弹性系统的稳定性与时变反馈镇定

Stabilization for vibrating plate with singular structural damping

We consider the dynamic elasticity equation, modeled by the Euler-Bernoulli plate equation, with a locally distributed singular structural (or viscoelastic ) damping in a boundary domain. Using a frequency domain method combined, based on the Burq's result, combined with an estimate of Carleman type we provide precise decay estimate showing that the energy of the system decays logarithmically as the type goes to the infinity.

Modeling and stabilization results for a charge or current-actuated active constrained layer (ACL) beam model with the electrostatic assumption

An infinite dimensional model for a three-layer active constrained layer (ACL) beam model, consisting of a piezoelectric elastic layer at the top and an elastic host layer at the bottom constraining a viscoelastic layer in the middle, is obtained for clamped-free boundary conditions by using a thorough variational approach. The Rao-Nakra thin compliant layer approximation is adopted to model the sandwich structure, and the electrostatic approach (magnetic effects are ignored) is assumed for the piezoelectric layer. Instead of the voltage actuation of the piezoelectric layer, the piezoelectric …

Stability results of a singular local interaction elastic/viscoelastic coupled wave equations with time delay

The purpose of this paper is to investigate the stabilization of a one-dimensional coupled wave equations with non smooth localized viscoelastic damping of Kelvin-Voigt type and localized time delay. Using a general criteria of Arendt-Batty, we show the strong stability of our system in the absence of the compactness of the resolvent. Finally, using frequency domain approach combining with a multiplier method, we prove a polynomial energy decay rate of order 1/t.

非光滑域上弹性薄板的多项式稳定性

Polynomial Stability of an Elastic Thin Plate on Non-Smooth Domain

研究非光滑域上弹性板的动力边界条件下的多项式稳定化，通过变分框架处理边界奇点引起的解正则性损失，证明小角域保持多项式衰减率，大角域通过反馈控制恢复稳定性。

关于小数据下完全浸没在粘性不可压缩流体中的弹性体模型

On a model of an elastic body fully immersed in a viscous incompressible fluid with small data

针对浸没在两层不可压缩粘性流体间的弹性体模型，由阻尼波动方程与Navier-Stokes方程耦合，建立了小初始数据下的全局时间适定性与指数衰减，并证明最终状态除水平小位移外必为平凡。

Stability results of an elastic/viscoelastic transmission problem of locally coupled waves with non smooth coefficients

We investigate the stabilization of a locally coupled wave equations with only one internal viscoelastic damping of Kelvin-Voigt type. The main novelty in this paper is that both the damping and the coupling coefficients are non smooth. First, using a general criteria of Arendt-Batty, combined with an uniqueness result, we prove that our system is strongly stable. Next, using a spectrum approach, we prove the non-exponential (uniform) stability of the system. Finally, using a frequency domain approach, combined with a piecewise multiplier technique and the construction of a new multiplier sati…

Local boundary feedback stabilization of a fuid-structure interaction problem under Navier slip boundary conditions with time delay

We consider a fluid-structure interaction system composed by a three-dimensional viscous incompressible fluid and an elastic plate located on the upper part of the fluid boundary. The fluid motion is governed by the Navier-Stokes system whereas the structure displacement satisfies the damped plate equation. We consider here the Navier slip boundary conditions. The main result of this work is the feedback stabilization of the strong solutions of the corresponding system around a stationary state for any exponential decay rate by means of a time delayed control localized on the fixed fluid bound…

A N-dimensional elastic\viscoelastic transmission problem with Kelvin-Voigt damping and non smooth coefficient at the interface

We investigate the stabilization of a multidimensional system of coupled wave equations with only one Kelvin Voigt damping. Using a unique continuation result based on a Carleman estimate and a general criteria of Arendt Batty, we prove the strong stability of the system in the absence of the compactness of the resolvent without any geometric condition. Then, using a spectral analysis, we prove the non uniform stability of the system. Further, using frequency domain approach combined with a multiplier technique, we establish some polynomial stability results by considering different geometric …

关于适合数值计算的 $\mathbb{R}^n$ 中参数化曲线的弹性流

On an elastic flow for parametrized curves in $\mathbb{R}^n$ suitable for numerical purposes

本文研究了适合数值计算的 $\mathbb{R}^n$ 中闭曲线弹性流变体的长时间性质，证明了该流在时间上全局存在。

压电智能复合材料的动力学与静电建模及相关镇定结果

Dynamic and electrostatic modeling for a piezoelectric smart composite and related stabilization results

该研究针对悬臂压电智能复合梁，基于麦克斯韦方程组建立全动力学与静电型两类Rao-Nakra、Mead-Marcus摄动模型，证明解的存在唯一性，分析$B^*-$型反馈镇定效果，发现静电模型所需控制器更少且稳定性更优。

迁移弹性流 II

Migrating elastic flows II

本文在自然边界条件下构造开曲线的迁移弹性流，解决了 Huisken 问题的一个变体，并将先前结果从非局部流推广到纯局部流。

Nonlinear modeling and preliminary stabilization results for a class of piezoelectric smart composite beams

Existing smart composite piezoelectric beam models in the literature mostly ignore the electro-magnetic interactions and adopt the linear elasticity theory. However, these interactions substantially change the controllability and stabilizability at the high frequencies, and linear models fail to represent and predict the governing dynamics since mechanical nonlinearities are pronounced in certain applications such as energy harvesting. In this paper, first, a consistent variational approach is used by considering nonlinear elasticity theory to derive equations of motion for a single-layer piez…

Strong Stabilization of a 3D Potential Flow via a Weakly Damped von Karman Plate

The elimination of aeroelastic instability (resulting in sustained oscillations of bridges, buildings, airfoils) is a central engineering and design issue. Mathematically, this translates to strong asymptotic stabilization of a 3D flow by a 2D elastic structure. The stabilization (convergence to the stationary set) of a aerodynamic wave-plate model is established here. A 3D potential flow on the half-space has a spatially-bounded von Karman plate embedded in the boundary. The physical model, then, is a Neumann wave equation with low regularity of coupling conditions. Motivated on empirical obs…

Variational methods for steady-state Darcy/Fick flow in swollen and poroelastic solids

Existence of steady states in elastic media at small strains with diffusion of a solvent or fluid due to Fick's or Darcy's laws is proved by combining usage of variational methods inspired from static situations with Schauder's fixed-point arguments. In the plain variant, the problem consists in the force equilibrium coupled with the continuity equation, and the underlying operator is non-potential and non-pseudomonotone so that conventional methods are not applicable. In advanced variants, electrically-charged multi-component flows through an electrically charged elastic solid are treated, em…