A Spectral Exponential Stability Criterion for Integral Difference Equations and Delay Differential Equations in various state spaces

TL;DR

This paper extends the spectral exponential stability criterion for integral and delay differential equations to various functional state spaces, including Lebesgue, Borel measurable, and bounded variation spaces.

Contribution

It generalizes the classical stability criterion, showing that root location conditions remain valid across multiple functional state spaces.

Findings

Stability criterion applies to Lebesgue spaces

Criterion valid for Borel measurable functions

Applicable to functions with bounded variation

Abstract

It is well-known that the exponential stability of Integral Difference Equations and Delay Difference Equations, in the usual state space of continuous functions, is equivalent to the location of the roots of its associated characteristic equation strictly in the open left half-plane (see e.g. [16, Chapter 9]). In this paper, we use results from [15, Chapter 4] to show that this characterization still holds for other functional state spaces: Lebesgue spaces, the space of Borel measurable bounded functions, and the space of functions with bounded variation.