Let S be a control system and let ”O” be an equilibrium state of S. Considered is the problem of the existence of a controller which drives every state of S asymptotically to O with inputs which themselves approach O. For linear, time-invariant, finite dimensional systems, necessary and sufficient conditions are known [see e.g. W. M. Wonham, Linear multivariable control. A geometric approach. Lect. Notes Econ. Math. Syst. 101, 344 p (1974; Zbl 0314.93007)]. This paper studies the problem for nonlinear systems and gives necessary and sufficient conditions for the existence of a controller in terms of new notions which generalize the usual one of stabilizability and detectability used in the linear case. The controller is not assumed to have any special structure but when one restricts the attention to the case where the original system is analytic or bilinear, simpler conditions are obtained.