Stabilization of some elastodynamic systems with localized Kelvin-Voigt damping

Abstract

We consider the dynamic elasticity equations with a locally distributed damping of Kelvin-Voigt type in a bounded domain. The damping is localized in a suitable open subset, of the domain under consideration, which satisfies the piecewise multipliers condition of Liu. Using multiplier techniques combined with the frequency domain method, we show that: i) the energy of this system decays polynomially when the damping coefficient is only bounded measurable, ii) the energy of this system decays exponentially when the damping coefficient as well as its gradient are bounded measurable, and the damping coefficient further satisfies a structural condition. These results generalize and improve, at the same time, on an earlier result of Liu and Rao involving the wave equation with Kelvin-Voigt damping; those authors proved the exponential decay of the energy provided that the damping region is a neighborhood of the whole boundary, and further restrictions are imposed on the damping coefficient.

Uniform stabilization of the Klein-Gordon system

“…The method then leads the authors to impose several technical conditions on the damping coefficient. In the work of Tebou in [35], such conditions on the damping coefficient as well as the conditions on the feedback control region are relaxed. However, the author still requires an inequality constraint on the gradient of the damping coefficient, which will not be required in our current study.…”

Stability of a star-shaped network with local Kelvin-Voigt damping and non-smooth coefficient at interface

“…The stability of this model was intensively studied in this last two decades: On high dimensional case: Liu and Rao [19] proved the exponential decay of the energy providing that the damping region is a neighborhood of the whole boundary, and further restrictions are imposed on the damping coefficient. Next, this result was generalized and improved by Tebou [25] when damping is localized in a suitable open subset, of the domain under consideration, which satisfies the piecewise multipliers condition of Liu. He shows that the energy of this system decays polynomially when the damping coefficient is only bounded measurable, and it decays exponentially when the damping coefficient as well as its gradient are bounded measurable, and the damping coefficient further satisfies a structural condition.…”

Local uniform stability for the semilinear wave equation in inhomogeneous media with locally distributed Kelvin–Voigt damping

“…The method then leads the authors to impose several tecnical conditions on the damping coefficient. In the work of Tebou in , such conditions on the damping coefficient as well as the conditions on the feedback control region are relaxed. However, the author still requires an inequality constraint on the gradient of the damping coefficient, which will not be required in our current study.…”