Robustly Stable Multivariate Polynomials

We consider stability and robust stability of polynomials with respect to a given arbitrary disjoint decomposition C n = Γ⊎Λ. The polynomial is called stable if it has no zeros in the region of instability Λ and robustly stable if it is stable and remains so under small variations of its coefficients. Inspired by the article Robust stability of multivariate polynomials. Part 1: Small coefficient perturbations by V. L. Kharitonov and J. A. Torres-Muñoz (Multidimens. Systems Signal Process., 10(1):21– 32, 1999), we generalise some of their results to arbitrary stability decompositions and develop some fundamental results on robustly stable polynomials. Among them is a characterisation of robust stability in terms of the stability of several other polynomials, which yields a test for robust stability based on stability tests. Finally, we consider the special situation that the region of instability is a Cartesian product and recover some results for the special situations of linear partial differential resp. difference equation with constant coefficients.