数学物理学报, 2023, 43(2): 355-376

带分数阶磁效应的压电梁在有/无热效应时的稳定性

安雁宁1, 刘文军,1,2,3,*, 孔奥文1

1南京信息工程大学数学与统计学院 南京 210044

2南京信息工程大学江苏省应用数学中心 南京210044

3南京信息工程大学江苏省系统建模与数据分析国际合作联合实验室 南京210044

Stability of Piezoelectric Beams with Magnetic Effects of Fractional Derivative Type and with/without Thermal Effects

An Yanning1, Liu Wenjun,1,2,3,*, Kong Aowen1

1School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044

2Center for Applied Mathematics of Jiangsu Province, Nanjing University of Information Science and Technology, Nanjing 210044

3Jiangsu International Joint Laboratory on System Modeling and Data Analysis, Nanjing University of Information Science and Technology, Nanjing 210044

通讯作者: *刘文军,E-mail: wjliu@nuist.edu.cn

收稿日期: 2021-09-29   修回日期: 2022-10-17  

基金资助: 国家自然科学基金(12271261)
江苏省重点研发计划(社会发展)(BE2019725)
江苏省青蓝工程项目和江苏省研究生科研与实践创新计划(KYCX22_1125)

Received: 2021-09-29   Revised: 2022-10-17  

Fund supported: National Natural Science Foundation of China(12271261)
Key Research and Development Program of Jiangsu Province (Social Development)(BE2019725)
Qing Lan Project of Jiangsu Province and the Postgraduate Research and Practice Innovation Program of Jiangsu Province(KYCX22_1125)

摘要

该文考虑了具有分数阶磁效应的一维压电梁系统的适定性及稳定性. 首先, 通过引入新函数将原系统转换为不含分数阶边界项的等价系统, 并利用Lumer-Philips定理证明了该系统的适定性. 然后, 基于谱分析证得无热效应的压电梁系统的非指数稳定性, 并借助Borichev-Tomilov定理[33]进一步推得系统是多项式稳定的. 此外, 该文又讨论了有热效应的压电梁系统的适定性, 并借助扰动泛函方法证明了压电梁系统在带有热效应时的指数稳定性.

关键词: 压电梁; 渐近行为; 分数阶导数; 半群理论

Abstract

In this paper, we consider the well-posedness and asymptotic behavior of a one-dimensional piezoelectric beam system with control boundary conditions of fractional derivative type, which represent magnetic effects on the system. By introducing two new matrixs to deal with control boundary conditions of fractional derivative type, we obtain a new equivalent system, so as to show the well-posedness of the system by using Lumer-Philips theorem. We then prove the lack of exponential stability by a spectral analysis, and obtain the polynomial stability of the system without thermal effects by using a result of Borichev and Tomilov (Math. Ann. 347 (2010), 455-478). To find a more stable system, we then consider the stability of the above system with thermal effects described by Fourier's law, and achieve the exponential stability for the system by using the perturbed functional method.

Keywords: Piezoelectric beams; Asymptotic behavior; Fractional derivative; Semigroup method

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本文引用格式

安雁宁, 刘文军, 孔奥文. 带分数阶磁效应的压电梁在有/无热效应时的稳定性[J]. 数学物理学报, 2023, 43(2): 355-376

An Yanning, Liu Wenjun, Kong Aowen. Stability of Piezoelectric Beams with Magnetic Effects of Fractional Derivative Type and with/without Thermal Effects[J]. Acta Mathematica Scientia, 2023, 43(2): 355-376

1 引言

压电材料是指在压力作用下能够产生电能的一类材料. 近年来, 压电材料已被广泛应用到智能器件的设计中, 最新的前沿应用有: 心脏起搏器[1]、纳米定位器[2]、健康监测仪[3]和能量收集器[4]等. 压电材料用以制造电子装置时, 机械扰动将以电的形式产生响应并附带产生磁效应. 换句话说, 具有压电梁的电子器件综合了机械、电以及磁三种效应的作用.

因此, 描述这三种效应的相互作用对于理解压电系统的稳定性是非常重要的[5]. 具有磁效应的压电梁模型是由Maxwell方程和Mindlin-Timoshenko板方程建立的, 其中Maxwell方程用于描述电磁耦合, Mindlin-Timoshenko理论则用来描述梁的力学行为[6,7].

Morris和Özer[8,9]首次考虑了综合三种效应的压电梁模型

$\begin{matrix} \label{other1.2}\begin{array}{ll}\rho v_{tt}-\alpha v_{xx}+\gamma\beta p_{xx}=0,&(x,t)\in(0,L)\times (0,T),\\\mu p_{tt}-\beta p_{xx}+\gamma\beta v_{xx}=0,&(x,t)\in(0,L)\times (0,T),\end{array}\end{matrix}$

其边界条件为

$\begin{matrix} \label{other1.2b}\begin{array}{ll}v(0,t)=0,\quad\alpha v_{x}(L,t)-\gamma\beta p_{x}(L,t)=0,& t\in (0,T),\\p(0,t)=0,\quad\beta p_{x}(L,t)-\gamma\beta v_{x}(L,t)+V(t)=0, & t\in (0,T).\end{array}\end{matrix}$

这里 $\rho$, $\alpha$, $\beta$, $\gamma$, $\mu$ 分别表示单位体积的质量密度、弹性刚度、梁的抗渗系数、压电系数和磁导率. 其中 $\alpha=\alpha_{1}+\gamma^{2}\beta$, $\alpha_{1}>0$ 表示由静力法和准静力法推导的Euler-Bernoulli小位移模型的弹性刚度[9]. $v(x,t)$$p(x,t)$ 分别表示梁在点$x$处的横向位移和沿横向电位移的总负荷, $V(t)=\frac{p_{t}(L,t)}{h}$ 是电子束电极上的电压. 他们证明了仅含一个边界控制项的系统(1.1)-(1.2)是非指数稳定的. 同时, 他们还得到了一个精确的边界可观测不等式.

