The finite inclusions theorem

This paper presents a novel, necessary and sufficient condition for a polynomial to have all its roots in an arbitrary convex region of the complex plane. The condition may be described as a variant of Nyquist´s stability theorem; however, unlike this theorem it only requires knowledge of the polynomial´s value at finitely many points along the region´s boundary. A useful corollary, the finite inclusions theorem (FIT), provides a simple sufficient condition for a family of polynomials to have all its roots in a given convex region. Since FIT only requires knowledge of the family´s value set at finitely many points along the region´s boundary, this corollary provides a new and convenient tool for the analysis and synthesis of robust controllers for parametrically uncertain systems