具有局部阻尼的二维 Mindlin-Timoshenko 板系统的镇定

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

 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)

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

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.

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

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

2 预备知识

\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*}

\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*}

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

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

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

\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*}

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

\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*}

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

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

\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*}

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

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

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

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

$(\psi_{01}, \varphi_{01}, \omega_{01}, \psi_{02}, \varphi_{02}, \omega_{02})\in{D(\mathcal{A})}$, 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

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

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

“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

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

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

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

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

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

\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*}

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

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

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

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

$(\psi_n, \varphi_{n}, \omega_n, p_n, q_n, r_n)\subset{D(\mathcal{A})}$, 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*}

$\mathcal{H}$ 中, 有

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

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

$[L^2(\Omega)]^3$ 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*}

\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*}

$$$p_n\to0,\ q_n\to0,\ r_n\to0.$$$

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

$H$ 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*}

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

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

$$$\lim_{n \to \infty} \int_{\Omega}(\rho_{1}|p_n|^2+\rho_{1}|q_n|^2+\rho_{2}|r_n|^2){\rm d}x{\rm d}y=\frac{1}{2},$$$
$$$\lim_{n \to \infty} a(\psi_n, \varphi_{n}, \omega_{n};\psi_n, \varphi_{n}, \omega_{n})=\frac{1}{2}.$$$

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

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

$$$\rho_{1}\lambda_{n}^{2}\varphi_n+L_2=-{\rm i}\lambda_{n}h_{2, n}-{h}_{5, n},$$$
$$$\rho_{2}\lambda_{n}^{2}\omega_n+L_3=-{\rm i}\lambda_{n}h_{3, n}-{h}_{6, n}.$$$

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

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

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

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

$$$\int_{\Omega}h{m}(x, y)\cdot\nabla \overline{\varphi}_n(\rho_{1}\lambda_{n}^{2}\varphi_n+L_2){\rm d}x{\rm d}y=o(1),$$$
$$$\int_{\Omega}h{m}(x, y)\cdot\nabla \overline{\omega}_n(\rho_{2}\lambda_{n}^{2}\omega_n+L_3){\rm d}x{\rm d}y=o(1).$$$

$$$\int_{\Omega}\rho_{1}\lambda_{n}^{2}h{m}(x, y)\cdot\nabla \overline{\psi}_n\psi_{n}{\rm d}x{\rm d}y=I_1,$$$
$$$\int_{\Omega}L_{1}h{m}(x, y)\cdot\nabla \overline{\psi}_n{\rm d}x{\rm d}y=I_2,$$$
$$$\int_{\Omega}\rho_{1}\lambda_{n}^{2}h{m}(x, y)\cdot\nabla \overline{\varphi}_n\varphi_{n}{\rm d}x{\rm d}y=I_1^{'},$$$
$$$\int_{\Omega}L_{2}h{m}(x, y)\cdot\nabla \overline{\varphi}_n{\rm d}x{\rm d}y=I_2^{'},$$$
$$$\int_{\Omega}\rho_{2}\lambda_{n}^{2}h{m}(x, y)\cdot\nabla\overline{\omega}_n\omega_{n}{\rm d}x{\rm d}y=I_1^{''},$$$
$$$\int_{\Omega}L_{3}h{m}(x, y)\cdot\nabla \overline{\omega}_n{\rm d}x{\rm d}y=I_2^{''},$$$

\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*}

\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*}

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

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

\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*}

\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*}

\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*}

\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*}

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

\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*}

\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*}

$I_2+I_2^{'}+I_{2}^{''}$ 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*}

\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*}

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

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

参考文献 原文顺序 文献年度倒序 文中引用次数倒序 被引期刊影响因子

Lagnese J E. Boundary Stabilization of Thin Plates. Philadelhia, PA: Society for Industrial and Applied Mathematics, 1989

Nicaise S.

Internal stabilization of a Mindlin-Timoshenko model by interior feedbacks

Mathematical Control and Related Fields, 2011, 1(3): 331-352

Rivera J E M, Oquendo H P O.

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

Funkcialaj Ekvacioj, 2003, 46(3): 363-382

Dalsen G V.

Stabilization of a thermoelastic Mindlin-Timoshenko plates model revisited

Zeitschrift fur angewandte Mathematik und Physik, 2013, 64(4): 1305-1325

Dalsen G V.

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

Zeitschrift fur angewandte Mathematik und Physik, 2015, 66(1): 113-128

Louis T.

Indirect stabilization of a Mindlin-Timoshenko plates

Journal of Mathematical Analysis and Applications, 2017, 449(2): 1880-1891

Zhang C G, Fu Y Z, Liu Y B.

Stability and optimality of 2-D Mindlin-Timoshenko plate system

Acta Math Sci, 2021, 41A(5): 1465-1491

Liu Y, Zhang C, Chen T.

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

Journal of Industrial and Management Optimization, 2022, 18(2): 1009-1034

Liu K.

Locally distributed control and damping for the conservative systems

SIAM Journal on Control and Optimization, 1997, 35(5): 1574-1590

Martinez P.

A new method to obtain decay rate estimates for dissipative systems

ESAIM: Control, Optimisation and Calculus of Variations, 1999, 4: 419-444

Liu K, Liu Z.

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

SIAM Journal on Control and Optimization, 1998, 36(3): 1086-1098

Fatiha A B.

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

Journal of Evolution Equations, 2006, 6(1): 95-112

Hong G, Hong H.

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

Journal of Evolution Equations, 2021, 20: 2239-2264

Pazy A. Semigroups of Linear Operators and Applications to Partial Differential Equations. New York: Springer Science and Business Media, 2012