Ramos等[10]考虑了带有磁效应的压电梁系统

$\begin{matrix} \label{other1.3} \begin{array}{ll} \rho v_{tt}-\alpha v_{xx}+\gamma\beta p_{xx}=0,&(x,t)\in(0,L)\times (0,T),\\ \mu p_{tt}-\beta p_{xx}+\gamma\beta v_{xx}=0,&(x,t)\in(0,L)\times (0,T), \end{array} \end{matrix}$

其边界条件为

$\begin{matrix} \label{other1.3b} \begin{array}{ll} v(0,t)=0,\quad \alpha v_{x}(L,t)-\gamma\beta p_{x}(L,t)+\frac{\xi_{1}}{h}v_{t}(L,t)=0,& t\in (0,T),\\ [3mm] p(0,t)=0,\quad \beta p_{x}(L,t)-\gamma\beta v_{x}(L,t)+\frac{\xi_{2}}{h}p_{t}(L,t)=0,& t\in (0,T). \end{array} \end{matrix}$

这里$\xi_{i}$$(i=1,2)$ 表示正的反馈增益常数. 作者首先将原始系统分为保守系统和辅助系统, 并利用乘子法证明了保守系统的边界可观测不等式. 之后, 他们证明了保守系统的可观测不等式与原系统的指数稳定性是等价的, 从而得到了系统(1.3)-(1.4)的指数稳定性.

最近, 一些研究者考虑了Timoshenko系统或混合系统在带有分数阶导数型控制边界条件下的稳定性[11-16]. 关于时间的分数阶导数项可视为系统的耗散项[17], 且具有很好的鲁棒性, 因此已被广泛应用于生物工程、电路、信号处理、化工过程和控制系统等领域[18]. 本文中, 分数阶导数型耗散项描述了一种磁效应控制器, 旨在提供系统稳定所需的边界耗散.

本文考虑$a\in(0,1)$阶的Caputo指数型分数阶导数 $\partial^{a,\eta}_{t}$, 定义为

$\begin{matrix}\partial^{a,\eta}_{t}f(t)=\frac{1}{\Gamma(1-a)}\int^{t}_{0}e^{-\eta(t-\tau)}(t-\tau)^{-a}\frac{\rm d}{{\rm d}\tau}f(\tau){\rm d}\tau,\end{matrix}$

这里$f\in L^{1}(I)$, $t>0$. 分数阶积分$I^{a,\eta}$是分数阶微分$\partial^{a,\eta}_{t}$的逆运算, 定义为

$\begin{matrix}I^{a,\eta}f(t)=\frac{1}{\Gamma(a)}\int^{t}_{0}e^{-\eta(t-\tau)}(t-\tau)^{a-1}f(\tau){\rm d}\tau,\end{matrix}$

这里$f\in W^{1,1}(I)$, $t>0$. 由定义可知$\partial^{a,\eta}_{t}f(t)=I^{1-a,\eta}f'(t)$.

本文的前一部分将考虑不带热效应的压电系统

$\begin{matrix} \label{problem1.1}\begin{array}{ll}\rho V_{tt}-\alpha V_{xx}+\gamma\beta P_{xx}=0, &(x,t)\in(0,L)\times (0,T),\\\mu P_{tt}-\beta P_{xx}+\gamma\beta V_{xx}=0, &(x,t)\in(0,L)\times (0,T),\end{array}\end{matrix}$

其边界条件为

$\begin{matrix} \label{1.2}\begin{array}{ll}V(0,t)=0,\quad\alpha V_{x}(L,t)-\gamma\beta P_{x}(L,t)=-l_{1}\partial^{a,\eta}_{t}V(L,t),& t\in (0,T),\\P(0,t)=0,\quad\beta P_{x}(L,t)-\gamma\beta V_{x}(L,t)=-l_{2}\partial^{a,\eta}_{t}P(L,t),& t\in (0,T),\end{array}\end{matrix}$

初始条件为

$\begin{matrix}\label{1.3}\left(V(x,0),V_{t}(x,0),P(x,0),P_{t}(x,0)\right)=\left(V_{0}(x),V_{1}(x),P_{0}(x),P_{1}(x)\right),x\in(0,L),\end{matrix}$

这里$V(x,t)$, $P(x,t)$ 分别表示梁在点$x$处的横向位移和沿横向方向电位移的总载荷, $l_{1},l_{2}>0$表示正的反馈增益常数$\!\!,\ \eta>0$, 常数$a\in(0,1)$. 本文将证明具有分数阶导数型磁效应的压电梁系统是多项式稳定的.

为了寻找一个更稳定的系统, 本文也考虑了压电梁系统在带有傅里叶律热效应时的稳定性. 其中, 热效应作用于梁的横向位移方程, 模型为

$\begin{matrix} \label{problem1.2} \begin{array}{ll} \rho V_{tt}-\alpha V_{xx}+\gamma\beta P_{xx}+\delta\theta_{x}=0, &(x,t)\in(0,L)\times (0,T),\\ \mu P_{tt}-\beta P_{xx}+\gamma\beta V_{xx}=0, &(x,t)\in(0,L)\times (0,T),\\ c\theta_{x}-k\theta_{xx}+\delta V_{xt}=0, &(x,t)\in(0,L)\times (0,T), \end{array} \end{matrix}$

其边界条件为

$\begin{matrix} \label{1.5} \begin{array}{ll} \theta_x(0,t)=\theta(L,t)=0,& t\in (0,T),\\ V(0,t)=0,\quad \alpha V_{x}(L,t)-\gamma\beta P_{x}(L,t)=-l_{1}\partial^{a,\eta}_{t}V(L,t),& t\in (0,T),\\ P(0,t)=0,\quad \beta P_{x}(L,t)-\gamma\beta V_{x}(L,t)=-l_{2}\partial^{a,\eta}_{t}P(L,t),& t\in (0,T), \end{array} \end{matrix} $

初始条件为

$\begin{matrix}\label{1.6} \left(V(x,0),V_{t}(x,0),P(x,0),P_{t}(x,0),\theta(x,0)\right)=\left(V_{0}(x),V_{1}(x),P_{0}(x),P_{1}(x),\theta_0(x)\right),x\in(0,L). \end{matrix}$

这里$\theta(x, t)$是温度(与固定恒定参考温度的差值), $\kappa$是热传导率$\!,$$c$是比热容$\!,$$\delta$是热膨胀模量与弹性模量的乘积.

