On the stability radii of continuous-time infinite Markov jump linear systems

In this paper we introduce the subject of stability radii for continuous-time infinite Markov jump linear systems (MJLS) with respect to unstructured perturbations. By means of the small-gain approach, a lower bound for the complex radius is derived along with a linear matrix inequality (LMI) optimization method which is new in this context. In this regard, we propose an algorithm to solve the optimization problem, based on a bisectional procedure, which is tailored in such a way that avoids the issue of scaling optimization. In addition, an easily computable upper bound for the real and complex stability radii is devised, with the aid of a spectral characterization of the problem. This seems to be a novel approach to the problem of robust stability, even when restricted to the finite case, which in turn allows us to obtain explicit formulas for the stability radii of two-mode scalar MJLS. We also introduce a connection between stability radii and a certain margin of stability with respect to perturbations on the transition rates of the Markov process. The applicability of the main results is illustrated with some numerical examples.