A class of stability regions for which a Kharitonov-like theorem holds

Families of complex polynomials whose coefficients lie within given intervals are discussed. In particular, the problem of determining if all polynomials in a family have the property that all of their roots lie within a given region is discussed. Towards this end, a notion of a Kharitonov region is defined. Roughly speaking, a Kharitonov region is a region in the complex plane with the following property: given any suitable family of polynomials, in order to determine if all polynomials in the family have all of their roots in the region, it suffices to check only the vertex polynomials of the family. The main result is a sufficient condition for a given region to be a Kharitonov region