Eigensets and invariant sets of switching dynamical systems

Abstract

We consider reachable sets of switching systems, which are families of linear ODE $\dot x(t) \, = \, \, A(t) \, x(t)$ with a function $A(\cdot)$ taking values on a given compact set of $d\times d$ matrices. An eigenset is a compact set $M \ne \{0\}$ that possesses the following property: the closure of the set of points reachable by trajectories $x(\cdot)$ starting from $M$ in time $t$ is equal to $e^{\, \alpha t}M, \, t\ge 0$. This concept introduced recently in the literature generalizes the notion of an eigenvector of a matrix to an arbitrary compact set of matrices. We prove the existence of eigensets, analyse their structure and properties, and find the corresponding ``eigenvalues'' $\alpha$. The relation of eigensets to the stability of the systems, to their Lyapunov exponents, invariant sets, and invariant norms is established. The question of which compact sets can be presented as eigensets of suitable systems is studied. In particular, for $d=2$, we show that every convex $n$-gone for $n=3,4,5$, is en eigenset, while for $n\ge 6$, this is not true.