Comments on: "Lyapunov matrices for a class of time delay systems" by V. L. Kharitonov

TL;DR

This paper critiques a previous method for constructing Lyapunov matrices for time delay systems, demonstrating that the core boundary value problem used is ill-posed and does not have a unique solution.

Contribution

It provides a mathematical proof that the boundary value problem in the referenced work is ill-posed, challenging the validity of that approach for Lyapunov matrix construction.

Findings

The boundary conditions in the original problem are linearly dependent.

The boundary value problem does not have a unique solution.

The previous method cannot reliably be used for Lyapunov matrix construction.

Abstract

We prove that an auxiliary two-point boundary value problem presented in V. L. Kharitonov, Lyapunov matrices for a class of time delay systems, Systems & Control Letters 55 (2006) 610-617 has linearly dependent boundary conditions, and consequently a unique solution does not exist. Therefore, the two-point boundary value problem presented therein fails to be a basis for constructing Lyapunov matrices for the class of time delay systems investigated.