On Lyapunov-Krasovskii characterizations of stability notions for discrete-time systems with unknown 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 unknown 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. Unknown and time-varying time-delays occur very frequently, for instance, in networked control systems. A numerical example is included which illustrates the proposed methodologies.