%0 Journal Article
%T Polynomials in Control Theory Parametrized by Their Roots
%A Baltazar Aguirre-Hern¨¢ndez
%A Jos¨¦ Luis Cisneros-Molina
%A Mart¨ªn-Eduardo Fr¨ªas-Armenta
%J International Journal of Mathematics and Mathematical Sciences
%D 2012
%I Hindawi Publishing Corporation
%R 10.1155/2012/595076
%X The aim of this paper is to introduce the space of roots to study the topological properties of the spaces of polynomials. Instead of identifying a monic complex polynomial with the vector of its coefficients, we identify it with the set of its roots. Vi¨¨te's map gives a homeomorphism between the space of roots and the space of coefficients and it gives an explicit formula to relate both spaces. Using this viewpoint we establish that the space of monic (Schur or Hurwitz) aperiodic polynomials is contractible. Additionally we obtain a Boundary Theorem. 1. Introduction It is well known that for the stability of a linear system it is required that all the roots of the corresponding characteristic polynomial have negative real part in other words, is a Hurwitz (stable) polynomial. There are various approaches to decide if a given polynomial is Hurwitz. Maybe the most popular of such methods is the Routh-Hurwitz criterion. Other important approaches are Lienard-Chipart conditions and the Hermite-Biehler Theorem (see Gantmacher [1], Lancaster and Tismenetsky, [2] and Bhattacharyya et al. [3]). On the other hand, to have the stability of a discrete time linear system it is necessary that all of the roots of the characteristic polynomial are strictly within the unit disc. A polynomial with this property is named Schur polynomial. Maybe Jury¡¯s test is the most studied criterion for checking if a given polynomial is a Schur polynomial [4], but also there exists the corresponding Hermite-Biehler Theorem for Schur polynomials [5] or we can mention as well the Schur stability test [3]. In addition to the stability of polynomials another important problem is the aperiodicity condition, which consists in obtaining from a (continuous or discrete) stable system a response that has no oscillations or has only a finite number of oscillations. Mathematically this requires that all the roots of the characteristic polynomial are distinct and on the negative real axis, for the case of continuous systems; and distinct and in the real interval for the discrete case. Criteria to decide if a system is aperiodic are given for instance in [6¨C12]. An important reference where Hurwitz and Schur stability and aperiodicity are studied is the book of Jury [13]. However, if a continuous or discrete system is modeling a physical phenomenon then it is affected by disturbances. Consequently it is convenient to think that there are uncertainties in the elements of the matrix and then there are uncertainties in the coefficients of the polynomial ; that is, we have a family of polynomials and
%U http://www.hindawi.com/journals/ijmms/2012/595076/