Determining the closest stable polynomial to an unstable one

Abstract

The following problem is considered: given a polynomial with zeros that do not lie on or inside the unit circle, find the closest polynomial with zeros that are all on or inside the unit circle. The measure of closeness used is the Euclidean distance in coefficient space. The direct formulation of this problem leads to a minimization problem with nonlinear constraints, and direct solution is difficult. The problem is approached by considering a related minimization problem with linear constraints. It is then hypothesized that only a finite number of solutions to the linear problem are candidate solutions to the given nonlinear problem. While a general proof of the hypothesis has not been found, numerical examples indicate that it may hold for a large number of cases.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>