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.