Systems with strong damping and their spectra. (4th October 2018)

• Record Type: Journal Article
• Authors: Jacob, Birgit; Tretter, Christiane; Trunk, Carsten; Vogt, Hendrik
• Is Part Of: Mathematical methods in the applied sciences. Volume 41:Number 16(2018)
• Page Start: 6546
• Page End: 6573
• Publication Date: 2018-10-04
• Subjects: abstract second‐order differential equation -- damped system -- numerical range -- operator matrix -- quadratic numerical range -- spectrum; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.5166
• Languages: English
• ISSNs: 0170-4214

Abstract

Abstract : We establish a new method for obtaining nonconvex spectral enclosures for operators associated with second‐order differential equations z ¨ ( t ) + D z ˙ ( t ) + A 0 z ( t ) = 0 in a Hilbert space. In particular, we succeed in establishing the existence of a spectral gap, which is the first result of this kind since the seminal results of Krein and Langer for oscillations of damped systems. While the latter and other spectral bounds are confined to dampings D that are symmetric and dominated by A 0, we allow for accretive D of equal strength as A 0 . To achieve these results, we prove new abstract spectral inclusion results that are much more powerful than classical numerical range bounds. Two different applications, small transverse oscillations of a horizontal pipe carrying a steady‐state flow of an ideal incompressible fluid and wave equations with strong (viscoelastic and frictional) damping, illustrate that our new bounds are explicit.