本文所研究问题的难点在于分数阶边界阻尼项的处理. 为此, 需引入两个新的函数将系统转化为等价的增广系统, 并利用Lumer-Philips定理推得模型是适定的. 之后, 通过将原系统分裂为无耗散系统和零初值系统, 并借助谱分析方法证得原系统的非指数稳定性. 最后, 利用Borichev-Tomilov定理, 给出了系统的多项式稳定性. 许多学者用不同的方法分析过其他系统的多项式稳定性[19-26]. 对带有热效应的系统(1.8)-(1.10), 本文利用扰动泛函方法得到了系统的指数稳定性结果. 有关热效应系统的指数稳定性结果可参见文献[27-30].

在下一节中, 将给出无热效应系统(1.5)-(1.7)的适定性证明. 在第3节中, 将证明无热效应系统的非指数稳定性. 在第4节中, 推得该系统的多项式稳定性. 在第5节中, 将考虑有热效应系统(1.8)-(1.10)的适定性. 在第6节中, 将会证明有热效应系统的指数稳定性.

2 无热效应压电梁系统的适定性

本节基于半群理论讨论系统(1.5)-(1.7)的适定性. 为将系统(1.5)-(1.7)转换为一个等价的增广模型, 首先引入如下定理和引理.

定理2.1[31] 设函数

$\mu(\xi)=|\xi|^{(2a-1)/2},\quad -\infty<\xi<+\infty,\quad 0<a<1.$

那么对于系统

$\partial_{t}\phi_i(\xi,t)+\left(\xi^{2}+\eta\right)\phi_i(\xi,t)-U_i(t)\mu(\xi)=0,\quad-\infty<\xi<+\infty,\quad\eta\geq0,\quad t>0,$
$\phi_i(\xi,0)=0,\quad-\infty<\xi<+\infty,$
$O_i(t)=\frac{\sin(a\pi)}{\pi}\int^{+\infty}_{-\infty}\mu(\xi)\phi_i(\xi,t){\rm d}\xi,$

$U_i$项和$O_i$项满足

$O_i(t)=I^{1-a,\eta}U_i(t), \quad i=1,2.$

$U_1(t)=V_t(L,t)$, $U_2(t)=P_t(L,t)$, 应用定理2.1和等式 $\partial^{a,\eta}_t f(t)=I^{1-a,\eta}f'(t)$, 可得

$\begin{eqnarray}\partial^{a,\eta}_tV(L,t)=I^{1-a,\eta}V_t(L,t)=I^{1-a,\eta}U_1=O_1(t)=\frac{\sin(a\pi)}{\pi}\int^{+\infty}_{-\infty}\mu(\xi)\phi_1(\xi,t){\rm d}\xi,\label{2.1}\\\partial^{a,\eta}_tP(L,t)=I^{1-a,\eta}P_t(L,t)=I^{1-a,\eta}U_2=O_2(t)=\frac{\sin(a\pi)}{\pi}\int^{+\infty}_{-\infty}\mu(\xi)\phi_2(\xi,t){\rm d}\xi.\label{2.101}\end{eqnarray}$

引理2.1[31]$\lambda\in E=\{\lambda\in C: {\rm Re}\lambda+\eta>0\}\cup\{\lambda\in{C}:Im\lambda\neq0\}$. 那么

$\int^{+\infty}_{-\infty}\frac{\mu^{2}(\xi)}{\xi^{2}+\eta+|\lambda|}{\rm d}\xi=\frac{\pi}{\sin(a\pi)}(\eta+|\lambda|)^{a-1}.$

应用等式(2.1)和(2.2), 原系统(1.5)-(1.7)可转化为

$\rho V_{tt}-\alpha V_{xx}+\gamma\beta P_{xx}=0, \ \ (x,t)\in (0,L)\times (0,T),$
$\partial_{t}\phi_{1}(\xi,t)+\left(\xi^{2}+\eta\right)\phi_{1}(\xi,t)-V_{t}(L,t)\mu(\xi)=0,\ \ \ \ (\xi,t)\in(-\infty,+\infty)\times(0,+\infty),$
$\mu P_{tt}-\beta P_{xx}+\gamma\beta V_{xx}=0,\ \ \ \ \ (x,t)\in(0,L)\times (0,T),$
$\partial_{t}\phi_{2}(\xi,t)+\left(\xi^{2}+\eta\right)\phi_{2}(\xi,t)-P_{t}(L,t)\mu(\xi)=0,\ \ \ \ \ (\xi,t)\in(-\infty,+\infty)\times(0,+\infty),$

其边界条件为

$V(0,t)=P(0,t)=0, t\in (0,T),$
$\alpha V_{x}(L,t)-\gamma\beta P_{x}(L,t)=-l_{1}\frac{\sin(a\pi)}{\pi}\int^{\infty}_{-\infty}\mu(\xi)\phi_{1}(\xi,t){\rm d}\xi, t\in (0,T),$
$\beta P_{x}(L,t)-\gamma\beta V_{x}(L,t)=-l_{2}\frac{\sin(a\pi)}{\pi}\int^{\infty}_{-\infty}\mu(\xi)\phi_{2}(\xi,t){\rm d}\xi, t\in (0,T),$

