On stability of parametrized families of polynomials and matrices (Q610875)

Summary: The Schur and Hurwitz stability problems for a parametric polynomial family as well as the Schur stability problem for a compact set of real matrix family are considered. It is established that the Schur stability of a family of real matrices \(\mathcal A\) is equivalent to the nonsingularity of the family \(\{A^{2} - 2tA+I:A\in \mathcal A,\;t\in [-1,1]\}\) if \(\mathcal A\) has at least one stable member. Based on the Bernstein expansion of a multivariable polynomial and extremal properties of a multilinear function, fast algorithms are suggested.