Input-output criteria for the trichotomic behaviors of discrete dynamical systems

Abstract

The aim of this paper is to develop a new and complete study regarding the detection of nonuniform, uniform and exponential trichotomies of discrete dynamical systems in infinite dimensional spaces, by means of control methods of input-output type. We show for the first time that an admissibility of nonuniform nature implies the existence of a nonuniform trichotomy and we deduce the structures of the trichotomy projections. After that, we prove criteria for uniform trichotomy of discrete variational systems. Next, for p,q∈[1,∞] we introduce the notion of uniform admissibility of a (ℓp,ℓq) pair and we obtain complete characterizations for uniform exponential trichotomy. Throughout our study, via illustrative examples, we show that the working hypotheses are minimal and cannot be removed. All the criteria are obtained in the most general framework, in which the base space is an arbitrary set and without any additional assumption on the system coefficients.