On the Lyapunov Matrix of Linear Delay Difference Equations in Continuous Time

TL;DR

This paper introduces the Lyapunov matrix for linear delay difference equations in continuous time, exploring its properties, discontinuities, and methods for construction and approximation in various delay scenarios.

Contribution

It provides new insights into the properties and construction of the Lyapunov matrix for delay difference equations, including cases with single, commensurate, and non-commensurate delays.

Findings

Properties of the Lyapunov matrix are established.

Discontinuities in the derivative are characterized.

Approximation methods for non-commensurate delays are proposed.

Abstract

The fundamental matrix and the delay Lyapunov matrix of linear delay difference equations are introduced. Some properties of the Lyapunov matrix, and the jump discontinuities of its derivative are proven, leading to its construction in the case of single delay or commensurate delays. An approximation is proposed for the non-commensurate case.