Chenmiao Zhang

Stabilizing switched systems

Switched systems are widely used across various fields. The linear discrete time switched system consists of a collection of linear subsystems and a switching rule that determines which subsystem is active at each time step. A fundamental question—whether a switched system is stabilizable, meaning that one can manually choose a switching rule for any initial state so that the trajectory converges to zero—remains challenging despite the linearity of the subsystems. To address this, several spectral radii have been introduced in the literature to better characterize stabilizability.

In this talk, we present a new lower bound for the stabilizability radius that applies to arbitrary matrix sets, revealing deeper connections between the stabilizability radius and the joint spectral subradius. We then focus on two-dimensional systems consisting of two matrices: one singular and one with complex eigenvalues. For this special case, we provide an exact formula for the stabilizability radius using knowledge from Diophantine approximation and analyze parameter sets with constant stabilizability radius using Hausdorff dimension.