Stabilization of the wave equation with localized compensating frictional and Kelvin-Voigt dissipating mechanisms
Date
2017-03-24
Authors
Cavalcanti, Marcelo M.
Domingos Cavalcanti, Valeria N.
Tebou, Louis
Journal Title
Journal ISSN
Volume Title
Publisher
Texas State University, Department of Mathematics
Abstract
We consider the wave equation with two types of locally distributed damping mechanisms: a frictional damping and a Kelvin-Voigt type damping. The location of each damping is such that none of them alone is able to exponentially stabilize the system; the main obstacle being that there is a quite big undamped region. Using a combination of the multiplier techniques and the frequency domain method, we show that a convenient interaction of the two damping mechanisms is powerful enough for the exponential stability of the dynamical system, provided that the coefficient of the Kelvin-Voigt damping is smooth enough and satisfies a structural condition. When the latter coefficient is only bounded measurable, exponential stability may still hold provided there is no undamped region, else only polynomial stability is established. The main features of this contribution are: (i) the use of the Kelvin-Voigt or short memory damping as opposed to the usual long memory type damping; this makes the problem more difficult to solve due to the somewhat singular nature of the Kelvin-Voigt damping, (ii) allowing the presence of an undamped region unlike all earlier works where a combination of frictional and viscoelastic damping is used.
Description
Keywords
Stabilization, Wave equation, Frictional damping, Kelvin-Voigt damping, Viscoelastic material, Localized damping
Citation
Cavalcanti, M. M., Domingos Cavalcanti, V. N., & Tebou, L. (2017). Stabilization of the wave equation with localized compensating frictional and Kelvin-Voigt dissipating mechanisms. <i>Electronic Journal of Differential Equations, 2017</i>(83), pp. 1-18.
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Attribution 4.0 International
Rights Holder
This work is licensed under a Creative Commons Attribution 4.0 International License.