Analysis of a Belgian Chocolate Stabilization Problem

We give a detailed numerical and theoretical analysis of a stabilization problem posed by V. Blondel in 1994. Our approach illustrates the effectiveness of a new “gradient sampling” algorithm for finding local optimizers of nonsmooth, nonconvex optimization problems arising in control, as well as the power of nonsmooth analysis for understanding variational problems involving polynomial roots and eigenvalues. I. I NTRODUCTION More than a decade ago, Blondel [Blo94, p.150] offered a prize of a kilogram of Belgian chocolate for the solution of the following stabilization problem. By a stable polynomial we mean a real polynomial with all its roots in the open left half-plane. Problem 1.1: Let a(s) = s 2 i 2‐s + 1 and b(s) = s 2 i 1. Find the range of real values for ‐ for which there exist stable polynomials x(s) and y(s) with deg(x) ‚ deg(y) such that ax+by