Segment Lemma and its applications

In this thesis, segment lemma and its applications (stability of segment polynomials) are discussed. The stability problem of polynomial segments connecting two stable polynomials has many applications. According to the Edge Theorem, the necessary and sufficient condition for a polynomial polytope to be stable is that the polynomial segments joining the extreme points of the polytope are stable. From these polynomial segments, an edge having a root at the stability boundary is determined by the segment lemma. The families of real and complex coefficient polynomials were examined by Hurwitz segment lemma. Schur segment lemmas are discussed and illustrated by examples. The stability of convex combinations of two stable monic polynomials with the help of Hurwitz matrix is investigated. Sturm sequence is used for the family of real and complex coefficient stable polynomials and an algorithm for stability of polynomials with complex coefficients is given. The Möbius transformation is discussed, with the help of this transformation Schur stable polynomial segment \left[p_1\left(z\right),p_2\left(z\right)\right] is transformed into Hurwitz stable polynomial segment \left[{\widetilde{p}}_1\left(z\right),{\widetilde{p}}_2\left(z\right)\right]. The polynomials with real coefficients in the sector stability problem have been reduced to complex coefficient polynomials by the transformation z=se^{\pmj(\frac{\pi}{2}-\phi)}. Robust stable polynomial polytopes are constructed. Keywords: Polynomial Polytope, Hurwitz and Schur Segment Lemma, Convex Combination, Möbius Transformation, Sector Stability