具有局部阻尼的二维 Mindlin-Timoshenko 板系统的镇定

具有局部阻尼的二维 Mindlin-Timoshenko 板系统的镇定

章春国^,¹^,^*, 孙宝楠¹, 付煜之¹, 于欣²

¹杭州电子科技大学数学系杭州 310018

²浙大宁波理工学院计算机与数据工程学院浙江宁波 315100

Stabilization of 2-D Mindlin Timoshenko Plate Systems with Local Damping

Zhang Chunguo^,¹^,^*, Sun Baonan¹, Fu Yuzhi¹, Yu Xin²

¹Department of Mathematics, Hangzhou Dianzi University, Hangzhou 310018

²School of Computer and Data Engineering, Ningbo Tech University, Zhejiang Ningbo 315100

通讯作者: *章春国, E-mail: cgzhang@hdu.edu.cn

收稿日期: 2023-04-29 修回日期: 2023-10-7

基金资助:

国家自然科学基金(61503103)
浙江省自然科学基金重点项目(LZ21A010001)

Received: 2023-04-29 Revised: 2023-10-7

Fund supported:

NSFC(61503103)
key project of Zhejiang Provencial Natural Science Foundation(LZ21A010001)

摘要

该文研究了具有局部阻尼的二维 Mindlin-Timoshenko 板系统, 首先将原系统转化成抽象柯西问题, 运用算子半群理论得到了系统的适定性. 借助系统频域稳定性结果, 通过引入几何光学条件结合乘子技巧得到了系统的一致指数稳定性.

关键词： Mindlin-Timoshenko 板系统; 局部阻尼; 分片乘子法; 一致指数稳定性

Abstract

In this paper, the two-dimensional Mindlin-Timoshenko plate system with local damping is studied. First, the original system is transformed into an abstract Cauchy problem, and the well posedness of the system is obtained by using operator semigroup theory. With the help of the frequency domain stability results of the linear system, the uniform exponential stability of the system is obtained by introducing geometric optical conditions and multiplier techniques.

Keywords： Mindlin-Timoshenko plate system; Local damping; Piecewise multiplier method; Uniform exponential stability.

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本文引用格式

章春国, 孙宝楠, 付煜之, 于欣. 具有局部阻尼的二维 Mindlin-Timoshenko 板系统的镇定[J]. 数学物理学报, 2024, 44(4): 946-959

Zhang Chunguo, Sun Baonan, Fu Yuzhi, Yu Xin. Stabilization of 2-D Mindlin Timoshenko Plate Systems with Local Damping[J]. Acta Mathematica Scientia, 2024, 44(4): 946-959

1 引言

众所周知, 弹性薄板具有广泛的工程应用背景, 如航天器、太阳帆板、旋转机翼以及螺旋桨桨叶等. 以弹性薄板为构件的柔性体, 用刚体动力学的理论和方法来研究弹性板的变形与振动. 在经典弹性板振动理论中, Kirchhoff 板模型是最为重要的经典薄板振动模型之一. 如果当弹性板具有一定的厚度且在板振动过程中考虑横向剪切效应时, 得到相应的板振动模型即为 Mindlin-Timoshenko 板模型 (系统), 因此 Mindlin-Timoshenko 板模型 (系统) 能更准确地描述具有一定厚度弹性板的振动.

近几年来, Mindlin-Timoshenko 系统的研究取得了许多成果. 文献 [1] 研究了具有边界阻尼的二维 Mindlin-Timoshenko 板系统的指数稳定性, Nicaise^[2] 研究了具有全局阻尼的 Mindlin-Timoshenko 系统的指数衰减性, 进一步, 当只有阻尼作用时, 该系统是多项式衰减的. 文献 [3 ⇓⇓⇓⇓-8] 研究了具有各种阻尼的 Mindlin-Timoshenko 系统或 Mindlin-Timoshenko 热弹性板系统的稳定性问题.

在工程技术应用中, 利用局部分布阻尼器对弹性振动系统进行抑振更为普遍. 因此, 分片乘子法作为有效的手段来解决问题. 文献 [9 ⇓⇓⇓-13] 的作者应用分片乘子法和应用 Liyapunov 函数法研究了弹性系统和粘弹性系统的稳定性.

本文将文献 [2] 推广到更一般情形, 更精确地说, 考虑如下的具有局部内部阻尼的二维 Mindlin-Timoshenko 板方程的初边值问题

$\begin{equation} \left\{\begin{array}{lll} \rho_{1}\psi_{tt}-D(\psi_{xx}+\frac{1-\mu}{2}\psi_{yy} +\frac{1+\mu}{2}\varphi_{xy}) +K(\psi+\omega_{x})+\chi_{\Omega_{0}}\psi_{t}=0, & \Omega\times\left (0, + \infty \right), \\[2mm] \rho_{1}\varphi_{tt}-D(\varphi_{yy}+\frac{1-\mu}{2}\varphi_{xx} +\frac{1+\mu}{2}\psi_{xy}) +K(\varphi+\omega_{y})+\chi_{\Omega_{0}}\varphi_{t}=0,& \Omega\times\left ( 0, + \infty \right ), \\[1mm] \rho_{2}\omega_{tt}-K(\psi_{x}+\omega_{xx}+\varphi_{y}+\omega_{yy})+\chi_{\Omega_{0}}\omega_{t} =0, \quad \quad\quad\quad\quad\quad\quad\quad & \Omega\times\left ( 0, + \infty \right ), \\ \psi=\varphi=\omega=0, & \Gamma\times(0, +\infty), \\[1mm] D(\psi_{x}+\mu\varphi_{y}, \frac{1-\mu}{2}(\psi_{y}+\varphi_{x}))\cdot{\nu} =0, & \Gamma\times(0, +\infty), \\[2mm] D(\frac{1-\mu}{2}(\psi_{y}+\varphi_{x}), \varphi_{y}+\mu\psi_{x})\cdot{\nu}=0,& \Gamma\times(0, +\infty), \\[2mm] K[\frac{\partial\omega}{\partial\mathbf{{\nu}}}+(\psi, \varphi)\cdot{\nu}] =0, & \Gamma\times(0, +\infty), \\[2mm] (\psi(x, y, 0), \varphi(x, y, 0), \omega(x, y, 0))^T=(\psi_{01}, \varphi_{01}, \omega_{01})^T, & (x, y)\in\Omega, \\ (\psi_{t}(x, y, 0), \varphi_{t}(x, y, 0), \omega_{t}(x, y, 0))^T=(\psi_{02}, \varphi_{02}, \omega_{02})^T, & (x, y)\in\Omega, \end{array}\right. \end{equation}$

(1.1)

其中 $\Omega$ 是 $\mathbb{R}^2$ ( ( $x$ , $y$ ) 平面) 中的有界区域, $\Omega_0(\subset\Omega)$ 是 $\Omega$ 中一个凸的子区域, 且 $\Omega$ 具有光滑的边界 $\Gamma=\partial \Omega$ , $\Omega_{0}$ 具有边界 $\Gamma_0$ , 正常数 $\rho_{1}$ , $\rho_{2}$ 分别表示单位质量, $\mu(0<\mu<\frac{1}{2})$ 是 Poisson's 比, $D$ , $K$ 分别为柔性弹性模量与剪切模量. $\psi$ , $\varphi$ 分别表示 $xz$ 平面与 $yz$ 平面的全转角, $\omega$ 表示横向位移, $\chi_{\Omega_{0}}$ 表示 $\Omega_0$ 上的特征函数, ${\nu}=(\nu_{1}, \nu_{2})$ 表示 $\Gamma$ 的单位外法向量.

本文应用线性算子半群理论、分片乘子法以及 Hilbert 空间中线性系统指数稳定的频域结果, 得到了系统的适定性以及指数稳定性. 本文的安排如下: 在第 2 节中, 通过适当的假设将系统转换为抽象 Cauchy 问题. 在第 3 节中, 应用 Pazy^[14] 的 Lumer-Phillips 定理证明了系统的适定性. 在第 4 节中, 运用分片乘子法以及系统稳定性的频域结果结合矛盾讨论, 证明了系统是指数稳定的.

