Uniform stabilization of some damped second order evolution equations with vanishing short memory. (6th February 2014)

• Record Type: Journal Article
• Authors: Tebou, Louis
• Is Part Of: ESAIM. Volume 20:Number 1(2014:Jan.)
• Page Start: 174
• Page End: 189
• Publication Date: 2014-02-06
• Subjects: Second order evolution equation, -- Kelvin–Voigt damping, -- hyperbolic equations, -- stabilization, -- boundary dissipation, -- localized damping, -- plate equations, -- elasticity equations, -- frequency domain method, -- resolvent estimates; System analysis -- Periodicals; Calculus of variations -- Periodicals; Mathematical analysis -- Periodicals; Mathematical optimization -- Periodicals; Control theory -- Periodicals; 515.64
• DOI: 10.1051/cocv/2013060
• Languages: English
• ISSNs: 1292-8119

Abstract

Abstract : We consider a damped abstract second order evolution equation with an additional vanishing damping of Kelvin–Voigt type. Unlike the earlier work by Zuazua and Ervedoza, we do not assume the operator defining the main damping to be bounded. First, using a constructive frequency domain method coupled with a decomposition of frequencies and the introduction of a new variable, we show that if the limit system is exponentially stable, then this evolutionary system is uniformly − with respect to the calibration parameter − exponentially stable. Afterwards, we prove uniform polynomial and logarithmic decay estimates of the underlying semigroup provided such decay estimates hold for the limit system. Finally, we discuss some applications of our results; in particular, the case of boundary damping mechanisms is accounted for, which was not possible in the earlier work mentioned above.