The author studies the stabilization of nonlinear control systems on \(\mathbb R^d\times M\) given by \[ \dot x(t)=f(x(t),y(t),u(t)),\qquad \dot y(t)=g(y(t),u(t)), \] where \(x\in\mathbb R^d\) and \(y\in M\), \(M\) is a Riemannian manifold and \(f,g\) are vector fields which are \(C^2\) in \(x,\) Lipschitz in \(y,\) and continuous in \(u.\) The control function \(u(\cdot)\) may be chosen from the set \(\mathcal U:=\{u:\mathbb R\to U\mid u(\cdot)\) measurable}, where \(U\subset\mathbb R^m\) is compact. The author’s interest lies in the stabilization of the \(x\)-component at a singular point \(x^{*},\) i.e., a point where \(f(x^{*},y,u)=0\) for all \((y,u)\in M\times U.\) Such singular situations do typically occur if the control enters in the parameters of an uncontrolled system at a fixed point, for instance, when the restoring force of a nonlinear oscillator is controlled. By using a discrete feedback law, results on the relation between asymptotic controllability and exponential stabilization are developed and an equivalence theorem is derived. The construction of the feedback is obtained by minimizing the Lyapunov exponent of the linearized system. For semilinear systems, asymptotic null controllability and exponential stabilizability by a discrete feedback turn out to be equivalent. For general nonlinear systems the equivalence between uniform exponential controllability and uniform exponential stabilizability is shown. An example illustrates that uniform exponential controllability is in fact a necessary condition for the applicability of linearization techniques. The obtained results can be applied to the stabilization problem of an inverted pendulum for which the suspension point is moved up and down periodically and the period of this motion can be controlled.