A standard finite dimensional nonlinear control system is considered, along with a state constraint set S and a target set Sigma. It is proven that open loop S-constrained controllability to Sigma implies closed loop S-constrained controllability to the closed delta-neighborhood of Sigma, for any specified delta > 0. When the S-constrained minimum time function to Sigma satisfies a local continuity condition, conclusions on closed loop S-constrained stabilizability ensue. The (necessarily discontinuous) feedback laws in question are implemented in the sample-and-hold sense and possess a robustness property with respect to state measurement errors. The feedback constructions involve the quadratic infimal convolution of a control Lyapunov function with respect to a certain modi. cation of the original dynamics. The modified dynamics in effect provide for constraint removal, while the convolution operation provides a useful semiconcavity property.