Resonance and marginal instability of switching systems

TL;DR

This paper investigates the phenomenon of marginal instability in linear switching systems, disproves recent conjectures about resonance conditions, and characterizes the conditions under which trajectories exhibit unbounded growth.

Contribution

It provides a new characterization of marginal instability, disproves conjectures linking resonance to instability, and analyzes growth rates of trajectories in switching systems.

Findings

Disproves conjectures linking resonance to marginal instability.

Provides an algorithmic verification method for marginal instability conditions.

Identifies possible fastest growth rates of trajectories, including sublinear growth.

Abstract

We analyse the so-called Marginal Instability of linear switching systems, both in continuous and discrete time. This is a phenomenon of unboundedness of trajectories when the Lyapunov exponent is zero. We disprove two recent conjectures of Chitour, Mason, and Sigalotti (2012) stating that for generic systems, the resonance is sufficient for marginal instability and for polynomial growth of the trajectories. We provide a characterization of marginal instability under some mild assumptions on the sys- tem. These assumptions can be verified algorithmically and are believed to be generic. Finally, we analyze possible types of fastest asymptotic growth of trajectories. An example of a pair of matrices with sublinear growth is given.