初始条件为

$\begin{matrix}\label{2.7}\left(V(x,0), V_{t}(x,0), \phi_{1}(0), P(x,0), P_{t}(x,0),\phi_{2}(0)\right)=\left(V_{0},V_{1},\phi_{01},P_{0},P_{1},\phi_{02}\right)\;x \in (0,L).\end{matrix}$

系统(2.3)-(2.10)的能量定义为

$\begin{matrix}\label{2.8}E(t)&=&\frac{1}{2}\int^{L}_{0}\left[\rho\left|V_{t}\right|^{2}+\alpha_{1}\left|V_{x}\right|^{2}+\mu\left|P_{t}\right|^{2}+\beta\left|\gamma V_{x}-P_{x}\right|^{2}\right] {\rm d}x\nonumber\\&&+\frac{\sin(a\pi)}{2\pi}\int^{\infty}_{-\infty}\left(l_{1}|\phi_{1}|^{2}+l_{2}|\phi_{2}|^{2}\right){\rm d}\xi.\end{matrix}$

将方程(2.3), (2.5)分别乘以 $ V_{t}$, $P_{t}$, 并在$(0,L)$上积分, 将方程(2.4), (2.6)分别乘以$l_{1}\frac{\sin(a\pi)}{\pi}\phi_{1}$, $l_{2}\frac{\sin(a\pi)}{\pi}\phi_{2}$, 并在 $\mathbb{R}$上积分. 将所得结果相加即得

$\begin{matrix}\label{2.9}\frac{\rm d}{{\rm d}t}E(t)=-\frac{\sin(a\pi)}{\pi}\int^{\infty}_{-\infty}(\xi^{2}+\eta)\left(l_{1}|\phi_{1}|^{2}+l_{2}|\phi_{2}|^{2}\right){\rm d}\xi.\end{matrix}$

令空间${\cal H}$

$\begin{matrix}{\cal H}:=H^{1}_{*}(0,L)\times {L}^{2}(0,L)\times{L}^{2}(-\infty,+\infty)\times {{H}}^{1}_{*}(0,L)\times {{L}}^{2}(0,L)\times{L}^{2}(-\infty,+\infty),\end{matrix}$

其中$ {{H}}^{1}_{*}(0,L)=\left\{f\in{{H}}^{1}(0,L):f(0)=0\right\}$. 空间${\cal H}$上的内积定义为

$\begin{matrix}\langle{U}_{1},{U}_{2}\rangle_{{\cal H}}&=&\int^{L}_{0}\left[\rho f_{1}\overline{f}_{2}+\mu g_{1}\overline{g}_{2}+\alpha_{1}V_{1,x}\overline{V}_{2,x}+\beta(\gamma V_{1,x}-P_{1,x})\overline{(\gamma V_{2,x}-P_{2,x})}\right]{\rm d}x \\&&+\frac{\sin(a\pi)}{\pi}\int^{+\infty}_{-\infty}\left(l_{1}\phi_{1,1}\overline{\phi}_{1,2}+l_{2}\phi_{2,1}\overline{\phi}_{2,2}\right){\rm d}x,\end{matrix}$

这里${{U}}_{i}=\left(V_{i},f_{i}, \phi_{1,i}, P_{i}, g_{i},\phi_{2,i}\right)\in {H},i=1,2$.

若定义算子, 使得${\cal A}:{\cal D}({\cal A})\subset {\cal H}\rightarrow {\cal H}$

$\begin{matrix} {\cal A}\left[\begin{array}{cc} V \\ f \\ \phi_{1} \\ P \\ g\\ \phi_{2}\end{array}\right]= \left[\begin{array}{cc} f \\ \frac{\alpha}{\rho}V_{xx} -\frac{\gamma\beta}{\rho}P_{xx} \\ -\left(\xi^{2}+\eta\right)\phi_{1}(\xi,t)+f(L,t)\mu(\xi)\\ g \\ -\frac{\gamma\beta}{\mu}V_{xx} + \frac{\beta}{\mu}P_{xx} \\ -\left(\xi^{2}+\eta\right)\phi_{2}(\xi,t)+g(L,t)\mu(\xi) \end{array}\right],\end{matrix}$

其中

$\begin{matrix} {\cal D}({\cal A}):&=&\Big\{{{U}}\in {\cal H}; V,P \in {H}^{2}(0,L)\cap H^{1}_{*}(0,L), f,g\in H^{1}_{*}(0,L),|\xi|\phi_{1},|\xi|\phi_{2}\in L^{2}(-\infty,+\infty),\\ & &-\left(\xi^{2}+\eta\right)\phi_{1}(\xi,t)+f(L,t)\mu(\xi),-\left(\xi^{2}+\eta\right)\phi_{2}(\xi,t)+g(L,t)\mu(\xi)\in L^{2}(-\infty,+\infty)\Big\}. \end{matrix}$

则系统 (2.3)-(2.10)可被记作

$\begin{matrix} \label{2.10} \left\{ \begin{array}{ll} {{U}}_{t}={{\cal A}}{{U}}\\ {{U}}(0)={{U}}_{0} \end{array} \right. \end{matrix}$

这里${{U}}=\left(V,f, \phi_{1}, P, g,\phi_{2}\right)$, ${{U}}_{0}=\left(V_{0},V_{1}, \phi_{1,0}, P_{0}, P_{1}, \phi_{2,0}\right)^{T}$.

引理2.2 算子${\cal A}$是耗散的, 且对任意${U}\in {\cal D({\cal A})}$, 有

$\begin{matrix}\label{2.11} {Re}\langle {{\cal A}}{U},{U}\rangle_{{H}}=-\frac{\sin(a\pi)}{\pi}\int^{\infty}_{-\infty}(\xi^{2}+\eta)\left(l_{1}|\phi_{1}|^{2}+l_{2}|\phi_{2}|^{2}\right){\rm d}\xi\leq 0. \end{matrix}$

对任意 ${U}\in {\cal D({\cal A})}$, 利用空间${\cal H}$上的内积定义, 易推得关系式(2.14). 证毕.

