On the asymptotic behavior of autonomous systems

Abstract

The aim of this paper is to present a new and a complete study concerning the asymptotic behavior of autonomous systems in terms of input–output techniques. We characterize the exponential dichotomy of autonomous systems modeled by C₀-semigroups in terms of admissibility type conditions, pointing out connections between the dichotomic splitting and the properties of the solutions of some associated control systems in certain function spaces. We deduce the structure and the uniqueness of the dichotomy projection and we obtain necessary and sufficient conditions for exponential dichotomy. Next, the main results are applied to the study of the stability and expansiveness of autonomous systems on the half-line. Throughout the paper, we present several examples which motivate the techniques and clarify the relevance of the arguments and the generality of the obtained results.