Sparse Stable Matrices

TL;DR

This paper investigates conditions under which sparse matrix spaces, representing network topologies, can contain stable matrices, providing graph-theoretic criteria for stability without requiring feedback channels.

Contribution

It introduces necessary and sufficient graph-based conditions for the existence of stable matrices within sparse matrix spaces, advancing understanding of network stability.

Findings

Established graph-theoretic criteria for stability in sparse matrices

Proved properties of sparse matrix spaces containing Hurwitz matrices

Identified conditions where feedback channels are or are not necessary for stability

Abstract

In the design of decentralized networked systems, it is useful to know whether a given network topology can sustain stable dynamics. We consider a basic version of this problem here: given a vector space of sparse real matrices, does it contain a stable (Hurwitz) matrix? Said differently, is a feedback channel (corresponding to a non-zero entry) necessary for stabilization or can it be done without. We provide in this paper a set of necessary and a set of sufficient conditions for the existence of stable matrices in a vector space of sparse matrices. We further prove some properties of the set of sparse matrix spaces that contain Hurwitz matrices. The conditions we exhibit are most easily stated in the language of graph theory, which we thus adopt in this paper.