定理2.2${U}_{0}\in {\cal D({\cal A})}$. 那么 ${U}(t)={\cal S}_{{\cal A}}(t){U}_{0}$存在唯一解且满足

${U}\in{C}\left([0,\infty);{\cal D}({\cal A})\right)\cap{C}^{1} \left([0,\infty); {\cal H}\right).$

通过验证算子${\cal A}$满足Lumer-Phillips定理的条件, 可以推得算子${\cal A}$$C_{0}$ -半群$\{{\cal S}_{{\cal A}}(t)\}_{t\geq0}$的无穷小生成元, 进而可证得此定理. 由关系式(2.14)知, 算子${\cal A}$是耗散、封闭且稠密的. 因此, 仅需证明 $0\in \rho({\cal A})$. 也即, 对任意的${F}=\left(F_{1},F_{2},F_{3},F_{4},F_{5},F_{6}\right)\in {\cal H}$, 存在${{U}}=\left(V,f, \phi_{1}, P, g, \phi_{2}\right)$使得

$-{\cal A}{U}={F}.$

等价地, 需要考虑下述系统解的存在唯一性

$-f=F_{1},$
$-{\alpha}V_{xx}+{\gamma\beta}P_{xx}={\rho}F_{2},$
$\left(\xi^{2}+\eta\right)\phi_{1}(\xi,t)-f(L)\mu(\xi)=F_{3},$
$-g=F_{4},$
$-{\beta}P_{xx}+{\gamma\beta}V_{xx}={\mu}F_{5},$
$\left(\xi^{2}+\eta\right)\phi_{2}(\xi,t)-g(L)\mu(\xi)=F_{6}.$

由方程(2.15)和(2.17)可得 $f,g\in H^{1}_{*}(0,L)$

$\begin{matrix} f=-F_{1},\quad \quad g=-F_{4}. \end{matrix}$

结合等式(2.17), (2.20)和(2.21), 有

$\begin{matrix} \phi_{1}=\frac{-F_{1}(L)\mu(\xi)+F_{3}}{\xi^{2}+\eta},\quad \quad \phi_{2}=\frac{-F_{4}(L)\mu(\xi)+F_{6}}{\xi^{2}+\eta}.\end{matrix}$

从引理2.1可推得$\phi_{i}\in L^{2}(R)$. 因此, 仅需证明下述系统的解是存在唯一的.

$\begin{matrix} \left\{ \begin{array}{ll} {\alpha}V_{xx}-{\gamma\beta}P_{xx}=-\rho F_{2},\\ \beta P_{xx}-\gamma\beta V_{xx}=-\mu F_{5},\\ \alpha V_{x}(L,t)-\gamma\beta P_{x}(L,t)=-l_{1}\frac{\sin(a\pi)}{\pi}\int^{\infty}_{-\infty}\mu(\xi)\frac{-F_{1}(L)\mu(\xi)+F_{3}(\xi)}{\xi^{2}+\eta}{\rm d}\xi,\\[3mm] \beta P_{x}(L,t)-\gamma\beta V_{x}(L,t)=-l_{2}\frac{\sin(a\pi)}{\pi}\int^{\infty}_{-\infty}\mu(\xi)\frac{-F_{4}(L)\mu(\xi)+F_{6}(\xi)}{\xi^{2}+\eta}{\rm d}\xi. \end{array} \right. \end{matrix}$

通过Lax-Milgram定理知, 系统(2.23)存在一个解$(V,P)\in H_{*}^{1}(0,L)\times H_{*}^{1}(0,L)$. 结合等式(2.21)-(2.23), 可得$0 \in \rho({ \cal A})$. 证毕.

3 无热效应压电梁系统的指数稳定性缺乏

本节将证明压电梁系统(2.3)-(2.10)的非指数稳定性. 首先, 给出会用到的一系列引理和定理.

引理3.1[13]$R\in \mathbb{R} $,有

${\cal K}=\Big\{\partial^{a,\eta}_{t}f(t):f'\in L^{2}(0,T),\int^{T}_{0}|f'(t)|^{2}{\rm d}t\leq R^{2}\Big\}.$

那么${\cal K}$$L^{2}(0,T)$中的一个紧子集.

定理3.1[15]${\cal H}$ 为一个Hilbert空间, ${\cal H}_{0}$ 为空间${\cal H}$的一个闭子集. 设$\{{\cal S}(t)\}_{t\geq0}$ 是一个定义在${\cal H}_{0}$ 上的压缩半群, $\{{\cal S}_{0}(t)\}_{t\geq0}$是一个定义在 ${\cal H}_{0}$上的酉群. 若 $\{{\cal S}(t)-{\cal S}_{0}(t)\}_{t\geq0}$ 是从 ${\cal H}_{0}$${\cal H}$的紧算子, 那么 $\{{\cal S}(t)\}_{t\geq0}$ 是非指数稳定的.

现引入泛函

${\cal J}(x,t)=\frac{1}{2}\big({\rho}\left|V_{t}\right|^{2}+{\alpha_{1}}\left|V_{x}\right|^{2}+{\mu}\left|P_{t}\right|^{2}+{\beta}\left|\gamma V_{x}-P_{x}\right|^{2}\big),$
$J(t)=\int^{L}_{0}\big(\rho q V_{t}V_{x}+\mu q P_{t}(P_{x}-\gamma V_{x})+\mu\gamma q P_{t} V_{x}\big){\rm d}x,$

这里$q$ 是一个仅与$x$有关的函数且 $q\in C^1[L]$.

为证明系统的非指数稳定性, 接下来将给出上述泛函与能量泛函$E(t)$之间的关系.

