Stabilizability and lower spectral radius for linear switched systems with singular matrices
Carl P. Dettmann, Chenmiao Zhang
TL;DR
This work addresses stabilizability for discrete-time linear switched systems containing singular matrices by introducing and analyzing the stabilizability radius $\tilde{\rho}$. It proves a universal lower bound $\tilde{\rho} \ge \frac{\check{\rho}}{m}$ and links $\tilde{\rho}$ to the joint spectral subradius $\check{\rho}$, highlighting cases of equality and implications for zero-stability. The authors provide a detailed 2D treatment for a system with a singular matrix and a rotation-like matrix, deriving an exact expression for $\tilde{\rho}$ via continued fractions and Diophantine approximation, and show that the parameter sets yielding a fixed $\tilde{\rho}$ have zero Hausdorff dimension. Through numerous examples, the paper demonstrates projection-like effects of singular matrices, strategies for constructing optimal switching laws, and the nuanced role of singular-value dynamics in stabilization, while identifying open questions about tighter bounds and finiteness properties.
Abstract
We investigate the stabilizability of linear discrete-time switched systems with singular matrices, focusing on the spectral radius in this context. A new lower bound of the stabilizability radius is proposed, which is applicable to any matrix set. Based on this lower bound, more relationships between the stabilizability radius and joint spectral subradius are established. Detailed analysis of the stabilizability radius of a special kind of two-dimensional switched system, consisting of a singular matrix and a rotation matrix, is presented. The Hausdorff dimensions of the parameter sets such that the stabilizability radius of these systems equals a constant are also presented. Other properties of switched systems with singular matrices are also discussed along with examples.