Approximating evolution operators of linear delay equations: a general framework for the convergence analysis

TL;DR

This paper introduces a unified framework for analyzing the convergence of discretization methods used to approximate the spectra of linear delay equations, aiding stability analysis of nonlinear systems.

Contribution

It provides a general convergence analysis based on fixed-point reformulation, unifies existing pseudospectral methods, and applies to weighted residual methods lacking prior formal analysis.

Findings

Unified convergence framework for delay equations

Application to pseudospectral discretization methods

Formal convergence proof for weighted residual methods

Abstract

We consider the problem of discretizing evolution operators of linear delay equations with the aim of approximating their spectra, which is useful in investigating the stability properties of (nonlinear) equations via the principle of linearized stability. We develop a general convergence analysis based on a reformulation of the operators by means of a fixed-point equation, providing a list of hypotheses related to the regularization properties of the equation and the convergence of the chosen approximation techniques on suitable subspaces. This framework unifies the proofs for some methods based on pseudospectral discretization, which we present here in this new form. To exemplify the generality of the framework, we also apply it to a method of weighted residuals found in the literature, which was previously lacking a formal convergence analysis.