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 z in the closed complex plane such that for every m -dimensional vector of parameters u^{0}\in Q , there exists an n -dimensional vector of parameters \upsilon^{0} \in P rendering F( \upsilon^{0}, u^{0}, z ) = 0 , where F is a given polynomial in z depending analytically and continuously on two sets of parameters \upsilon and u , and Q and P are the Cartesian products of the given domains of definition of each of the parameters u_{i} and \upsilon_{i} , 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.