# Stabilising the Metzler matrices with applications to dynamical systems

## TL;DR

This paper develops efficient algorithms to find the closest stable Metzler matrices to unstable ones, using spectral methods, with applications in dynamical systems and large-scale matrices up to dimension 2000.

## Contribution

It introduces a generalized spectral optimization approach for stabilizing Metzler matrices, extending existing methods to larger matrices and diverse applications.

## Key findings

- Explicit solutions and algorithms for closest stable/unstable Metzler matrices.
- Demonstrated efficiency on matrices up to dimension 2000.
- Applications to dynamical systems and switching systems.

## Abstract

Metzler matrices play a crucial role in positive linear dynamical systems. Finding the closest stable Metzler matrix to an unstable one (and vice versa) is an important issue with many applications. The stability considered here is in the sense of Hurwitz, and the distance between matrices is measured in $l_\infty,\ l_1$, and in the max norms. We provide either explicit solutions or efficient algorithms for obtaining the closest (un)stable matrix. The procedure for finding the closest stable Metzler matrix is based on the recently introduced selective greedy spectral method for optimizing the Perron eigenvalue. Originally intended for non-negative matrices, here is generalized to Metzler matrices. The efficiency of the new algorithms is demonstrated in examples and by numerical experiments in the dimension of up to 2000. Applications to dynamical systems, linear switching systems, and sign-matrices are considered.