2 预备知识

定义系统 (1.1) 在 $t$ 时刻的能量为

$\begin{align*} E(t)={}&\frac{1}{2}\int_{\Omega}^{}[D(|\psi_{x}|^{2}+|\varphi_{y}|^{2}+2\mu\psi_{x}\varphi_{y}+\frac{1-\mu}{2}|\psi_{y}+\varphi_{x}|^{2})\nonumber\\ &+K(|\psi+\omega_{x}|^{2}+|\varphi+\omega_{y}|^{2})]{\rm d}x{\rm d}y+\int_{\Omega}^{}[\rho_{1}(|\psi_{t}|^{2}+|\varphi_{t}|^{2})+\rho_{2}|\omega_{t}|^{2}]{\rm d}x{\rm d}y. \end{align*}$

对于充分光滑的 $(\psi, \varphi, \omega)$ 和 $(\widehat{\psi}, \widehat{\varphi}, \widehat{\omega})$ , 定义如下线性泛函

$\begin{equation} \left\{\begin{array}{lll} L_{1}=L_{1}\{\psi, \varphi, \omega\}=D(\psi_{xx}+\frac{1-\mu}{2}\psi_{yy} +\frac{1+\mu}{2}\varphi_{xy})-K(\psi+\omega_{x}), \\[3mm] L_{2}=L_{2}\{\psi, \varphi, \omega\}=D(\varphi_{yy}+\frac{1-\mu}{2}\varphi_{xx} +\frac{1+\mu}{2}\psi_{xy})-K(\varphi+\omega_{y}), \\[1mm] L_{3}=L_{3}\{\psi, \varphi, \omega\}=K(\psi_{x}+\omega_{xx}+\varphi_{y}+\omega_{yy}), \\[1mm] B_{1}=B_{1}\{\psi, \varphi, \omega\}=D(\psi_{x}+\mu\varphi_{y}, \frac{1-\mu}{2}(\psi_{y}+\varphi_{x}))\cdot{\nu}, \\[3mm] B_{2}=B_{2}\{\psi, \varphi, \omega\}=D(\frac{1-\mu}{2}(\psi_{y}+\varphi_{x}), \varphi_{y}+\mu\psi_{x})\cdot{\nu}, \\[3mm] B_{3}=B_{3}\{\psi, \varphi, \omega\}=K[\frac{\partial \omega }{\partial {\nu}} +(\psi, \varphi)\cdot{\nu}] \end{array}\right. \end{equation}$

(2.1)

和双线性泛函

$\begin{align*} a({\psi, \varphi, \omega};{\widehat{\psi}, \widehat{\varphi}, \widehat{\omega}})= {}&D\int_{\Omega}[\psi_{x}\overline{\widehat{\psi}}_{x}+\varphi_{y}\overline{\widehat{\varphi}}_{y} +\mu\psi_{x}\overline{\widehat{\varphi}}_{y}+\mu\overline{\widehat{\psi}}_{x}\varphi_{y} \\ & +\frac{1-\mu}{2}(\psi_{y}+\varphi_{x})(\overline{\widehat{\psi}}_{y}+\overline{\widehat{\varphi}}_{x})]{\rm d}x{\rm d}y\\&+K\int_{\Omega}^{}[(\psi+\omega_{x})(\overline{\widehat{\psi}}+\overline{\widehat{\omega}}_{x})+(\varphi+\omega_{y})(\overline{\widehat{\varphi}}+\overline{\widehat{\omega}}_{y})]{\rm d}x{\rm d}y, \end{align*}$

(2.2)

经过简单计算, 我们得到

$\begin{equation} a({\psi, \varphi, \omega};{\widehat{\psi}, \widehat{\varphi}, \widehat{\omega}})=\overline{a(\widehat{\psi}, \widehat{\varphi}, \widehat{\omega};{\psi, \varphi, \omega})}. \end{equation}$

(2.3)

进一步, 对所有充分光滑的 $(\psi, \varphi, \omega)$ 和 $(\widehat{\psi}, \widehat{\varphi}, \widehat{\omega})$ , 应用格林公式, 我们得到如下恒等式

$\begin{align*}&\ \int_{\Omega}(\overline{\widehat{\psi}}L_1+\overline{\widehat{\varphi}}L_2+\overline{\widehat{\omega}}L_3){\rm d}x{\rm d}y\nonumber\\ &=\int_{\Omega}[D\overline{\widehat{\psi}}\mathrm{div}(\psi_{x}+\mu\varphi_{y}, \frac{1-\mu}{2}(\psi_{y}+\varphi_{x}))+D\overline{\widehat{\varphi}}\mathrm{div}(\frac{1-\mu}{2}(\psi_{y}+\varphi_{x}), \mu\psi_{x}+\varphi_{y})\\ &\quad\ +K\overline{\widehat{\omega}}\mathrm{div}(\psi+\omega_{x}, \varphi+\omega_{y})]{\rm d}x{\rm d}y-K\int_{\Omega}\overline{\widehat{\psi}}(\psi+\omega_{x})+\overline{\widehat{\varphi}}(\varphi+\omega_{y}){\rm d}x{\rm d}y\\ &=-\int_{\Omega}[D(\overline{\widehat{\psi}}_{x}, \overline{\widehat{\psi}}_{y})\cdot(\psi_{x}+\mu\varphi_{y}, \frac{1-\mu}{2}(\psi_{y}+\varphi_{x}))+D(\overline{\widehat{\varphi}}_{x}, \overline{\widehat{\varphi}}_{y})\cdot(\frac{1-\mu}{2}(\psi_{y}+\varphi_{x}), \mu\psi_{x}+\varphi_{y})\\ &\quad\ +K(\overline{\widehat{\omega}}_{x}, \overline{\widehat{\omega}}_{y})\cdot(\psi+\omega _{x}, \varphi+\omega_{y})]{\rm d}x{\rm d}y-K\int_{\Omega}[\overline{\widehat{\psi}}(\psi+\omega_{x})+\overline{\widehat{\varphi}}(\varphi+\omega_{y})]{\rm d}x{\rm d}y\\ &\quad\ +\int_{\Gamma}[D\overline{\widehat{\psi}}(\psi_{x}+\mu\varphi_{y}, \frac{1-\mu}{2}(\psi_{y}+\varphi_{x}))\cdot{\nu}]{\rm d}S+\int_{\Gamma}[D\overline{\widehat{\varphi}}(\frac{1-\mu}{2}(\psi_{y}+\varphi_{x}), \mu\psi_{x}+\varphi_{y})\cdot{\nu}]{\rm d}S\\ &\quad\ +\int_{\Gamma}[k\overline{\widehat{\omega}}(\psi+\omega_{x}, \varphi+\omega_{y})\cdot{\nu}]{\rm d}S\\ &=-a(\psi, \varphi, \omega;\widehat{\psi}, \widehat{\varphi}, \widehat{\omega}), \end{align*}$

即有以下等式成立

$\begin{equation} a(\psi, \varphi, \omega;\widehat{\psi}, \widehat{\varphi}, \widehat{\omega})+\int_{\Omega}(\widehat{\psi}L_1+\widehat{\varphi}L_2+\widehat{\omega}L_3){\rm d}x{\rm d}y=0. \end{equation}$

(2.4)

由 (2.1)式, 我们将系统 (1.1) 改写为

$\begin{equation} \left\{\begin{array}{lll} \rho_{1}\psi_{tt}-L_1=-\chi_{\Omega_{0}}\psi_{t},& \Omega\times\left ( 0, + \infty \right ), \\ \rho_{1}\varphi_{tt}-L_2=-\chi_{\Omega_{0}}\varphi_{t}, & \Omega\times\left ( 0, + \infty \right ), \\ \rho_{2}\omega_{tt}-L_3=-\chi_{\Omega_{0}}\omega_{t}, & \Omega\times\left ( 0, + \infty \right ), \\ \psi=\varphi=\omega=0,& \Gamma\times\left ( 0, + \infty \right ), \\ B_1=B_2=B_3=0, & \Gamma\times\left ( 0, + \infty \right ), \\ (\psi(x, y, 0), \varphi(x, y, 0), \omega(x, y, 0))^T=(\psi_{01}, \varphi_{01}, \omega_{01})^T, & (x, y)\in\Omega, \\ (\psi_{t}(x, y, 0), \varphi_{t}(x, y, 0), \omega_{t}(x, y, 0))^T=(\psi_{02}, \varphi_{02}, \omega_{02})^T, & (x, y)\in\Omega. \end{array}\right. \end{equation}$

(2.5)

引入函数空间

$W=\left \{ (\psi, \varphi, \omega)\in[H^1(\Omega)]^3\vert \psi=\varphi=\omega =0, B_j\left \{ \psi, \varphi, \omega\right \} =0(j=1, 2, 3), \text{在}\ \Gamma\ \text{上} \right \},$

其中 $H^1(\Omega)$ 是一阶 Sobolev 空间^[15], 赋予范数

$\begin{align*} \left \| (\psi, \varphi, \omega)^T\right \| _{W}^{2} ={}&\int_{\Omega}[D(|\psi_{x}|^2+|\varphi_{y}|^2+\mu\psi_{x}\overline{\varphi}_{y}+\mu\overline{\psi}_{x}\varphi_{y}+\frac{1-\mu}{2}|\psi_{y}+\varphi_{x}|^2)]{\rm d}x{\rm d}y\nonumber\\ &+K\int_{\Omega}(|\psi+\omega_{x}|^2+|\varphi+\omega_{y}|^2){\rm d}x{\rm d}y, \nonumber\end{align*}$

具有内积 $(\cdot, \cdot)_{W}$ .

设空间 $H=L^2_{\rho_{1}}(\Omega)\times{L^2_{\rho_{1}}(\Omega)}\times{L^2_{\rho_{2}}(\Omega)}$ , $\psi_{t}=p, \varphi_t=q, \omega_t=r$ , 并赋予范数

$\left \| (p, q, r)^T \right \| ^{2}_{H}=\int _{\Omega}[\rho_1(|p|^2+|q|^2)+\rho_2|r|^2]{\rm d}x{\rm d}y,$

具有内积 $(\cdot, \cdot)_{H}$ , 则 $W$ 和 $H$ 都是 (复) Hilbert 空间.

记 $\mathcal{H}=W\times{H}$ , 其范数为

$\left \| (\psi, \varphi, \omega, p, q, r)^T \right \|^{2}_{\mathcal{H}}=\left \| (\psi, \varphi, \omega ) ^T\right \|^{2}_{W}+\left \| (p, q, r)^T \right \|^{2}_{H},$

具有内积 $(\cdot, \cdot)_{\mathcal{H}}$ , 因此 $\mathcal{H}$ 也是一个 (复) Hilbert 空间.

