Variational Analysis of the Abscissa Mapping for Polynomials∗

The abscissa mapping on the affine variety Mn of monic polynomials of degree n is the mapping that takes a monic polynomial to the maximum of the real parts of its roots. This mapping plays a central role in the stability theory of matrices and dynamical systems. It is well known that the abscissa mapping is continuous on Mn, but not Lipschitz continuous. Furthermore, its natural extension to the linear space Pn of polynomials of degree n or less is not continuous. In our analysis of the abscissa mapping, we use techniques of modern nonsmooth analysis described extensively in Variational Analysis (R. T. Rockafellar and R. J.-B. Wets, Springer-Verlag, Berlin, 1998). Using these tools, we completely characterize the subderivative and the subgradients of the abscissa mapping, and establish that the abscissa mapping is everywhere subdifferentially regular. This regularity permits the application of our results in a broad context through the use of standard chain rules for nonsmooth functions. Our approach is epigraphical, and our key result is that the epigraph of the abscissa map is everywhere Clarke regular.