We expand the class of systems to which back-stepping is applicable, by allowing states that are used as virtual controls to appear in non-invertible maps. Representing these maps as products of C/sup 1/ bijective maps and sign definite C/sup 0/ gains, we develop a recursive design which is robust to multiplicative uncertainties. When the linearization of the system is controllable, our design achieves global asymptotic stability, otherwise it guarantees global practical stability. The designed feedback system possesses desirable inverse optimality properties.