Stability radii of polynomial matrices

which are square invertible and have zeros – i.e. roots of detP (λ) – inside a given region Γ. We say that P (λ) is Γ-stable and call Γ the stability region. The complex stability radius rC of such polynomial matrices is the norm of the smallest perturbation ∆P (λ) . = ∆P0 +∆P1λ+ · · ·∆Pkλ k needed to “destabilize” P (λ) + ∆P (λ) and hence causing at least one zero of P (λ) + ∆P (λ) to leave the region Γ. If we measure the perturbations via the norm of a constant matrix ∆ depending on the coefficients of ∆P (λ) : ‖∆‖ . = g (∆P0,∆P1, · · · ,∆Pk) (1) then we have the expression rC = inf{‖∆‖ : ∃ root(P (λ) + ∆P (λ)) ∈ Γc}, (2) where Γc is the complement of Γ. The two regions that are typically considered for Γ are the open left half plane and the open unit disc, which are both open and connected sets of the complex plane. By continuity of zeros of perturbed matrices, the root “leaving” Γ must actually lie on its boundary ∂Γ, which can be parameterized by a real variable ω. In this paper we prove for Hölder norms that r C = sup λ∈∂Γ ‖G(λ)‖p, G(λ) = d(λ) · P (λ) , (3)