引理3.2 存在一个常数$C$使得

$\begin{matrix} \left|\int^{t}_{0}\left[q(L){\cal J}(L,\tau)-q(0){\cal J}(0,\tau)\right]{\rm d}\tau - \int^{t}_{0} \int^{L}_{0}q'(x){\cal J}(x,\tau){\rm d}x{\rm d}\tau\right|=\left|J(0)-J(t)\right|\leq C E(0). \end{matrix}$

对方程(2.3)两边乘以$qV_{x}$, 并使用

$\rho qV_{tt}V_{x}=\rho\frac{\partial}{\partial t}\left(qV_{t}V_{x}\right)-\rho q V_{t}V_{xt}= \rho\frac{\partial}{\partial t}\left(qV_{t}V_{x}\right)-q\frac{\partial}{\partial x} \left(\frac{\rho}{2}\left|V_t\right|^2 \right),$

则有

$\begin{matrix} \rho\frac{\partial}{\partial t}\left(qV_{t}V_{x}\right)- q \frac{\partial}{\partial x}\left(\frac{\rho}{2}|V_{t}|^{2}+\frac{\alpha_{1}}{2}|V_{x}|^{2}\right)=\gamma\beta q\left(\gamma V_{xx}-P_{xx}\right)V_{x}. \end{matrix}$

对方程(2.5)两边乘以$qP_{x}$, 且使用

$ qP_{tt}P_{x}=\frac{\partial}{\partial t}\left(qP_{t}P_{x}\right)- q P_{t}P_{xt} =\frac{\partial}{\partial t}\left(qP_{t}P_{x}\right)-q\frac{\partial}{\partial x} \left(\frac{1}{2}\left|P_t\right|^2 \right),$

可得

$\begin{matrix} \mu\frac{\partial}{ \partial t}\left(qP_{t}P_{x}\right)- q \frac{\partial}{\partial x}\left(\frac{\mu}{2}|P_{t}|^{2}\right)=-\beta q\left(\gamma V_{xx}-P_{xx}\right)P_{x}. \end{matrix}$

将等式(3.2)与等式(3.3)相加, 可得

$\begin{matrix}&& \frac{\partial}{\partial t}\left(\rho q V_{t}V_{x}+\mu q P_{t}(P_{x}-\gamma V_{x})+\mu\gamma q P_{t} V_{x}\right)\\&=&q\frac{\partial}{ \partial x}\left(\frac{\rho}{2}\left|V_{t}\right|^{2}+\frac{\alpha_{1}}{2}\left|V_{x}\right|^{2}+\frac{\mu}{2}\left|P_{t}\right|^{2}+\frac{\beta}{2}\left|\gamma V_{x}-P_{x}\right|^{2}\right). \end{matrix}$

对上述等式在$x$方向上从$0$$L$进行积分, 并使用分部积分, 则有

$\begin{matrix} \frac{\rm d}{{\rm d}t}J(t)-\left[q(x){\cal J}\right]|^{L}_{0}+\int^{L}_{0}q'(x){\cal J}(x,t){\rm d}x=0. \end{matrix}$

对上述等式在$t$方向上从$0$$t$进行积分, 得

$\begin{matrix} \left|\int^{t}_{0}\left[q(L){\cal J}(L,\tau)-q(0){\cal J}(0,\tau)\right]{\rm d}\tau -\int^{t}_{0} \int^{L}_{0}q'(x){\cal J}(x,\tau){\rm d}x{\rm d}\tau\right|=\left|J(0)-J(t)\right|. \end{matrix}$

注意到, $J$中所含的$V_{x},V_{t},P_{t},\gamma V_{x}-P_{x}$也出现能量$E(t)$中, 因此应用Young不等式易得

$\begin{matrix} \left|J(0)-J(t)\right|\leq \left|J(0)\right|+\left|J(t)\right|\leq C E(t)+C E(0). \end{matrix}$

其中, $C$是一个仅依赖于 $\alpha,\beta,\gamma,\rho,L,\|q\|_{\infty}$的常数. 由等式(2.12)可知能量 $E(t)$ 是耗散的, 因此有$E(t)\leq E(0)$. 证毕.

下面考虑无耗散的压电梁系统

$\rho \widetilde{V}_{tt}-\alpha \widetilde{V}_{xx}+\gamma\beta \widetilde{P}_{xx}=0, (x,t)\in(0,L)\times (0,T),$
$\mu \widetilde{P}_{tt}-\beta \widetilde{P}_{xx}+\gamma\beta\widetilde{V}_{xx}=0, (x,t)\in(0,L)\times (0,T),$

其边界条件为

$\widetilde{V}(0,t)=0,\quad \alpha \widetilde{V}_{x}(L,t)-\gamma\beta \widetilde{P}_{x}(L,t)=0, t\in (0,T),$
$\widetilde{P}(0,t)=0,\quad\beta \widetilde{P}_{x}(L,t)-\gamma\beta \widetilde{V}_{x}(L,t)=0, t\in (0,T),$

初始条件与原系统一致, 也即

$\begin{matrix}\left(\widetilde{V}(x,0),\widetilde{V}_{t}(x,0),\widetilde{P}(x,0),\widetilde{P}_{t}(x,0)\right)=\left(V_{0}(x),V_{1}(x),P_{0}(x),P_{1}(x)\right),x\in(0,L).\end{matrix}$

问题(3.4)-(3.8) 定义在空间 ${\cal H}_{0}$上,

${\cal H}_{0}:=H^{1}_{*}(0,L)\times {L}^{2}(0,L)\times\{0\}\times {{H}}^{1}_{*}(0,L)\times {{L}}^{2}(0,L)\times\{0\}.$

将带有初值$U_{0}=\left(V_{0}, V_{1}, 0, P_{0}, P_{1}, 0\right)\in{\cal H}_{0}$的系统$\{{\cal S}_{0}(t)\}$的解记作

${{\cal S}_{0}(t)U_{0}}=\left(\widetilde{V}, \widetilde{V}_{t}, 0, \widetilde{P}, \widetilde{P}_{t}, 0\right).$

注意到$\|{{\cal S}_{0}(t)U_{0}}\|^{2}=\|U_{0}\|^{2}$, 也即, $\{{\cal S}_{0}(t)\}$ 在空间 ${\cal H}_{0}$上定义了一个酉群.

