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

TL;DR

This paper establishes that controllability to the origin in discrete-time nonlinear systems guarantees the existence of state-feedback controls that stabilize the system in finite steps, linking controllability properties to stabilizability.

Contribution

It proves that N-step controllability and asymptotic controllability with rank condition imply finite-step stabilization via state feedback for discrete nonlinear systems.

Findings

N-step controllability implies finite-time stabilization.

Asymptotic controllability plus rank condition ensures finite-step convergence.

State feedback laws can be designed based on controllability properties.

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.