Solution Estimates for the Discrete Lyapunov Equation in a Hilbert Space and Applications to Difference Equations

Gil’, Michael

doi:10.3390/axioms8010020

Volume 8, Issue 1

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Axioms 2019, 8(1), 20; https://doi.org/10.3390/axioms8010020

Solution Estimates for the Discrete Lyapunov Equation in a Hilbert Space and Applications to Difference Equations

Michael Gil’

Department of Mathematics, Ben Gurion University of the Negev, P.0. Box 653, Beer-Sheva 84105, Israel

Received: 15 January 2019 / Revised: 1 February 2019 / Accepted: 2 February 2019 / Published: 6 February 2019

(This article belongs to the Special Issue New Trends in Differential and Difference Equations and Applications)

Full-Text | PDF [317 KB, uploaded 27 February 2019]

Abstract

The paper is devoted to the discrete Lyapunov equation

X - A^{*} X A = C

, where A and C are given operators in a Hilbert space

H

and X should be found. We derive norm estimates for solutions of that equation in the case of unstable operator A, as well as refine the previously-published estimates for the equation with a stable operator. By the point estimates, we establish explicit conditions, under which a linear nonautonomous difference equation in

H

is dichotomic. In addition, we suggest a stability test for a class of nonlinear nonautonomous difference equations in

H

. Our results are based on the norm estimates for powers and resolvents of non-self-adjoint operators. View Full-Text

Keywords: discrete Lyapunov equation; difference equations; Hilbert space; dichotomy; exponential stability

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).