scalar control systems defined on R² are generically linearizable. Since there are several possible interpretations of the term ‘generic’ and since, until now, no rigorous proof of genericity for any use of the term has been offered, I thought it worthwhile to investigate in what sense, if any, this ‘folklore theorem’ is valid. Contrary to the prevailing intuition, linearizable systems fail to be dense in any topology for which evaluation of vector fields at points is a continous operation. This includes, for example, both the weak and strong C^∞-topologies.What, then, are the global properties of the class of linearizable systems? In this note, it is also shown that this class has no interior points in any reasonable, complete topology making feedback operations continuous. In particular, linearizable systems in R² are far from being structurally stable in the weak C^∞-topology. In the other hand, one of the main results presented here is that globally linearizable systems are open in the strong, or Whitney, topology. Thus, globally linearizable planar systems are structurally stable. If the well-known local conditions actually implied global linearizability, this result would be trivial. Recent work of Boothby shows, however, that global linearization is far more subtle. For this reason, in the course of the proof it is necessary to modify the existing global linearization criteria. In particular, a ‘nonvanishing’ criterion for irjectivity of smooth maps on R″, sharpening the so-called ‘ratio condition’, is presented. This result appears to be of independent interest and leads to improvements of the global linearization condition in R″, due to Hunt, Su and Meyer, and of the global non-linear observability conditions in R″, due to Kou, Elliott and Tarn.