论文标题
通过反馈诺伊曼控制的线性MGT方程的边界稳定
Boundary Stabilization of the linear MGT equation with Feedback Neumann control
论文作者
论文摘要
Jordan-Moore-Gibson-Thompson（JMGT）\ cite {Christov_heat_2005，Jordan_NonlineAr_2008，Straughan_heat_2014}方程是一种基准模型，描述了在杂物液中非线性声波的传播的基础模型。这是一个三阶（时间）动态，可以说明热信号传播的有限速度（请参阅\ cite {coulouvrat_equations_1992，crighton_model_1979，jordan_nonlinalear_2008，jordan_second-socond-sound_2014，kaltenbacher_jordan-moore_jordan-morethan-mother_2008。在本文中，我们研究了{\ it关键案例}中线性化版本（也称为MGT-方程式）的边界稳定，其中构型的构型在其中扩散效应的较小性会导致保守的动力学\ cite {kaltenbacher_well_wellposedness_2011}。通过在自然几何条件下以非空的，相对开放的边界的相对开放的边界形式进行的{\ it {反馈}}形式的单个测量，我们能够获得统一的指数稳定性结果，除了相对于空间依赖的粘弹性参数而言，这些稳定性结果是均匀的。对MGT方程边界稳定区域的独立兴趣的结果为研究{\ it Infinite Horizon} \ cite {Bucci_feedback_2019}的最佳边界反馈控制提供了必要的第一步。
The Jordan-Moore-Gibson-Thompson (JMGT)\cite{christov_heat_2005,jordan_nonlinear_2008,straughan_heat_2014} equation is a benchmark model describing propagation of nonlinear acoustic waves in heterogeneous fluids at rest. This is a third-order (in time) dynamics which accounts for a finite speed of propagation of heat signals (see \cite{coulouvrat_equations_1992,crighton_model_1979,jordan_nonlinear_2008,jordan_second-sound_2014,kaltenbacher_jordan-moore-gibson-thompson_2019}). In this paper, we study a boundary stabilization of linearized version (also known as MGT-equation) in the {\it critical case}, configuration in which the smallness of the diffusion effects leads to conservative dynamics \cite{kaltenbacher_wellposedness_2011}. Through a single measurement in {\it{feedback}} form made on a non-empty, relatively open portion of the boundary under natural geometric conditions, we were able to obtain uniform exponential stability results that are, in addition, uniform with respect to the space-dependent viscoelasticity parameter which no longer needs to be assumed positive and in fact can be degenerate and taken to be zero on the whole domain. This result, of independent interest in the area of boundary stabilization of MGT equations, provides a necessary first step for the study of optimal boundary feedback control on {\it infinite horizon} \cite{bucci_feedback_2019}.
