Date: Tuesday, March 9, 12:30-1:20, PDL C-401

Speaker: Jonathan Cross, Math, University of Washington

Spectral Abscissa Optimization using Polynomial Stability Conditions

The abscissa map $\mathsf{a}$ of a polynomial is 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. Indeed, determining the parameters for an optimally stable system may be reduced to the following optimization problem: \[ \min_{x \in \mathbb{R}^k} \; \mathsf{a}(p_x) \] where $p_x$ is a family of polynomials whose coefficients are functions of $x$. Using classical polynomial stability criteria, we reformulate this problem into a particular type of polynomial optimization problem, which we study from an epigraphical viewpoint. From this, we compute variational properties of the abscissa map, and relate the factorization of $p_x$ to the geometry of the constraints in the polynomial optimization problem. Finally, we present a new algorithm for solving abscissa map minimization problems and give some numerical results.