定义 $\mathcal{H}$ 上的线性算子 $\mathcal{A}$

$D(\mathcal{A})=\left\{\begin{array}{lll}(\psi, \varphi, \omega, p, q, r)\in{\mathcal{H}}\vert(p, q, r)\in W, \\ (\rho_{1}^{-1}(L_{1}-\chi_{\Omega_{0}}p), \rho_{1}^{-1}(L_2-\chi_{\Omega_{0}}q), \rho_{2}^{-1}(L_3-\chi_{\Omega_{0}}r))\in{H}\end{array}\right\},$

$\begin{equation} \mathcal{A}(\psi, \varphi, \omega, p, q, r)=(p, q, r, \rho_{1}^{-1}(L_{1}-\chi_{\Omega_{0}}p), \rho_{1}^{-1}(L_2-\chi_{\Omega_{0}}q), \rho_{2}^{-1}(L_3-\chi_{\Omega_{0}}r)). \end{equation}$

(2.6)

于是系统 (1.1) 改写为 $\mathcal{H}$ 中的抽象 Cauchy 问题

$\begin{equation} \frac{\mathrm{d} U}{\mathrm{d} t} =\mathcal{A}U, U(0)=U_0, \end{equation}$

(2.7)

其中, $U=(\psi, \varphi, \omega, p, q, r)$ , $U_0=U(0)=(\psi_{01}, \varphi_{01}, \omega_{01}, \psi_{02}, \varphi_{02}, \omega_{02}).$

3 适定性

定理 3.1 线性算子 $\mathcal{A}$ 生成 $\mathcal{H}$ 上一个压缩 $C_0$ -半群 ${\rm e}^{t\mathcal{A}}.$

证首先, 由算子 $\mathcal{A}$ 和空间 $\mathcal{H}$ 的定义, 对任意的 $U=(\psi, \varphi, \omega, p, q, r)$ 有

$\begin{align*}&\ \mathrm{Re}(\mathcal{A}U, U)_{\mathcal{H}}\\ &= \mathrm{Re}\Big[\int_{\Omega_{0}}(\overline{p}(L_{1}-p)+\overline{q}(L_{2}-q)+\overline{r}(L_{3}-r)){\rm d}x{\rm d}y\Big] +\mathrm{Re}\Big[\int_{\Omega\setminus\Omega_{0}}(\overline{p}L_{1}+\overline{q}L_{2}+\overline{r}L_{3}){\rm d}x{\rm d}y\Big] \nonumber\\ &\quad \ +\mathrm{Re}\Big[D\int_{\Omega_{0}}(p_{x}\overline{\psi}_{x}+q_{y}\overline{\varphi}_{y}+\mu{p_{x}}\overline{\varphi}_{y}+\mu{q_y\overline{\psi}_x}+\frac{1-\mu}{2}(p_y+q_x)(\overline{\psi}_y+\overline{\varphi}_x)){\rm d}x{\rm d}y\Big] \nonumber\\ &\quad\ +\mathrm{Re}\left \{ K\int_{\Omega_{0}}\Big[(p+r_x)(\overline{\psi}+\overline{\omega}_{x})+(q+r_y)(\overline{\varphi}+\overline{\omega}_{y})\Big]{\rm d}x{\rm d}y \right \} \nonumber\\ &\quad\ +\mathrm{Re}\Big[D\int_{\Omega\setminus\Omega_{0}}(p_x\overline{\psi}_{x}+q_y\overline{\varphi}_{y}+\mu{p_x\overline{\varphi}_y}+\mu{q_y\overline{\psi}_x}+\frac{1-\mu}{2}(p_y+q_x)(\overline{\psi}_{y}+\overline{\varphi}_{x})){\rm d}x{\rm d}y\Big]\nonumber\\ &\quad\ +\mathrm{Re}\left \{ K\int_{\Omega\setminus\Omega_{0}}[(p+r_x)(\overline{\psi}+\overline{\omega}_{x})+(q+r_y)(\overline{\varphi}+\overline{\omega}_{y})]{\rm d}x{\rm d}y \right \} \nonumber\\ &=\mathrm{Re}[-a(\psi, \varphi, \omega;p, q, r)+a(p, q, r;\psi, \varphi, \omega)]-\mathrm{Re}\int_{\Omega_{0}}(\overline{p}p+\overline{q}q+\overline{r}r){\rm d}x{\rm d}y\nonumber\\ &=-\int_{\Omega_{0}}(|p|^2+|q|^2+|r|^2){\rm d}x{\rm d}y\le0. \nonumber \end{align*}$

因此, 我们有

$\begin{equation} \mathrm{Re}(\mathcal{A}U, {U})_{\mathcal{H}}=-\int_{\Omega_{0}}(|p|^2+|q|^2+|r|^2){\rm d}x{\rm d}y\le0. \end{equation}$

(3.1)

这就说明了线性算子 $\mathcal{A}$ 是耗散的.

接着, 我们将证明: 存在充分小的 $\lambda >0$ , 使得 $\lambda{I-\mathcal{A}}$ 是满射.

即对给定的 $\lambda>0$ 以及任意的 $G=({g}_{1}, {g}_2, {g}_3, g_4, {g}_5, {g}_{6})\in\mathcal{A}$ , 存在 $(\psi, \varphi, \omega, p, q, r)\in{D(\mathcal{A})}$ 使得

$(\lambda{I}-\mathcal{A})(\psi, \varphi, \omega, p, q, r)=({g}_{1}, {g}_2, {g}_3, g_4, {g}_5, {g}_{6}),$

即

$\begin{equation} \left\{\begin{array}{lll} \lambda(\psi, \varphi, \omega)-(p, q, r)=({g}_{1}, {g}_2, {g}_3), \\ \lambda(p, q, r)-[\rho_{1}^{-1}(L_1-\chi_{\Omega_{0}}p), \rho_{1}^{-1}(L_2-\chi_{\Omega_{0}}q), \rho_{2}^{-1}(L_3-\chi_{\Omega_{0}}r)]=(g_4, {g}_5, {g}_{6}). \end{array}\right. \end{equation}$

(3.2)

因此, $(\psi, \varphi, \omega)\in{W}$ 满足

$\begin{align*}&(\lambda^{2}+\lambda{\rho_{1}^{-1}\chi_{\Omega_{0}}}, \lambda^{2}+\lambda{\rho_{1}^{-1}\chi_{\Omega_{0}}}, \lambda^{2}+\lambda{\rho_{2}^{-1}\chi_{\Omega_{0}}})(\psi, \varphi, \omega)^T-(\rho_{1}^{-1}L_1, \rho_{1}^{-1}L_2, \rho_{2}^{-1}L_3)^T\\ =\ &(\lambda+\rho_{1}^{-1}\chi_{\Omega_{0}}, \lambda+\rho_{1}^{-1}\chi_{\Omega_{0}}, \lambda+\rho_{2}^{-1}\chi_{\Omega_{0}})(g_1, g_2, g_3)^T+(g_4, g_5, g_6)^T. \end{align*}$

(3.3)

为得到结果, 我们将运用 Lax-Milgram 定理证明 (3.2) 式存在唯一弱解.

对于任意 $(\psi_1, \varphi_1, \omega_1)\in{W}$ 与 (3.2)式在 $\mathcal{H}$ 中作内积, 有

$\begin{equation} a_{\lambda}(\psi, \varphi, \omega;\psi_1, \varphi_1, \omega_1)=L_{G}(\psi_1, \varphi_1, \omega_1), \end{equation}$

(3.4)

其中

$\begin{align*} &\ a_{\lambda}(\psi, \varphi, \omega;\psi_1, \varphi_1, \omega_1)\\&=\int_{\Omega}[(\lambda^2\rho_{1}+\lambda\chi_{\Omega_{0}})(\psi\overline{\psi}_1+\varphi\overline{\varphi}_1)+(\lambda^2\rho_{2}+\lambda\chi_{\Omega_{0}})\omega\overline{\omega}_1]{\rm d}x{\rm d}y+a(\psi, \varphi, \omega;\psi_1, \varphi_1, \omega_1), \nonumber \end{align*}$

以及

$\begin{align*} L_{G}(\psi_1, \varphi_1, \omega)=&\int_{\Omega} \left \{ \rho_{1}[(\lambda +\rho_{1}^{-1}\chi_{\Omega _{0}})g_1+{g_4}]\overline{\psi}_{1}+ \rho_{1}[(\lambda +\rho_{1}^{-1}\chi_{\Omega _{0}})g_2+{g_5}]\overline{\varphi}_{1}\right. \nonumber \\ &\left. + \rho_{2}[(\lambda +\rho_{2}^{-1}\chi_{\Omega _{0}})g_3+{g_6}]\overline{\omega}_{1} \right \}{\rm d}x{\rm d}y. \nonumber \end{align*}$

显然, $L_{G}(\psi_1, \varphi_1, \omega_1)$ 是空间 $W$ 上的连续线性泛函, 且 $a_{\lambda}(\psi, \varphi, \omega;\psi_1, \varphi_1, \omega_1)$ 是空间 $W$ 上的共轭双线性连续泛函. 接下来验证 $a_{\lambda}(\psi, \varphi, \omega;\psi_1, \varphi_1, \omega_1)$ 在 $W$ 上满足强制性. 事实上, 对任意的 $(\psi, \varphi, \omega)\in{W}$ , 有

$\begin{align*} &a_{\lambda}(\psi, \varphi, \omega;\psi_1, \varphi_1, \omega_1)\nonumber\\ =&\int_{\Omega_{0}}[(\lambda^2\rho_1+\lambda\chi_{\Omega _{0}})(\psi\overline{\psi}_{1}+\varphi\overline{\varphi}_{1})+(\lambda^2\rho_{2}+\lambda\chi_{\Omega _{0}})\omega\overline{\omega}_{1}]{\rm d}x{\rm d}y+a(\psi, \varphi, \omega;\psi_1, \varphi_1, \omega_{1}), \end{align*}$

(3.5)

因此, 由式 (3.5) 得

$\begin{equation} \mathrm{Re}a_{\lambda}(\psi, \varphi, \omega;\psi, \varphi, \omega)\ge{a(\psi, \varphi, \omega;\psi, \varphi, \omega)}=\left \| (\psi, \varphi, \omega ) \right \|^{2}_{W}, \end{equation}$

(3.6)

故 $a_{\lambda}(\psi, \varphi, \omega;\psi, \varphi, \omega)$ 是强制的, 由 Lax-Milgram 定理知系统(3.2) 存在唯一解.

