Some Results On Stability Of Time-Varying Discrete-Time Systems

This paper discusses the problem of the uniformly (global) asymptotic stability (UAS and UGAS) for general discrete-time nonlinear time-varying systems, under some certain output-dependent conditions in the spirit of the Krasovskii-LaSalle theorem. The celebrated Krasovskii-LaSalle theorem is extended from two directions.One is using the weak zero-state detectability property associated with reduced limiting systems of the system in question to generalize the condition that the maximal invariance set contained in the zero locus of the time-derivative of Lyapunov function is the zero set. This is achieved by the construction of a simple and intuitive criterion using the inequality of the output function and modified detectability conditions. The result can be viewed as an extension of the Krasovskii-LaSalle theorem to general time-varying systems in the version of discrete-time.The other one is using a bounded output-energy condition to relax the assumption in which the time derivative of the Lyapunove function is negative semi-definite. To achieve this, a unified criterion is proposed using a modified detectability condition and certain inequalities. Then, the UAS and UGAS properties of the origin can be guaranteed by employing these two improved conditions related to certain output function for uniformly Lyapunov stable systems. The proposed conditions turn out to be also necessary under some mild assumptions and thus, give a new characterization of UAS (and UGAS).