Absolute stability in delay equations
An equilibrium of a delay equation is said to be absolutely stable if it is asymptotically stable for all delays. As is well known, this is equivalent to the characteristic equation having all its roots in the left half-plane for all delays. The author considers delay equations where the characteristic equation is of the form \(P(z)+Q(z)e^{-\tau z}=0\) (\(\tau\) is the delay, p,Q are square matrices) and proves the following result. Theorem. Let P and Q be analytic in some open set containing Re \(z\geq 0\) and let P(z)\(\neq 0\), Re \(z\geq 0\), \(P(-iy)=P(iy)\), \(Q(-iy)=Q(iy)\), \(| Q(iy)| <| P(iy)|,\) \(0\leq y<\infty\), \(\lim_{| z| \to \infty,Re z\geq 0}| Q(z)P^{-1}(z)| =0.\) Then the equation is absolutely stable.
scientific article; zbMATH DE number 4036258