On Stability of Parametrized Families of Polynomials and Matrices

Abstract

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 $\left\{A^2-2tA+I:A\in \mathcal{A},t\in \left[-1,1\right]\right\}$ 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.