特别地, 我们取 $(\psi_1, \varphi_1, \omega_{1})\in(D(\Omega_{0}))^3$ , 则由(3.4) 式, 在 $(D^{'}(\Omega_{0}))^3$ 中分部积分得

$\begin{align*}&(\lambda^2+\lambda{\rho_{1}^{-1}\chi_{\Omega_{0}}}, \lambda^2+\lambda{\rho_{1}^{-1}\chi_{\Omega _{0}}}, \lambda^2+\lambda{\rho_{2}^{-1}\chi_{\Omega _{0}}})(\psi, \varphi, \omega)^T-(\rho_{1}^{-1}L_1, \rho_{1}^{-1}L_2, \rho_{2}^{-1}L_3)^T\\=\ &(\lambda+\rho_{1}^{-1}\chi_{\Omega_{0}}, \lambda+\rho_{1}^{-1}\chi_{\Omega_{0}}, \lambda+\rho_{2}^{-1}\chi_{\Omega_{0}})(g_1, {g_2}, g_3)^T+(g_4, {g_5}, g_6)^T, \end{align*}$

(3.7)

从而

$(\rho_{1}^{-1}(L_1-\chi_{\Omega _{0}}p), (\rho_{1}^{-1}(L_2-\chi_{\Omega _{0}}q), (\rho_{2}^{-1}(L_3-\chi_{\Omega _{0}}r))\in{H},$

取 $(p, q, r)=\lambda(\psi, \varphi, \omega)-(g_1, g_2, g_3)$ , 即得 $\lambda{I}-\mathcal{A}$ 是满射.

因此, 对于任意的 $(x, y)\in{\Omega_{0}}$ , $\mathcal{A}$ 是双射且 $\mathcal{A}^{-1}\in{\mathcal{L}(\mathcal{H})}$ , 则由 Lumer-Phillips 定理可得 $\mathcal{A}$ 是 $\mathcal{H}$ 上 $C_0$ -压缩半群 ${\rm e}^{t\mathcal{A}}$ 的无穷小生成元. 证毕.

定理 3.2 对于任意的 $(\psi_{01}, \varphi_{01}, \omega_{01}, \psi_{02}, \varphi_{02}, \omega_{02})\in{\mathcal{H}}$ , 则控制系统 (1.1) 存在唯一的弱解

$(\psi, \varphi, \omega, p, q, r)={\rm e}^{t\mathcal{A}}(\psi_{01}, \varphi_{01}, \omega_{01}, \psi_{02}, \varphi_{02}, \omega_{02}),$

且 $(\psi, \varphi, \omega)\in{C([0, +\infty);W)\cap{C^1([0, +\infty);H)}}$ ;

若 $(\psi_{01}, \varphi_{01}, \omega_{01}, \psi_{02}, \varphi_{02}, \omega_{02})\in{D(\mathcal{A})}$ , 则控制系统 (1.1) 存在唯一的强解

$(\psi, \varphi, \omega)\in{C^1([0, +\infty);W)\cap{C^2([0, +\infty);H)}}.$

4 指数稳定

注意到边界 $\partial (\Omega\setminus \Omega_{0})=\Gamma_{0}\cup\Gamma$ , 为了得到系统的指数稳定性, 我们需要引入如下几何光学条件.

假设 1 (几何光学条件) 设 ${\nu_{0}}$ 是 $\Gamma_{0}$ ( $\Omega\setminus\Omega_0$ 的部分边界) 的单位外法向量, 在 $\mathbb{R}^{2}$ 中, 设点 $(x_0, y_0)$ 是 ${\Omega_{0}}$ 内任一固定点, 对于任意的 $(x, y)\in\Gamma_{0}$ , 及向量

${m}{(x, y)=(x, y)-(x_0, y_0)},$

都有

${m}\cdot{{\nu}}_{0}\le{0}.$

“几何光学条件”指的是区域 (其强调的是区域的边界). 例子示意如图1所示.

图1

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图1 示意图

引理 4.1 线性算子 $\mathcal{A}$ 具有紧的预解式.

证定义算子 $\mathcal{A}_{1}, \mathcal{A}_{2}, A$ 如下

$\begin{array}{c} D\left(\mathcal{A}_{1}\right)=\left\{\begin{array}{l} (\psi, \varphi, \omega, p, q, r) \in \mathcal{H} \mid(p, q, r) \in W,\left(\rho_{1}^{-1} L_{1}, \rho_{1}^{-1} L_{2}, \rho_{2}^{-1} L_{3}\right) \in H, \\ \psi=\varphi=\omega=0, \text { 在 } \Gamma \text { 上, }\left(B_{1}, B_{2}, B_{3}\right)=(0,0,0), \text { 在 } \Gamma \text { 上. } \end{array}\right\}, \\ \mathcal{A}_{1}(\psi, \varphi, \omega, p, q, r)=\left(p, q, r, \rho_{1}^{-1} L_{1}, \rho_{1}^{-1} L_{2}, \rho_{2}^{-1} L_{3}\right), \end{array}$

和

$\begin{array}{c} D\left(\mathcal{A}_{2}\right)=\left\{\begin{array}{l} (\psi, \varphi, \omega, p, q, r) \in \mathcal{H} \mid(p, q, r) \in W,\left(-\rho_{1}^{-1} \chi_{\Omega_{0}} p,-\rho_{1}^{-1} \chi_{\Omega_{0}} q,-\rho_{2}^{-1} \chi_{\Omega_{0}} r\right) \in H \\ \psi=\varphi=\omega=0 \text { 在 } \Gamma \text { 上. } \end{array}\right\}, \\ \mathcal{A}_{2}(0,0,0, p, q, r)=\left(0,0,0, \rho_{1}^{-1} \chi_{\Omega_{0}} p, \rho_{1}^{-1} \chi_{\Omega_{0}} q, \rho_{2}^{-1} \chi_{\Omega_{0}} r\right), \end{array}$

以及

$D(A)=\left \{ (\psi, \varphi, \omega)\in(H^2(\Omega))^3\vert(\rho_{1}^{-1}L_1, \rho_{1}^{-1}L_2, \rho_{2}^{-1}L_3)\in{H}, \psi=\varphi=\omega=0,\, \ \text{在}\ \Gamma\ \text{上}.\right \},$

$A(\psi, \varphi, \omega)=-(\rho_{1}^{-1}L_1, \rho_{1}^{-1}L_2, \rho_{2}^{-1}L_3),$

因此

$\mathcal{A}=\mathcal{A}_1+\mathcal{A}_2, \mathcal{A}_1=\begin{pmatrix} 0& I\\ -A &0 \end{pmatrix},$

这里 $I$ 是恒等算子.

