Lyapunov-like solutions to stability problems for discrete-time systems onasymptotically contractive sets

This paper presents new criteria for stability properties of discrete-time non-stationary systems. The criteria are based on the concept of asymptotic ally contractive sets. As a result, general necessary conditions are establ ished for asymptotic stability of the zero equilibrium state, the instantan eous asymptotic stability domain of which can be either time-invariant or t ime-varying and then possibly asymptotically contractive. it is shown that the classical Lyapunov stability conditions including the invariance princi ple by LaSalle cannot be applied to the stability test as soon as the syste m instantaneous domain of asymptotic stability is asymptotically contractiv e. In order to investigate asymptotic stability of the zero state in such a case novel criteria are established. Under the criteria the total first ti me difference of a system Lyapunov function may be non-positive only and st ill can guarantee asymptotic stability of the zero state. The results are i llustrated by examples.