Stability radii for some propagation models

The simple idea of Hinrichsen & Pritchard (1986a,b), who defined the structured stability radius, has proved to be unexpectedly fruitful, generating a large amount of work and making interesting connections (Hinrichsen & Pritchard 1988, 1989; Hinrichsen & Motscha 1988; Hinrichsen et al. 1989; Hinrichsen & Son 1991; Pritchard & Townley 1989, 1990; Rasvan 1993). In this paper we shall discuss stability radii for a class of applications concerning systems with propagation. Such systems are described by simple lossless telegraph equations and include the practical cases of physical systems containing lossless steam, gas, or water pipes or electrical circuits containing lossless LC transmission lines. We shall state in a natural way the problem of a structured stability radius for such models. To estimate such radii we shall use a frequency-domain approach of Popov (1973) and Yakubovich (1978), as suggested by the known formulae of Hinrichsen & Pritchard. Our specific models lead to a general differential difference equation coupled with a diference equation.lt could be possible to express such systems as abstract evolution equations (see for instance the paper of Weiss (1994) on regular systems) and then use the general results by Pritchard & Townley (1990). We do not intend here to compete with such approach. Our aim here is to point out some examples of naturally occuring, practical problems where the stability radius has to be considered and then to draw attention to the possibility of approaching the problem with •methods similar to the ones used in absolute stability. It is hoped that existence of several approaches to the problem will stimulate further research concerning comparisons between them, both from the view-point of the tightness of estimates as well as from the view-point of simplicity, directness, and accessibility.