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

Abstract: The problem of state-feedback stabilizability of discrete-time nonlinear systems 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 a state feedback control law for which the trajectory of the closed-loop system converges to the origin in $N$ steps. Second, if the system is asymptotically controllable to the origin and satisfies the controllability rank condition at the origin, then there is a state feedback control law for which the trajectory of the closed-loop system converges to the origin in finite steps.