On an inverse tridiagonal eigenvalue problem and its application to synchronization of network motion

TL;DR

This paper addresses an inverse eigenvalue problem for tridiagonal Laplacian matrices and explores its application in ensuring the asymptotic stability of synchronous network motion, highlighting limitations under symmetry constraints.

Contribution

It introduces a method to construct tridiagonal Laplacian matrices with prescribed eigenvalues and applies this to analyze network synchronization stability.

Findings

Constructed specific tridiagonal Laplacian matrices with desired eigenvalues.

Established conditions for asymptotic stability of synchronous orbit.

Identified limitations when imposing symmetry on the Laplacian matrix.

Abstract

In this work, motivated by the study of stability of the synchronous orbit of a network with tridiagonal Laplacian matrix, we first solve an inverse eigenvalue problem which builds a tridiagonal Laplacian matrix with eigenvalues $\lambda_1=0<\lambda_2<\cdots <\lambda_N$ and null-vector $\boldsymbol{e} = \begin{bmatrix} 1 \\ \vdots \\ 1 \end{bmatrix}$. Then, we show how this result can be used to guarantee -- if possible -- that a synchronous orbit of a connected tridiagonal network associated to the matrix $L$ above is asymptotically stable, in the sense of having an associated negative Master Stability Function (MSF). We further show that there are limitations when we also impose symmetry for $L$.