引理3.3$\widetilde{E}(t)$为系统 (3.4)-(3.8)的能量. 那么存在一个常数$C_T$, 使得

$\int^{T}_{0}\left(\left|\widetilde{{V}}_{t}(L,t)\right|^{2}+\left|\widetilde{{P}}_{t}(L,t)\right|^{2}\right){\rm d}t\leq C_{T}\widetilde{E}(0). $

将方程(3.4)和(3.5)分别乘以$x\widetilde{V}_{x}$$x\widetilde{P}_{x}$, 并利用与引理3.2相同的计算技巧, 得

$\int^{T}_{0}\left(\left|\widetilde{{V}}_{t}(L,t)\right|^{2}+\left|\widetilde{{P}}_{t}(L,t)\right|^{2}\right){\rm d}t\leq\int^{T}_{0}\widetilde{E}(t){\rm d}t+C\widetilde{E}(T)+C\widetilde{E}(0),$

这里$C$是一个常数, $\widetilde{E}(t)$是系统 (3.4)-(3.8)的能量. 结合无耗散系统的能量总是保持不变这一事实, 可得

$\begin{matrix}\int^{T}_{0}\left(\left|\widetilde{{V}}_{t}(L,t)\right|^{2}+\left|\widetilde{{P}}_{t}(L,t)\right|^{2}\right){\rm d}t\leq C_{T}\widetilde{E}(0),\end{matrix}$

这里$C_{T}$ 是一个常数.证毕.

定理3.2$C_{0}$ -半群 $\{{\cal S}(t)\}_{t\geq 0}$ 是非指数稳定的.

下面借助定理3.1来证明无热效应的压电梁系统的非指数稳定性. 为此, 需要说明 ${{\cal S}(t)}-{{\cal S}_{0}(t)}$是紧算子.对任一有界的初值列

$U^{n}_{0}=\left(V^{n}_{0}, V^{n}_{1}, 0, P^{n}_{0}, P^{n}_{1}, 0\right)\in{\cal H}_0,$

系统(2.3)-(2.10)的解被记作$U^{n}={\cal S}(t){U^{n}_{0}}=\left(V^{n}, V^{n}_{t}, \phi^{n}_{1}, P^{n}, P^{n}_{t}, \phi^{n}_{2}\right)\in{\cal H}$, 系统(3.4)-(3.8)的解被记作$\widetilde{U}^{n}={\cal S}_{0}(t){U^{n}_{0}}=\left(\widetilde{V}^{n}, \widetilde{V}^{n}_{t}, 0, \widetilde{P}^{n},\widetilde{P}^{n}_{t}, 0\right)\in{\cal H}_0$.

$\widehat{V}^{n}_{x}=V^{n}_{x}-\widetilde{V}^{n}_{x},\quad \widehat{P}^{n}_{x}=P^{n}_{x}-\widetilde{P}^{n}_{x},\quad\widehat{V}^{n}_{t}=V^{n}_{t}-\widetilde{V}^{n}_{t},\quad\widehat{P}^{n}_{t}=P^{n}_{t}-\widetilde{P}^{n}_{t},$那么, 可得$\left({\cal S}(t)-{\cal S}_{0}(t)\right)U^{n}_{0}=\left(\widehat{V}^{n}, \widehat{V}^{n}_{t}, \phi^{n}_{1}, \widehat{P}^{n}, \widehat{P}^{n}_{t}, \phi^{n}_{2}\right)\in{\cal H},$$\left(\widehat{V}^{n}, \widehat{V}^{n}_{t}, \phi^{n}_{1}, \widehat{P}^{n}, \widehat{P}^{n}_{t}, \phi^{n}_{2}\right)$ 满足系统

$\rho \widehat{V}^{n}_{tt}-\alpha \widehat{V}^{n}_{xx}+\gamma\beta \widehat{P}^{n}_{xx}=0, (x,t)\in (0,L)\times (0,T), $
$\partial_{t}\phi^{n}_{1}(\xi,t)+\left(\xi^{2}+\eta\right)\phi^{n}_{1}(\xi,t)-{V}^{n}_{t}(L,t)\mu(\xi)=0, (\xi,t)\in(-\infty,+\infty)\times(0,+\infty),$
$\mu \widehat{P}^{n}_{tt}-\beta \widehat{P}^{n}_{xx}+\gamma\beta \widehat{V}^{n}_{xx}=0, (x,t)\in(0,L)\times (0,T),$
$\partial_{t}\phi^{n}_{2}(\xi,t)+\left(\xi^{2}+\eta\right)\phi^{n}_{2}(\xi,t)-{P}^{n}_{t}(L,t)\mu(\xi)=0, (\xi,t)\in(-\infty,+\infty)\times(0,+\infty),$

其边界条件为

$\widehat{V}^{n}(0,t)=0,\quad\alpha \widehat{V}^{n}_{x}(L,t)-\gamma\beta \widehat{P}^{n}_{x}(L,t)=-l_{1}\partial^{a,\eta}_{t}V^{n}(L,t), t\in (0,T),$
$\widehat{P}^{n}(0,t)=0,\quad\beta \widehat{P}^{n}_{x}(L,t)-\gamma\beta \widehat{V}^{n}_{x}(L,t)=-l_{2}\partial^{a,\eta}_{t}P^{n}(L,t), t\in (0,T),$

