Local feedback stabilization and bifurcation control of nonlinear aystems are studied for the case in which the critical linearized system possesses a simple zero eigenvalue. Sufficient conditions are obtained for local stabilizability of the equilibrium point at criticality and for local stabilizability of bifurcated equilibria. These conditions involve assumptions on the controllability of the critical mode for the linearized system. Explicit stabilizing feedback controls are constructed. The Projection Method of analysis of stationary bifurcations is employed. This work complements an earlier study by the same authors (Systems Control Lett. 7 (1986) 11-17) of stabilization and bifurcation control in the (Hopf bifurcation) case of two pure imaginary eigenvalues of the linearized system at criticality.