The Largest Stability Hypercube for Families of Polynomials with Linear Uncertainty

Let ϕ(s) = ϕ₀(s) = ϕ₀(s) + a₁ ϕ₁(s) + a₂ϕ₂(s) +...+ a_kϕ_k(s) be a polynomial with coefficients that depend linearly on real parameters a_i, 1 < i < k. Let ϕ₀(s) be stable of degree n and ϕ_i, 1 < i < k, of degree less than n. Assume that the a_i are allowed to take values in the k dimensional hypercube Ω_λa = {(a₁,...,a_k)εR^k: a^-_i ≤' a_i ≤ λ a⁺_i}, where a^-1_i < 0, a⁺_i > 0, 1 ≤ i ≤ k are fixed and λ ≤ 0. In this paper we consider the problem of how to compute the largest value λ* such that the family is stable in Ω_λa for 0 ≤ λ < λ*. Considering the problem in the frequency domain, a function of frequency can be constructed whose infimum is λ*. In this paper we show that to compute λ* one need only consider the values of this function at a finite number of frequencies. The number of frequencies is polynomial in k, and the frequencies themselves are explicitly determined from the given data.