类似于文献 [9,引理 3.2] 的证明, 容易验证算子 $A$ 是正定自伴随算子, 所以算子 $\mathcal{A}_1$ 是斜自伴的, 即 $1\in{\rho(\mathcal{A}_1)}$ , ( $\rho(\mathcal{A}_1)$ 表示算子 $\mathcal{A}_1$ 的预解集). 假设 $\left \{ X_n \right \}\subset \mathcal{H}$ 有界, 则存在正常数 $M$ , 使得 $\left \| X_n \right \| _\mathcal{H}\le M$ , 令 $Y_n=(I-\mathcal{A}_1)X_n$ , 则

$\left \| (I-\mathcal{A}_1)Y_n \right \| ^{2}_{\mathcal{H}}\le M,$

即

$\begin{align*} \left \| (I-\mathcal{A}_1)Y_n \right \| ^{2}_{\mathcal{H}}\nonumber &=(Y_n-\mathcal{A}_1Y_n, Y_n-\mathcal{A}_1Y_n)_\mathcal{H}\nonumber\\ &=(Y_n-\mathcal{A}_1Y_n, Y_n)_{\mathcal{H}}-(Y_n-\mathcal{A}_1Y_n, \mathcal{A}_1Y_n)_{\mathcal{H}}\nonumber\\ &=(Y_n, Y_n)_{\mathcal{H}}-(\mathcal{A}_1Y_n, Y_n)_{\mathcal{H}}-(Y_n, \mathcal{A}_1Y_n)_\mathcal{H}+(\mathcal{A}_1Y_n, \mathcal{A}_1Y_n)_\mathcal{H}\nonumber\\ &=(Y_n, Y_n)_{\mathcal{H}}+(\mathcal{A}_1Y_n, \mathcal{A}_1Y_n)_\mathcal{H}\nonumber\\ &\le{M}, \nonumber \end{align*}$

根据 Sobolev 嵌入定理知 $W$ 紧嵌入 $\mathcal{H}$ , 故 $\left \{ Y_n\right \}$ 具有收敛子列, 这就证明了 $(I-\mathcal{A}_{1})^{-1}$ 是紧的, 则对于任意的 $\lambda\in{\rho(\mathcal{A}_1)}$ , 当 $\mathrm{Re}\lambda\ge0$ 时, 有

$\left \| (\lambda I-\mathcal{A}_1)^{-1} \right \| \le\frac{1}{\mathrm{Re}\lambda },$

且

$(\lambda I-\mathcal{A}_1)^{-1}=[I-(1-\lambda)(I-\mathcal{A}_1)^{-1}]^{-1}(I-\mathcal{A}_{1})^{-1},$

$\lambda I-\mathcal{A}=(\lambda {I}-\mathcal{A}_1)[I-(\lambda{I}-\mathcal{A}_1)^{-1}\mathcal{A}_2],$

所以, 当 $\mathrm{Re}\lambda\ge3\left \| \mathcal{A}_2 \right \|$ 时, $[I-(\lambda{I}-\mathcal{A}_1)^{-1}\mathcal{A}_2]$ 存在且有界, 这就说明了 $\mathcal{A}$ 的预解式是紧的.

引理 4.2 $\rho(\mathcal{A})\supset \left \{ {\rm i}\lambda \vert\lambda\in{\mathbb{R}} \right \}.$

证若结论不成立, 由于 $0\in{\rho(\mathcal{A})}$ , 则存在 $\lambda\ne0, \lambda\in{\mathbb{R}}$ , 使得 $\rm i\lambda\notin\rho(\mathcal{A})$ , 由引理 4.1 知 $\mathcal{A}$ 只有点谱. 如果 ${\rm i}\lambda\in\sigma_p(\mathcal{A})$ ( $\sigma_p(\mathcal{A})$ 为 $\mathcal{A}$ 的点谱), 则存在 $(\psi, \varphi, \omega, p, q, r)\in{D(\mathcal{A})}$ , 且 $(\psi, \varphi, \omega, p, q, r)\ne0$ , 使得 $({\rm i}\lambda-\mathcal{A})(\psi, \varphi, \omega, p, q, r)=0$ , 即

$\begin{equation} \left\{\begin{array}{lll} {\rm i}\lambda\psi=p,\ {\rm i}\lambda\varphi=q,\ {\rm i}\lambda\omega=r, \\ {\rm i}\lambda{p}=\rho_{1}^{-1}(L_1-\chi_{\Omega_{0}}p), {\rm i}\lambda{q}=\rho_{1}^{-1}(L_2-\chi_{\Omega_{0}}q), {\rm i}\lambda{r}=\rho_{2}^{-1}(L_3-\chi_{\Omega_{0}}r). \end{array}\right. \end{equation}$

(4.1)

由 (3.1) 式有

$\begin{align} 0&=\mathrm{Re}(({\rm i}\lambda-\mathcal{A})(\psi, \varphi, \omega, p, q, r), (\psi, \varphi, \omega, p, q, r))_{\mathcal{H}}\nonumber\\ &=-\mathrm{Re}(\mathcal{A}(\psi, \varphi, \omega, p, q, r), (\psi, \varphi, \omega, p, q, r))_{\mathcal{H}}\nonumber\\ &=\int_{\Omega}(\chi_{\Omega _{0}}|p|^2+\chi_{\Omega _{0}}|q|^2+\chi_{\Omega _{0}}|r|^2){\rm d}x{\rm d}y\nonumber\\ &=\int_{\Omega_{0}}(|p|^2+|q|^2+|r|^2){\rm d}x{\rm d}y, \nonumber \end{align}$

因而在 $L^2(\Omega_0)$ 中 $p=q=r=0$ , 又由 (4.1)式第一式可得, 在 $L^2(\Omega_0)$ 中 $\psi=\varphi=\omega=0$ . 根据常微分方程初值问题的唯一性, 可以得到结论: 在 $L^2(\Omega)$ 中 $(\psi, \varphi, \omega, p, q, r)=0$ . 故得矛盾, 则引理 4.2 得证.

引理 4.3^[16] Hilbert 空间 $\mathcal{H}$ 上的压缩 $C_0$ -半群 ${\rm e}^{t\mathcal{A}}$ 指数稳定的充要条件是

$\begin{equation} \rho(\mathcal{A})\supset {\left \{ {\rm i}{\lambda} \vert\lambda \in{\mathbb{R}} \right \}}, \end{equation}$

(4.2)

$\begin{equation} \mathrm{sup}\left \{ \left \| (\rm i\lambda -\mathcal{A})^{-1} \right \| \vert\lambda \in{\mathbb{R}} \right \}<+\infty. \end{equation}$

(4.3)

定理 4.4 在假设1下, $\mathcal{A}$ 生成压缩 $C_0$ -半群 ${\rm e}^{t\mathcal{A}}$ 指数稳定.

证由引理 4.2, 故只需证明式 (4.3). 下面将采用反证法.

假设式 (4.3) 不成立, 即

$\mathrm{sup}\left \{ \left \| ({\rm i}\lambda -\mathcal{A})^{-1} \right\| \vert\lambda \in{\mathbb{R}} \right \} =+\infty.$

由共鸣性定理和预解式的连续性可知, 存在 $\lambda_n \in{\mathbb{R}}$ 及

$(\psi_n, \varphi_{n}, \omega_n, p_n, q_n, r_n)\subset{D(\mathcal{A})}$ , 使得当 $|\lambda _n|\to\infty$ 时, 有

$\begin{align*} &\left \| (\psi_n, \varphi_{n}, \omega_n, p_n, q_n, r_n) \right \| _{\mathcal{H}}\\ =\ &a(\psi_n, \varphi_{n}, \omega_n;\psi_{n}, \varphi_{n}, \omega_n)+\int_{\Omega}(\rho_1|p_n|^2+\rho_1|q_n|^2+\rho_2|r_n|^2){\rm d}x{\rm d}y=1, \end{align*}$

(4.4)

在 $\mathcal{H}$ 中, 有

$\begin{equation} ({\rm i}\lambda_n-\mathcal{A})(\psi_n, \varphi_{n}, \omega_n, p_n, q_n, r_n)=(h_{1, n}, {h}_{2, n}, h_{3, n}, {h}_{4, n}, {h}_{5, n}, {h}_{6, n})\longrightarrow0, \end{equation}$

(4.5)

其中, 在 $[H^1(\Omega)]^3$ 中, 有

$\begin{equation} {\rm i}\lambda_n(\psi_{n}, \varphi_{n}, \omega_n)-(p_n, q_n, r_n)=(h_{1, n}, {h}_{2, n}, h_{3, n})\longrightarrow0, \end{equation}$

(4.6)

在 $[L^2(\Omega)]^3$ 中, 有

$\begin{align*} &\ {\rm i}\lambda_n(p_n, q_n, r_n)-(\rho_{1}^{-1}(L_1-\chi_{\Omega _{0}}p_n), \rho_{1}^{-1}(L_2-\chi_{\Omega _{0}}q_n), \rho_{2}^{-1}( L_3-\chi_{\Omega _{0}}r_n))\\ &=({h}_{4, n}, {h}_{5, n}, {h}_{6, n})\longrightarrow0, \end{align*}$

(4.7)

且

$\begin{align*} &\ \mathrm{Re}(({\rm i}\lambda_n-\mathcal{A})(\psi_n, \varphi_{n}, \omega_{n}, p_n, q_n, r_n), (\psi_n, \varphi_{n}, \omega_{n}, p_n, q_n, r_n))_\mathcal{H}\\ &=-\mathrm{Re}(\mathcal{A}(\psi_n, \varphi_{n}, \omega_{n}, p_n, q_n, r_n), (\psi_n, \varphi_{n}, \omega_{n}, p_n, q_n, r_n))_\mathcal{H}\\ &=\int_{\Omega_{0}}(|p_n|^2+|q_n|^2+|r_n|^2){\rm d}x{\rm d}y\\ &=o(1), \end{align*}$

(4.8)

因此, 在 $L^2(\Omega_{0})$ 中, 有

$\begin{equation} p_n\to0,\ q_n\to0,\ r_n\to0. \end{equation}$

(4.9)

由于 $W$ 嵌入 $H$ 是稠的, 则在空间 $H$ 中, 对 (4.6) 式用 $(p_n, q_n, r_n)$ 做内积得

$\begin{equation} \int_{\Omega_{0}}{\rm i}\lambda_n(\rho_{1}\psi_n\overline{p}_n, \rho_{1}\varphi_n\overline{q}_n, \rho_{2}\omega_{n}\overline{r}_n)-(\rho_{1}|p_n|^2+\rho_{1}|q_n|^2+\rho_{2}|r_n|^2){\rm d}x{\rm d}y=o(1), \end{equation}$

