Numerical Range and Quadratic Numerical Range for Damped Systems

TL;DR

This paper develops new spectral enclosures for non-selfadjoint operator matrices in damped systems using quadratic numerical range, providing tighter bounds and explicit estimates that improve upon previous results.

Contribution

It introduces novel spectral bounds for damped systems leveraging quadratic numerical range, applicable under weak assumptions on damping operators.

Findings

Spectral enclosures can have bounded imaginary parts.

New bounds improve earlier sectorial and selfadjoint D results.

Explicit bounds demonstrated in fluid flow example.

Abstract

We prove new enclosures for the spectrum of non-selfadjoint operator matrices associated with second order linear differential equations $\ddot{z}(t) + D \dot{z} (t) + A_0 z(t) = 0$ in a Hilbert space. Our main tool is the quadratic numerical range for which we establish the spectral inclusion property under weak assumptions on the operators involved; in particular, the damping operator only needs to be accretive and may have the same strength as $A_0$. By means of the quadratic numerical range, we establish tight spectral estimates in terms of the unbounded operator coefficients $A_0$ and $D$ which improve earlier results for sectorial and selfadjoint $D$; in contrast to numerical range bounds, our enclosures may even provide bounded imaginary part of the spectrum or a spectral free vertical strip. An application to small transverse oscillations of a horizontal pipe carrying a steady-state flow of an ideal incompressible fluid illustrates that our new bounds are explicit.