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On Uniform Global Asymptotic Stability Nonlinear Discrete-Time Systems

The paper presents new characterizations of the uniform global asymptotic stability (UGAS) for nonlinear and time-varying discrete-time systems. Under mild assumptions, it is shown that weak zero-state detectability (WZSD) is equivalent to UGAS for globally uniformly stable systems. On the other hand, WZSD is further simplified by employing the notion of reduced limiting systems. Then, a second characterization of UGAS is proposed in terms of the detectability condition of the reduced limiting systems associated with the original system. As a result, we derive a generalized, discrete-time version of the well-known Krasovskii-LaSalle theorem but for time-varying, not necessarily periodic, systems.