Uniform stabilization of some damped second order evolution equations with vanishing short memory

Tebou, Louis

ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 1, pp. 174-189.

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.

Classification : 93D15, 35L10, 35Q74, 37L15, 74K20
Mots clés : second order evolution equation, Kelvin-Voigt damping, hyperbolic equations, stabilization, boundary dissipation, localized damping, plate equations, elasticity equations, frequency domain method, resolvent estimates

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