The abscissa map takes a polynomial to the maximum of the real parts of its roots. This map plays an important role in control theory because it describes the asymptotic stability of dynamical systems. In many applications one can determine the parameters for an optimally stable system by minimizing the abscissa mapping over a parametrized family of polynomials.

Using the classical Routh, Hurwitz, and Liénard-Chipart polynomial stability criteria, we reformulate the problem of optimizing the abscissa mapping into a variety of constrained polynomial optimization problems. We then study these problems from an epigraphical viewpoint. This perspective allows us to compute variational properties of the abscissa map and to study the geometry of the related semi-algebraic constraint region. For example, we show to how to compute the subdifferential of the abscissa mapping from the polynomial stability criteria, and also show how to dissect the geometry of the constraint region based on certain matrix factorization properties of the underlying Hurwitz matrix. Finally, we present a new algorithm for the minimization of the abscissa map that uses the theoretical properties that we develop and give some numerical results.