A note on simple randomly switched linear systems

TL;DR

This paper investigates the long-term behavior of a planar process that switches randomly between two stable linear systems, revealing conditions under which the process can blow up despite individual stability.

Contribution

It introduces a construction of a planar process with switching between Hurwitz matrices and analyzes how switching rates influence stability and blow-up phenomena.

Findings

Stable systems can lead to blow-up with certain switching rates

Connection between tail behavior of invariant measure and process explosion

Switching dynamics can destabilize otherwise stable systems

Abstract

We construct a planar process that switches randomly between the flows of two linear systems built from two Hurwitz matrices (all eigenvalues have negative real parts). The goal here is to study the long time behaviour according to the switching rates. We will see that, even if the two systems are stable, it is possible to obtain a blow up if we choose the switching rates wisely. Finally we will see a connection, between the tail of the invariant measure (when the switching times follow an exponential law) and the existence of a deterministic control that makes the process explode.