Eigensets of switching dynamical systems

TL;DR

This paper investigates eigensets in linear switching dynamical systems, establishing their existence, properties, and the conditions under which certain convex sets can serve as eigensets, extending the eigenvector concept to set-valued dynamics.

Contribution

It introduces the concept of eigensets for switching systems, proves their existence, analyzes their structure, and explores which convex sets can be eigensets.

Findings

Eigensets exist for linear switching systems.

The structure and properties of eigensets are characterized.

Conditions for convex sets to be eigensets are identified.

Abstract

Reachability sets of linear switching dynamical systems (systems of ODE with time-dependent matrices that take values from a given compact set) are analysed. An eigenset is a non-trivial compact set M that possesses the following property: the closure of the set of points reachable by trajectories starting in M in time t is equal to exp(at)M. This concept introduced in a recent paper of E.Viscovini is an analogue of an eigenvector for compact sets of matrices. We prove the existence of eigensets, analyse their structure and properties, and find ``eigenvalues'' a for an arbitrary system. The question which compact sets, in particular, which convex sets and polyhedra, can be presented as eigensets of suitable systems, is studied.