Asymptotic controllability of discrete-time systems with disturbances

This paper demonstrates that the existence of a smooth control Lyapunov function for discrete-time systems with disturbances is equivalent to the system is uniformly globally asymptotic controllability with respect to a closed set. It is obtained that this control Lyapunov function may be used to construct a feedback law to stabilize the closed-loop system. In addition, it is shown that for periodic discrete systems, the resulted control Lyapunov functions are also time periodic.