On Lyapunov-Krasovskii Characterizations of Stability Notions for Discrete-Time Systems with Uncertain Time-Varying Time-Delays

In this paper we present Lyapunov characterizations of the global asymptotic stability, the input-to-state stability and the incremental input-to-state stability for discrete-time nonlinear systems with uncertain and time-varying time-delays.We provide necessary and sufficient conditions for the global asymptotic stability and for the input-to-state stability and sufficient conditions for the incremental input-to-state stability, in terms of discrete-time Lyapunov-Krasovskii functionals. In particular, the Lyapunov characterization of the notion of incremental input-to-state stability is motivated by problems concerning the construction of discrete abstractions approximating discrete-time nonlinear systems. Uncertain and time-varying time-delays occur very frequently, for instance, in networked control systems. Examples are included which illustrate the proposed methodologies.