On the differentiability of feedback linearization and the Hartman-Grobman theorem for control systems

We consider a natural extension of the Hartman-Grobmann theorem to control systems, namely we investigate when is it possible to topologically conjugate the trajectories of a smooth single-input nonlinear control system, near an equilibrium, to those of its linear approximation when the latter is reachable. This turns out to be equivalent to smooth static-feedback linearizability which is highly non-generic. In the positive direction, we state a weak form of the Hartman-Grobman theorem for control systems whose input has a prescribed dynamics