Exponential Stability Criteria for Nonautonomous Difference Systems

The aim of this paper is to characterize the exponential stability of linear systems of difference equations with slowly varying coefficients. Our approach is based on the generalization of the freezing method for difference equations combined with new estimates for the norm of bounded linear operators. The main novelty of this work is that we use estimates for the absolute values of entries of a matrix-valued function, instead of bounds on its eigenvalues. By this method, new explicit stability criteria for linear nonautonomous systems are derived. 1. Introduction In the theory of difference equations it is well known that the placement of eigenvalues in the complex plane of a time-invariant linear system is a necessary and sufficient condition to ensure the stability or exponential stability. However, in time-varying systems, the stability and exponential stability are not characterized by the spectrum of transition matrices (see, e.g., [1, 2]). Desoer [1] illustrated the same instability characteristic of a class of discrete-time-varying systems, but remedied the situation considering bounded and sufficiently slowly varying coefficients. More explicitly, Desoer considered the system in (the Euclidean -dimensional space): where and for all ( denotes the class of -matrices with real elements), and his assumptions are as follows:(a)there is a finite such that (b)for some , (c) is sufficiently small. Under this set of conditions it is proven that system (1) is exponentially stable. Actually, Desoer uses (a) and (b) to generate a bound of the form where and depends on , and but is independent of , and then uses a Lyapunov argument to show that system (1) is exponentially stable if is small enough. Without the restriction on the rate of variation on , the system (1) may have exponentially increasing solutions. Thus, there must be an additional condition on in order to get stability. It is well known that the Lyapunov function method serves as a main technique to reduce a given complicated system into a relatively simpler system, and it provides useful applications to control theory, but finding Lyapunov functions is still a difficult task (see, e.g., [1, 3–6]). By contrast, many methods different from Lyapunov functions have been successfully applied to the stability analysis of discrete-time systems (see, e.g., [1, 2, 5, 7–10]). Recently, Gil and Medina [11, 12] and Medina [13–15] begun the study of stability and stabilizability theory for discrete-time systems by means of new estimates for the powers of matrix-valued and operator-valued functions. In the