Abstract and Applied Analysis
Volume 2010 (2010), Article ID 687951, 16 pages
doi:10.1155/2010/687951
Research Article

On Stability of Parametrized Families of Polynomials and Matrices

Handan Akyar, Taner Büyükköroğlu, and Vakıf Dzhafarov

Department of Mathematics, Faculty of Science, Anadolu University, 26470 Eskisehir, Turkey

Received 1 June 2010; Revised 2 August 2010; Accepted 13 September 2010

Academic Editor: Ülle Kotta

Copyright © 2010 Handan Akyar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 A is equivalent to the nonsingularity of the family {A2−2tA+I:A∈A,t∈[−1,1]} if 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.