Stability radii of infinite-dimensional positive systems

We show that for innnite-dimensional discrete-time positive systems the complex and real stability radius coincide. Furthermore, we provide a simple formula for the complex stability radius of positive systems by the associated transfer function. We illustrate our results with an example dealing with a simple type of diierential-diierence equations. 1. Introductory remarks This work is motivated by a paper of Hinrichsen and Son on stability radii of nite-dimensional discrete-time positive systems, see HS]. The stability radius for linear systems, introduced by Hinrichsen and Pritchard, is a measure for the stability robustness of a stable system, see HP]. It is deened as the smallest (in norm) complex or real perturbation which destabilizes the system. In general, the complex and real stability radius diier. It is therefore natural to investigate, for which kinds of systems these two radii coincide. This note is a contribution to this problem and is motivated by the following nite-dimensional result due to Hinrichsen and Son, see HS, p. 13]. Consider the discrete-time system x(t + 1) = Ax(t), t 2 N 0 , where A 2 R nn +. the positive stability radius, respectively. Then we have