The Applied Mathematics and Engineering Physics Seminar will feature Dr. Jay Elsinger on Tuesday, April 2 from 12-1 p.m. in IST 1014.
The title of the seminar is: An Elementary Approach to Kharitonov’s Theorem: Polynomials, Perturbations, and a Proof – Oh My!
Abstract:
The continuity of the roots of a monic polynomial on its coefficients (over the complex numbers) can be proved using an elementary real analysis approach using essentially just the Bolzano-Weierstrass Theorem. Other approaches require higher-level techniques from either complex analysis or topology, which are not readily accessible to undergraduates. Fortunately, our elementary approach does apply for nonmonic polynomials that occur in many important areas such as singular perturbation theory, perturbation theory for generalized eigenvalue problems, and, as we will see in this talk, in the stability theory of polynomials – an active area of modern research. Furthermore, the possibility of a change in the degree of the perturbed polynomial from that of the unperturbed polynomial requires a special consideration that does not occur for monic polynomials. Yet, we are still able to provide a short and simple proof on the continuity of the roots of nonmonic polynomials as a function of their coefficients using only elementary results from analysis. Finally, we show how to apply these results to give a new elementary proof of a breakthrough result in the robust stability of polynomials from the late 1970s, the so-called Kharitonov’s Theorem, that originally used more sophisticated techniques which can be circumvented now using our approach. This is joint work with Aaron Welters and Anthony Stefan at Florida Tech.
For more information, please contact Dr. Aaron Bardall.