Generalized zero sets of multiparameter polynomials and feedback stabilization

A new theorem is stated and proved, which enables one to find the set of points in the closed complex plane such that for every -dimensional vector of parameters , there exists an -dimensional vector of parameters rendering , where is a given polynomial in depending analytically and continuously on two sets of parameters and , and and are the Cartesian products of the given domains of definition of each of the parameters and , respectively. A numerical example is provided. The new theorem is used to answer the question whether there exists a feedback matrix, with possible constraints on its entries, which stabilizes a linear system with any number of inputs and outputs. If such a matrix exists, a procedure is outlined to find one. A numerical example is provided, which shows that this new method is computationally simpler than previous procedures.