(4.10)

在 $H$ 中, 对 (4.7)式用 $(\psi_n, \varphi_n, \omega_n)$ 作内积得

$\begin{align*} &\ ((p_n({\rm i}\lambda_n+\rho_{1}^{-1}\chi_{\Omega _{0}})-\rho_{1}^{-1}L_1, q_n({\rm i}\lambda_n+\rho_{1}^{-1}\chi_{\Omega _{0}})-\rho_{1}^{-1}L_2, r_n({\rm i}\lambda_n+\rho_{2}^{-1}\chi_{\Omega _{0}})-\rho_{2}^{-1}L_3), \nonumber\\&\ (\psi_n, \varphi_n, \omega_{n}))_H\nonumber\\&=\int_{\Omega}{\rm i}\lambda_n(\rho_{1}p_n\overline{\psi}_{n}+\rho_{1}q_n\overline{\varphi}_{n}+\rho_{2}r_n\overline{\omega}_{n})+\chi_{\Omega _{0}}(p_n\overline{\psi}_{n}+q_n\overline{\varphi}_{n}+r_n\overline{\omega}_n)\nonumber\\&\quad-(L_1\overline{\psi}_n+L_2\overline{\varphi}_n+L_3\overline{\omega}_n){\rm d}x{\rm d}y\\ &=\int_{\Omega}{\rm i}\lambda_n(\rho_{1}p_n\overline{\psi}_{n}+\rho_{1}q_n\overline{\varphi}_{n}+\rho_{2}r_n\overline{\omega}_{n})-(\rho_{1}p_n\overline{\psi}_{n}+\rho_{1}q_n\overline{\varphi}_{n}+\rho_{2}r_n\overline{\omega}_{n}){\rm d}x{\rm d}y\\ &\quad\ +\int_{\Omega}\chi_{\Omega _{0}}(p_n\overline{\psi}_{n}+q_{n}\overline{\varphi}_{n}+r_n\overline{\omega}_{n}){\rm d}x{\rm d}y\\ &={\rm i}\lambda_n\int_{\Omega}(\rho_{1}p_n\overline{\psi}_{n}+\rho_{1}q_n\overline{\varphi}_{n}+\rho_{2}r_n\overline{\omega}_{n}){\rm d}x{\rm d}y+a(\psi_n, \varphi_{n}, \omega_{n};\psi_n, \varphi_{n}, \omega_{n})\\ &=o(1), \end{align*}$

即

$\begin{equation} {\rm i}\lambda_{n}\int_{\Omega}(\rho_{1}\psi_{n}\overline{p}_n+\rho_{1}\varphi_{n}\overline{q}_n+\rho_{2}\omega\overline{r}_{n}){\rm d}x{\rm d}y+a(\psi_n, \varphi_{n}, \omega_{n};\psi_n, \varphi_{n}, \omega_{n})=o(1), \end{equation}$

(4.11)

故由 (4.10) 和 (4.11)式相加并取实部得

$\begin{equation} a(\psi_n, \varphi_{n}, \omega_{n};\psi_n, \varphi_{n}, \omega_{n})-\int_{\Omega}(\rho_{1}|p_n|^2+\rho_{1}|q_n|^2+\rho_{2}|r_n|^2){\rm d}x{\rm d}y=o(1), \end{equation}$

(4.12)

由 (4.4) 和 (4.12)式可得

$\begin{equation} \lim_{n \to \infty} \int_{\Omega}(\rho_{1}|p_n|^2+\rho_{1}|q_n|^2+\rho_{2}|r_n|^2){\rm d}x{\rm d}y=\frac{1}{2}, \end{equation}$

(4.13)

$\begin{equation} \lim_{n \to \infty} a(\psi_n, \varphi_{n}, \omega_{n};\psi_n, \varphi_{n}, \omega_{n})=\frac{1}{2}. \end{equation}$

(4.14)

由 (4.6) 和 (4.13)式得到

$\begin{equation} \psi_{n}\to0, \varphi_{n}\to0, \omega_{n}\to0, \text{在}\ L^2(\Omega)\text{中}. \end{equation}$

(4.15)

将(4.6)式代入 (4.7)式, 并消去 $(p_n, q_n, r_n)$ 得

$\begin{equation} \rho_{1}\lambda_{n}^{2}\psi_n+L_1=-{\rm i}\lambda_{n}h_{1, n}-{h}_{4, n}, \end{equation}$

(4.16)

同理有

$\begin{equation} \rho_{1}\lambda_{n}^{2}\varphi_n+L_2=-{\rm i}\lambda_{n}h_{2, n}-{h}_{5, n}, \end{equation}$

(4.17)

$\begin{equation} \rho_{2}\lambda_{n}^{2}\omega_n+L_3=-{\rm i}\lambda_{n}h_{3, n}-{h}_{6, n}. \end{equation}$

(4.18)

接下来, 我们将运用分片乘子法证明预期的结果.

取 $h=h(x, y)\in{C^1(\Omega)}$ , $0\le{h}\le{1}$ 且

$h(x, y)=\begin{cases} 1, & \text (x, y)\in{\Omega\setminus \Omega_0}, \\ 0, & \text (x, y)\in{\overline{\Omega}_0. } \end{cases}$

用 $h(x, y){m}(x, y)\cdot\nabla \overline{\psi}_n$ 乘以 (4.16)式, 并在 $\Omega$ 上积分得

$\begin{equation} \int_{\Omega}h{m}(x, y)\cdot\nabla \overline{\psi}_n(\rho_{1}\lambda_{n}^{2}\psi_n+L_1){\rm d}x{\rm d}y=o(1), \end{equation}$

(4.19)

同理有

$\begin{equation} \int_{\Omega}h{m}(x, y)\cdot\nabla \overline{\varphi}_n(\rho_{1}\lambda_{n}^{2}\varphi_n+L_2){\rm d}x{\rm d}y=o(1), \end{equation}$

(4.20)

$\begin{equation} \int_{\Omega}h{m}(x, y)\cdot\nabla \overline{\omega}_n(\rho_{2}\lambda_{n}^{2}\omega_n+L_3){\rm d}x{\rm d}y=o(1). \end{equation}$

(4.21)

为了书写方便起见, 置

$\begin{equation} \int_{\Omega}\rho_{1}\lambda_{n}^{2}h{m}(x, y)\cdot\nabla \overline{\psi}_n\psi_{n}{\rm d}x{\rm d}y=I_1, \end{equation}$

(4.22)

$\begin{equation} \int_{\Omega}L_{1}h{m}(x, y)\cdot\nabla \overline{\psi}_n{\rm d}x{\rm d}y=I_2, \end{equation}$

(4.23)

$\begin{equation} \int_{\Omega}\rho_{1}\lambda_{n}^{2}h{m}(x, y)\cdot\nabla \overline{\varphi}_n\varphi_{n}{\rm d}x{\rm d}y=I_1^{'}, \end{equation}$

(4.24)

$\begin{equation} \int_{\Omega}L_{2}h{m}(x, y)\cdot\nabla \overline{\varphi}_n{\rm d}x{\rm d}y=I_2^{'}, \end{equation}$

(4.25)

$\begin{equation} \int_{\Omega}\rho_{2}\lambda_{n}^{2}h{m}(x, y)\cdot\nabla\overline{\omega}_n\omega_{n}{\rm d}x{\rm d}y=I_1^{''}, \end{equation}$

(4.26)

$\begin{equation} \int_{\Omega}L_{3}h{m}(x, y)\cdot\nabla \overline{\omega}_n{\rm d}x{\rm d}y=I_2^{''}, \end{equation}$

(4.27)

其中

$\begin{align*} I_1={}&-\int_{\Omega\setminus\Omega_{0}}\mathrm{div}(\rho_{1}\lambda_{n}^{2}\psi_{n}{m})\overline{\psi}_{n}{\rm d}x{\rm d}y+\int_{\Gamma_0}\rho_{1}\lambda_{n}^{2}|\psi_{n}|^{2}{m}\cdot{\nu_0}{\rm d}S+\int_{\Gamma}\rho_{1}\lambda_{n}^{2}|\psi_{n}|^{2}{m}\cdot{\nu}{\rm d}S\nonumber\\ ={}&-\int_{\Omega\setminus\Omega_{0}}[\rho_{1}\lambda_{n}^{2}|\psi_n|^2\mathrm{div}{m}+{m}\cdot\nabla (\rho_{1}\lambda_{n}^{2}\psi_n)\overline{\psi}_{n}]{\rm d}x{\rm d}y\\ &+\int_{\Gamma_0}\rho_{1}\lambda_{n}^{2}|\psi_{n}|^{2}{m}\cdot{\nu_{0}}{\rm d}S+\int_{\Gamma}\rho_{1}\lambda_{n}^{2}|\psi_{n}|^{2}{m}\cdot{\nu}{\rm d}S\\ ={}&-\rho_{1}\int_{\Omega\setminus\Omega_{0}}(|\lambda_{n}\psi_{n}|^{2}\mathrm{div}{m}+\lambda_{n}^{2}\overline{\psi}_{n}{m}\cdot\nabla \psi_{n}){\rm d}x{\rm d}y\\ &+\rho_{1}\int_{\Gamma_0}|\lambda_{n}\psi_{n}|^{2}{m}\cdot{\nu_0}{\rm d}S+\rho_{1}\int_{\Gamma}|\lambda_{n}\psi_{n}|^{2}{m}\cdot{\nu}{\rm d}S\\ ={}&\rho_{1}\int_{\Omega\setminus\Omega_{0}}(|p_{n}|^{2}\mathrm{div}{m}-\lambda_{n}^{2}\overline{\psi}_n{m}\cdot\nabla \psi_{n}){\rm d}x{\rm d}y-\rho_{1}\int_{\Gamma_0}|p_{n}|^{2}{m}\cdot{\nu_0}{\rm d}S-\rho_{1}\int_{\Gamma}|p_{n}|^{2}{m}\cdot{\nu}{\rm d}S\\ ={}&\rho_{1}\int_{\Omega\setminus\Omega_{0}}(|p_{n}|^{2}\mathrm{div}{m}-\lambda_{n}^{2}\overline{\psi}_{n}{m}\cdot\nabla \psi_{n}){\rm d}x{\rm d}y-\rho_{1}\int_{\Gamma_0}|p_{n}|^{2}{m}\cdot{\nu_0}{\rm d}S, \end{align*}$

