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The problem of state-feedback stabilizability of discrete-time nonlinearsystems has been considered in this note. Two assertions have been proved.First, if the system is $N$-step controllable to the origin, then there is astate feedback control law for which the trajectory of the closed-loop systemconverges to the origin in $N$ steps. Second, if the system is asymptoticallycontrollable to the origin and satisfies the controllability rank condition atthe origin, then there is a state feedback control law for which the trajectoryof the closed-loop system converges to the origin in finite steps.

Controllability to the origin implies state-feedback stabilizability for discrete-time nonlinear systems