Linear reachability versus global stabilization

In this paper the problem of whether or not "linear reachability" implies global stabilizability of nonlinear control systems is studied. Although it is not hard to imagine that the pointwise reachability does not imply the stabilizability of a nonlinear system, constructing counter examples is not as trivial as one might at first think. In this paper four examples are used to show why, in general, the reachability of (A(x), b(x)) everywhere does not necessarily imply the global stabilizability of the nonlinear system in any conventional sense and what additional hypotheses may be needed.