Strong stability of linear delay-difference equations

TL;DR

This paper extends the stability analysis of linear delay-difference equations to include integral terms, providing new criteria for strong stability and illustrating these results with numerical examples.

Contribution

It generalizes the notion of strong stability to equations with integral delay terms and extends Melvin Criterion for scalar equations.

Findings

Equivalent local and global strong stability conditions

Characterization of stability via total variation of the defining function

Numerical illustrations confirming theoretical results

Abstract

This paper considers linear delay-difference equations, that is, equations relating the state at a given time with its past values over a given bounded interval. After providing a well-posedness result and recalling Hale--Silkowski Criterion for strong stability in the case of equations with finitely many pointwise delays, we propose a generalization of the notion of strong stability to the more general class of linear delay-difference equations with an integral term defined by a matrix-valued measure. Our main result is an extension of Melvin Criterion for the strong stability of scalar equations, showing that local and global strong stability are equivalent, and that they can be characterized in terms of the total variation of the function defining the equation. We also provide numerical illustrations of our main result.