Uniformly exponentially stable approximations for a class of damped systems with unbounded feedbacks

TL;DR

This paper demonstrates that adding a numerical viscosity term to semi-discrete schemes ensures uniform exponential stability for approximations of certain infinite-dimensional systems with unbounded feedbacks, overcoming high-frequency issues.

Contribution

It introduces a modified numerical scheme with viscosity that guarantees uniform exponential stability in the discretization of damped systems with unbounded feedbacks.

Findings

Numerical viscosity stabilizes semi-discrete approximations.

Exponential decay is preserved uniformly across discretizations.

High frequency spurious components are effectively controlled.

Abstract

In this paper we study time semi-discrete approximations of a class of exponentially stable infinite dimensional systems with unbounded feedbacks. It has recently been proved that for time semi-discrete systems, due to high frequency spurious components, the exponential decay property may be lost as the time step tends to zero. We prove that adding a suitable numerical viscosity term in the numerical scheme, one obtains approximations that are uniformly exponentially stable with respect to the discretization parameter