Lyapunov Type Theorems for Exponential Stability of Linear Skew-Product Three-Parameter Semiflows with Discrete Time

Davor Dragičević,Ciprian Preda

doi:10.3390/axioms9020047

Davor Dragičević, Ciprian Preda

Open Access

https://doi.org/10.3390/axioms9020047

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Abstract

For linear skew-product three-parameter semiflows with discrete time acting on an arbitrary Hilbert space, we obtain a complete characterization of exponential stability in terms of the existence of appropriate Lyapunov functions. As a nontrivial application of our work, we prove that the notion of an exponential stability persists under sufficiently small linear perturbations.

Highlights

The main objective of this paper is to obtain a complete characterization of exponential stability for linear skew-product semiflows with discrete time acting on an arbitrary Hilbert space in terms of the existence of appropriate Lyapunov functions
As a nontrivial application of our work, we prove that the notion of an exponential stability persists under sufficiently small linear perturbations
We use Theorems 1 and 2 to prove that the notion of exponential stability persists under sufficiently small linear perturbations

Summary

Introduction

The main objective of this paper is to obtain a complete characterization of exponential stability for linear skew-product semiflows with discrete time acting on an arbitrary Hilbert space in terms of the existence of appropriate Lyapunov functions. We use this characterization to prove that the notion of an exponential stability persists under sufficiently small linear perturbations. In the context of nonautonomous dynamics, the relationship between exponential dichotomies and the existence of appropriate Lyapunov functions was first considered by Maizel [6] His results were further developed by Coppel [7,8] as well as Muldowney [9]. There has been a renewed interest in this topic

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