即

$\begin{align*} &\int_{\Omega\setminus\Omega_{0}}\rho_{1}\lambda_{n}^{2}{m}\cdot\nabla\overline{\psi}_n\psi_{n}{\rm d}x{\rm d}y+\int_{\Omega\setminus\Omega_{0}}\rho_{1}\lambda_{n}^{2}{m}\cdot\nabla\psi_{n} \overline{\psi}_{n}{\rm d}x{\rm d}y\nonumber\\ =\ &\rho_{1}\int_{\Omega\setminus\Omega_{0}}|p_{n}|^{2}\mathrm{div}{m}{\rm d}x{\rm d}y-\rho_{1}\int_{\Gamma_0}|p_{n}|^{2}{m}\cdot{\nu_0}{\rm d}S, \end{align*}$

两边取实部得

$\begin{equation} \mathrm{Re}I_{1}=\frac{1}{2}\rho_{1}\bigg(\int_{\Omega\setminus\Omega_{0}}|p_{n}|^{2}\mathrm{div}{m}{\rm d}x{\rm d}y-\int_{\Gamma_0}|p_{n}|^{2}{m}\cdot{\nu_0}{\rm d}S\bigg), \end{equation}$

(4.28)

同理, 对 (4.24) 和 (4.26)式采用相同的方法, 有

$\begin{equation} \mathrm{Re}I_{1}^{'}=\frac{1}{2}\rho_{1}\bigg(\int_{\Omega\setminus\Omega_{0}}|q_{n}|^{2}\mathrm{div}{m}{\rm d}x{\rm d}y-\int_{\Gamma_0}|q_{n}|^{2}{m}\cdot{\nu_0}{\rm d}S\bigg), \end{equation}$

(4.29)

$\begin{equation} \mathrm{Re}I_{1}^{''}=\frac{1}{2}\rho_{2}\bigg(\int_{\Omega\setminus\Omega_{0}}|r_{n}|^{2}\mathrm{div}{m}{\rm d}x{\rm d}y-\int_{\Gamma_0}|r_{n}|^{2}{m}\cdot{\nu_0}{\rm d}S\bigg), \end{equation}$

(4.30)

由 (4.28), (4.29) 和 (4.30)式得

$\begin{align*} \mathrm{Re}(I_1+I_{1}^{'}+I_{1}^{''})=&\frac{1}{2}\bigg\{ \int_{\Omega\setminus\Omega_{0}}[(\rho_{1}|p|^2+\rho_{1}|q|^2+\rho_{2}|r|^2)\mathrm{div}{m}]{\rm d}x{\rm d}y \\ & -\int_{\Gamma_0}[(\rho_{1}|p|^2+\rho_{1}|q|^2+\rho_{2}|r|^2){m}\cdot{\nu_0}]{\rm d}S \bigg \}. \end{align*}$

(4.31)

另一方面, 对 (4.23) 式应用 Green 公式

$\begin{align*} I_{2}={}&\int_{\Omega\setminus\Omega_{0}}D\mathrm{div}(\psi_{n, x}+\mu\varphi_{n, y}, \frac{1-\mu}{2}(\psi_{n, y}+\varphi_{n, x}))h{m}\cdot\nabla \overline{\psi}_{n}{\rm d}x{\rm d}y\\ &-\int_{\Omega\setminus\Omega_{0}}K(\psi_{n}+\omega_{n, x})\cdot\nabla (h{m}\cdot\nabla \overline{\psi}_n){\rm d}x{\rm d}y\\ ={}&-\int_{\Omega\setminus\Omega_{0}}D(\psi_{n, x}, +\mu\varphi_{n, y}, \frac{1-\mu}{2}(\psi_{n, y}+\varphi_{n, x}))\nabla (h{m}\cdot\nabla \overline{\psi}_{n}){\rm d}x{\rm d}y\\ &+\int_{\Gamma_0}Dh({m}\cdot\nabla \overline{\psi}_{n})[(\psi_{n, x}+\mu\varphi_{n, y}, \frac{1-\mu}{2}(\psi_{n, y}+\varphi_{n, x}))\cdot{\nu_0}]{\rm d}S\\ &+\int_{\Gamma}Dh({m}\cdot\nabla \overline{\psi}_{n})[(\psi_{n, x}+\mu\varphi_{n, y}, \frac{1-\mu}{2}(\psi_{n, y}+\varphi_{n, x}))\cdot{\nu}]{\rm d}S\\ &-\int_{\Omega\setminus\Omega_{0}}K(\psi_{n}+\omega_{n, x})(h{m}\cdot\nabla \overline{\psi}_{n}){\rm d}x{\rm d}y, \end{align*}$

(4.32)

由边界条件以及函数 $h$ 的定义, 则

$\begin{align*} I_{2}={}&-\int_{\Omega\setminus\Omega_{0}}D(\psi_{n, x}+\mu\varphi_{n, y}, \frac{1-\mu}{2}(\psi_{n, y}+\varphi_{n, x}))\cdot\nabla ({m}\cdot\nabla \overline{\psi}_{n}){\rm d}x{\rm d}y\\ &-\int_{\Omega\setminus\Omega_{0}}K(\psi_{n}+\omega_{n, x})({m}\cdot\nabla \overline{\psi}_{n}){\rm d}x{\rm d}y, \end{align*}$

(4.33)

同理

$\begin{align*} I_{2}^{'}={}&-\int_{\Omega\setminus\Omega_{0}}D(\frac{1-\mu}{2}(\psi_{n, y}+\varphi_{n, x}), \varphi_{n, y}+\mu\psi_{n, x})\cdot\nabla ({m}\cdot\nabla \overline{\varphi}_{n}){\rm d}x{\rm d}y\\ &-\int_{\Omega\setminus\Omega_{0}}K(\varphi_{n}+\omega_{n, y})({m}\cdot\nabla \overline{\varphi}_{n}){\rm d}x{\rm d}y, \end{align*}$

(4.34)

以及

$\begin{equation} I_{2}^{''}=-K\int_{\Omega\setminus\Omega_{0}}(\psi_{n}+\omega_{n, x}, \varphi_{n}+\omega_{n, y})\cdot\nabla ({m}\cdot\nabla \overline{\omega}_{n}){\rm d}x{\rm d}y, \end{equation}$

(4.35)

所以

$\begin{align*} I_2+I_2^{'}+I_{2}^{''}={}&\int_{\Omega}(L_1h{m}\cdot\nabla \overline{\psi}_{n}+L_2h{m}\cdot\nabla\overline{\varphi}_{n}+L_3h{m}\cdot\nabla\overline{\omega}_{n}){\rm d}x{\rm d}y\\ ={}&-\int_{\Omega\setminus\Omega_{0}}D(\psi_{n, x}+\mu\varphi_{n, y}, \frac{1-\mu}{2}(\psi_{n, y}+\varphi_{n, x}))\cdot\nabla({m}\cdot\nabla\overline{\psi}_{n}){\rm d}x{\rm d}y\\ &-\int_{\Omega\setminus\Omega_{0}}D(\frac{1-\mu}{2}(\psi_{n, y}+\varphi_{n, x}), \varphi_{n, y}+\mu\psi_{n, x})\cdot\nabla ({m}\cdot\nabla \overline{\varphi}_{n}){\rm d}x{\rm d}y\\ &-\int_{\Omega\setminus\Omega_{0}}K[(\psi_{n}+\omega_{n, x})({m}\cdot\nabla\overline{\psi}_{n})+(\psi_{n}+\omega_{n, y})({m}\cdot\nabla\overline{\varphi}_{n})]{\rm d}x{\rm d}y\\ &-\int_{\Omega\setminus\Omega_{0}}K(\psi_{n}+\omega_{n, x}, \varphi_{n}+\omega_{n, y})\cdot\nabla({m}\cdot\nabla\overline{\omega}_{n}){\rm d}x{\rm d}y\\ ={}&-a(\psi_{n}, \varphi_{n}, \omega_{n};h{m}\cdot\nabla\psi_{n}, h{m}\cdot\nabla\varphi_{n}, h{m}\cdot\nabla\omega_{n}). \end{align*}$

(4.36)

