Switched Linear Systems: Stability and the Convergence of Random Products

© 2011 International PressIn this paper we provide conditions for the stability of discrete time switched linear systems. We accomplish this by calculating the mean and covariance of the set of matrices obtained by using all possible switching sequences. The theory of switched linear systems has received considerable attention in the systems theory literature in the last two decades. However, for discrete time switched systems the literature is much older going back to at least the early 1960's with the publication of the paper of Furstenberg and Kesten in the area of products of random matrices, or, if you like, the random products of matrices. The way that we have approached this problem is to consider the switched linear system as evolving on a partially ordered network that is, in fact, a tree.This allows us to make use of the developments of 50 years of study on random products that exist in the statistics literature

Quantitative ergodicity for some switched dynamical systems

We provide quantitative bounds for the long time behavior of a class of Piecewise Deterministic Markov Processes with state space Rd \times E where E is a finite set. The continuous component evolves according to a smooth vector field that switches at the jump times of the discrete coordinate. The jump rates may depend on the whole position of the process. Under regularity assumptions on the jump rates and stability conditions for the vector fields we provide explicit exponential upper bounds for the convergence to equilibrium in terms of Wasserstein distances. As an example, we obtain convergence results for a stochastic version of the Morris-Lecar model of neurobiology