其中

$\begin{align*} &-a(\psi_{n}, \varphi_{n}, \omega_{n};h{m}\cdot\nabla\psi_{n}, h{m}\cdot\nabla\varphi_{n}, h{m}\cdot\nabla\omega_{n})\nonumber\\ =&-\int_{\Omega} \Bigg\{Dh\Bigg[\nabla \overline{\psi}_n\begin{pmatrix} m_{1, x} &m_{1, y} \\ m_{2, x} &m_{2, y} \end{pmatrix}\begin{pmatrix} \psi_{n, x}+\mu\varphi_{n, y}\\\frac{1-\mu}{2}(\psi_{n, y}+\varphi_{n, x}) \end{pmatrix}\\&+\nabla \overline{\varphi}_{n}\begin{pmatrix} m_{1, x} &m_{1, y} \\ m_{2, x} &m_{2, y} \end{pmatrix}\begin{pmatrix}\frac{1-\mu}{2}(\psi_{n, y}+\varphi_{n, x})\\ \mu\psi_{n, x}+\varphi_{n, y} \end{pmatrix}\Bigg]{\rm d}x{\rm d}y \\ &+Kh\nabla \overline{\omega}_n\begin{pmatrix} m_{1, x} &m_{1, y} \\ m_{2, x} &m_{2, y} \end{pmatrix}\begin{pmatrix} \psi_{n}+\omega_{n, x}\\ \varphi_{n}+\omega_{n, y} \end{pmatrix}\Bigg\}{\rm d}x{\rm d}y\\ &-\int_{\Omega}\left[Dh(m_1, m_{2})\begin{pmatrix} \overline{\psi}_{n, xx} &\overline{\psi}_{n, xy} \\ \overline{\psi}_{n, xy} &\overline{\psi}_{n, yy} \end{pmatrix}\begin{pmatrix} \psi_{n, x}+\mu\varphi_{n, y}\\ \frac{1-\mu}{2}(\psi_{n, y}+\varphi_{n, x}) \end{pmatrix}\right]{\rm d}x{\rm d}y\\ &-\int_{\Omega}\left[Dh(m_1, m_{2})\begin{pmatrix} \overline{\varphi}_{n, xx} &\overline{\varphi}_{n, xy} \\ \overline{\varphi}_{n, xy} &\overline{\varphi}_{n, yy} \end{pmatrix}\begin{pmatrix} \frac{1-\mu}{2}(\psi_{n, y}+\varphi_{n, x})\\ \mu\psi_{n, x}+\varphi_{n, y} \end{pmatrix}\right]{\rm d}x{\rm d}y\\ &-K\int_{\Omega}(\psi_{n}+\omega_{n, x})(h{m}\cdot\nabla\overline{\psi}_{n}){\rm d}x{\rm d}y-K\int_{\Omega}(\varphi_{n}+\omega_{n, y})(h{m}\cdot\nabla\overline{\varphi}_{n}){\rm d}x{\rm d}y\\ &-K\int_{\Omega}h(m_1, m_2)\begin{pmatrix} \overline{\omega}_{n, xx} &\overline{\omega}_{n, xy} \\ \overline{\omega}_{n, xy} &\overline{\omega}_{n, yy} \end{pmatrix}\begin{pmatrix} \psi_n+\omega_{n, x}\\ \varphi_n+\omega_{n, y} \end{pmatrix}{\rm d}x{\rm d}y. \end{align*}$

将 $I_2+I_2^{'}+I_{2}^{''}$ 取实部得

$\begin{align*}\mathrm{Re}(I_2+I_2^{'}+I_{2}^{''})={}&\frac{1}{2}\mathrm{Re}[(I_2+I_2^{'}+I_{2}^{''})+(\overline{I}_2+\overline{I_2^{'}}+\overline{I_{2}^{''}})]\\ ={}&\frac{1}{2}\mathrm{Re}[a(h{m}\cdot\nabla\psi_{n}, h{m}\cdot\nabla\varphi_{n}, h{m}\cdot\nabla\omega_{n};\psi_{n}, \varphi_{n}, \omega_{n})\\ &-a(\psi_{n}, \varphi_{n}, \omega_{n};h{m}\cdot\nabla\psi_{n}, h{m}\cdot\nabla\varphi_{n}, h{m}\cdot\nabla\omega_{n})]\\ ={}&0, \end{align*}$

(4.37)

结合 (4.28)-(4.31) 和 (4.37) 式, 我们有

$\begin{align*} o(1)={}&\frac{1}{2}\bigg \{ \int_{\Omega\setminus\Omega_{0}}\mathrm{div}{m}(\rho_{1}|p_n|^2+\rho_{1}|q_n|^2+\rho_{2}|r_n|^2){\rm d}x{\rm d}y \\ & -\int_{\Gamma_0}(\rho_{1}|p_n|^2+\rho_{1}|q_n|^2+\rho_{2}|r_n|^2){m}\cdot{\nu_0}{\rm d}S\bigg\}, \end{align*}$

(4.38)

根据几何光学条件 ${m}\cdot{\nu_0}\le0,$ 有

$\begin{equation} p_n\to0, q_n\to0, r_n\to0,\ \text{在}\ L^2(\Omega \setminus \Omega _0)\ \text{中}, \end{equation}$

(4.39)

故由 (4.9) 与 (4.39) 式推得

$\lim_{n \to \infty} \int_{\Omega}(\rho_{1}|p_n|^2+\rho_{1}|q_n|^2+\rho_{2}|r_n|^2){\rm d}x{\rm d}y=0.$

与 (4.13) 式矛盾, 从而完成定理 4.4 证明.

参考文献

原文顺序

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1989

... 近几年来, Mindlin-Timoshenko 系统的研究取得了许多成果. 文献 [1] 研究了具有边界阻尼的二维 Mindlin-Timoshenko 板系统的指数稳定性, Nicaise^[2] 研究了具有全局阻尼的 Mindlin-Timoshenko 系统的指数衰减性, 进一步, 当只有阻尼作用时, 该系统是多项式衰减的. 文献 [3⇓⇓⇓⇓-8] 研究了具有各种阻尼的 Mindlin-Timoshenko 系统或 Mindlin-Timoshenko 热弹性板系统的稳定性问题. ...

Internal stabilization of a Mindlin-Timoshenko model by interior feedbacks

2011

... 本文将文献 [2] 推广到更一般情形, 更精确地说, 考虑如下的具有局部内部阻尼的二维 Mindlin-Timoshenko 板方程的初边值问题 ...

Asymptotic behavior of a Mindlin-Timoshenko plate with viscoelastic dissipation on the boundary

2003

Stabilization of a thermoelastic Mindlin-Timoshenko plates model revisited

2013

Polynomial decay rate of a thermoelastic Mindlin-Timoshenko plates model with Dirichlet boundary conditions

2015

Indirect stabilization of a Mindlin-Timoshenko plates

2017

二维 Mindlin-Timoshenko 板系统的稳定性与最优性

2021

二维 Mindlin-Timoshenko 板系统的稳定性与最优性

2021

Stabilization of 2-d Mindlin-Timoshenko plates with localized acoustic boundary feedback

2022

Locally distributed control and damping for the conservative systems

1997

... 在工程技术应用中, 利用局部分布阻尼器对弹性振动系统进行抑振更为普遍. 因此, 分片乘子法作为有效的手段来解决问题. 文献 [9⇓⇓⇓-13] 的作者应用分片乘子法和应用 Liyapunov 函数法研究了弹性系统和粘弹性系统的稳定性. ...

... 类似于文献 [9,引理 3.2] 的证明, 容易验证算子

$A$

是正定自伴随算子, 所以算子

$\mathcal{A}_1$

是斜自伴的, 即

$1\in{\rho(\mathcal{A}_1)}$

, (

$\rho(\mathcal{A}_1)$

表示算子

$\mathcal{A}_1$

的预解集). 假设

$\left \{ X_n \right \}\subset \mathcal{H}$

有界, 则存在正常数

$M$

, 使得

$\left \| X_n \right \| _\mathcal{H}\le M$

, 令

$Y_n=(I-\mathcal{A}_1)X_n$

, 则 ...

A new method to obtain decay rate estimates for dissipative systems

1999

Exponential decay of energy of the Euler-Bernoulli beam with locally distributed Kelvin-Voigt damping

1998

Piecewise multiplier method and nonlinear integral inequalities for Petrowsky equation with nonlinear dissipation

2006

Stabilization of transmission system of Kirchhoff plates and wave equations with a localized Kelvin-Voigt damping

2021

2012

... 本文应用线性算子半群理论、分片乘子法以及 Hilbert 空间中线性系统指数稳定的频域结果, 得到了系统的适定性以及指数稳定性. 本文的安排如下: 在第 2 节中, 通过适当的假设将系统转换为抽象 Cauchy 问题. 在第 3 节中, 应用 Pazy^[14] 的 Lumer-Phillips 定理证明了系统的适定性. 在第 4 节中, 运用分片乘子法以及系统稳定性的频域结果结合矛盾讨论, 证明了系统是指数稳定的. ...

1975

... 其中

$H^1(\Omega)$

是一阶 Sobolev 空间^[15], 赋予范数 ...

Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces

1985

... 引理 4.3^[16] Hilbert 空间

$\mathcal{H}$

上的压缩

$C_0$

-半群

${\rm e}^{t\mathcal{A}}$

指数稳定的充要条件是 ...

〈

〉