初始条件为

$\begin{matrix} \left(\widehat{V}^{n}(x,0), \widehat{V}^{n}_{t}(x,0), \phi_1(0), \widehat{P}^{n}(x,0), \widehat{P}^{n}_{t}(x,0), \phi_2(0)\right)=\left(0,0,0,0,0,0\right),\; x \in (0,L). \end{matrix}$

系统(3.10)-(3.16)的能量定义为

$\begin{matrix} \widehat{E}(t)&=&\frac{1}{2}\int^{L}_{0}\left[\rho\left|\widehat{V}^{n}_{t}\right|^{2}+\alpha_{1}\left|\widehat{V}^{n}_{x}\right|^{2}+\mu\left|\widehat{P}^{n}_{t}\right|^{2}+\beta\left|\gamma \widehat{V}^{n}_{x}-\widehat{P}^{n}_{x}\right|^{2}\right] {\rm d}x \\ & &+\frac{\sin(a\pi)}{2\pi}\int^{\infty}_{-\infty}\left(l_{1}|\phi^{n}_{1}|^{2}+l_{2}|\phi^{n}_{2}|^{2}\right){\rm d}\xi. \end{matrix}$

将方程(3.10), (3.12), (3.11) 和(3.13)分别乘以 $ \overline{\widehat{V}^{n}_{t}}$,$\overline{\widehat{P}^{n}_{t}}$, $l_{1}\frac{\sin(a\pi)}{\pi}\phi^{n}_{1}$$l_{2}\frac{\sin(a\pi)}{\pi}\phi^{n}_{2}$, 并分别在 $(0,L)$$\mathbb{R}$上积分, 且使用边界条件(3.14)-(3.15), 可得

$\begin{matrix} \frac{\rm d}{{\rm d}t}\widehat{E}(t)=-l_{1}\partial^{a,\eta}_{t}V^{n}(L,t)\widehat{V}^{n}_{t}(L,t)-l_{2}\partial^{a,\eta}_{t}P^{n}(L,t)\widehat{P}^{n}_{t}(L,t). \end{matrix}$

$\widehat{V}^{n}_{t}=V^{n}_{t}-\widetilde{V}^{n}_{t}, \widehat{P}^{n}_{t}=P^{n}_{t}-\widetilde{P}^{n}_{t}$代入到上式中, 并在$[t]$上进行积分, 利用初始条件(3.16), 可得

$\begin{matrix}\label{3.70}&& \widehat{E}(t)+\int^t_0\frac{\sin(a\pi)}{\pi}\int^{\infty}_{-\infty}(\xi^{2}+\eta)\left(l_{1}|\phi_{1}|^{2}+l_{2}|\phi_{2}|^{2}\right){\rm d}\xi {\rm d}\tau\\ &=&\int^{t}_{0} \left[l_{1}\partial^{a,\eta}_{t}V^{n}(L,\tau)\widetilde{{V}}^{n}_{t}(L,\tau)+l_{2}\partial^{a,\eta}_{t}P^{n}(L,\tau)\widetilde{{P}}^{n}_{t}(L,\tau)\right]{\rm d}\tau. \end{matrix}$

设初始值是${\cal H}_{0}$中的有界列. 系统 (3.10)-(3.16)的解记为 $\left(\widehat{V}^{n},\widehat{V}^{n}_{t},\phi^{n}_{1},\widehat{P}^{n},\widehat{P}^{n}_{t},\phi^{n}_{2}\right).$ 由于 ${\cal S}(t)$ 是耗散的, 而${\cal S}_{0}(t)$是无耗散的, 因此系统(3.10)-(3.16)的能量 $\widehat{E}(t)$是不增且有界的. 那么, 上述解序列必有一个弱收敛子列, 将其仍记作 $\left(\widehat{V}^{n},\widehat{V}^{n}_{t},\phi^{n}_{1},\widehat{P}^{n},\widehat{P}^{n}_{t},\phi^{n}_{2}\right).$

接下来证明该序列是强收敛的. 引理3.2与引理3.3表明$V^{n}_{t}(L,t)$, $P^{n}_{t}(L,t)$, $\widetilde{V}^{n}_{t}(L,t)$, $\widetilde{P}^{n}_{t}(L,t)$在空间$L^{2}(0,T)$中是有界的. 使用引理3.1 进一步推得$\partial^{a,\eta}_{t}V^{nk}(L,t)$, $\partial^{a,\eta}_{t}P^{nk}(L,t)$存在强收敛子列. 因此, 等式(3.17) 的右端是强收敛的, 解序列$\left(\widehat{V}^{nk},\widehat{V}^{nk}_{t},\phi^{nk}_{1},\widehat{P}^{nk},\widehat{P}^{nk}_{t},\phi^{nk}_{2}\right)$依范数收敛. 由以上证明过程可知, ${\cal S}(t)-{\cal S}_{0}(t)$可以将任一有界列映射到强收敛列, 因此${\cal S}(t)-{\cal S}_{0}(t)$ 是从 ${\cal H}_{0}$${\cal H}$的紧算子. 由定理3.1, 该定理得证. 证毕.

4 无热效应压电梁系统的多项式稳定性

本节将证明系统(2.3)-(2.11)的多项式稳定性. 为此, 需引入Borichev-Tomilov定理[33].

定理4.1[33]$\{{\cal S}(t)\}_{t\geq0}$ 是一个定义在Hilbert空间$H$上的有界$C_{0}$ -半群, ${\cal A}$$\{{\cal S}(t)\}_{t\geq0}$的无穷小生成元, 满足 $i \mathbb{R} \subset \rho ({\cal A})$. 那么对任意$k>0$, 下述结论等价

(1) $\big\|\big(i\lambda I -{\cal A}\big)^{-1}\big\|_{{\cal L}({\cal H})}=o\big(|\lambda |^{k}\big),\lambda\rightarrow \infty$;

(2) $\big\|{\cal S}(t){\cal A}^{-1}\big\|_{{\cal L}({\cal H})}=o\big(t^{-\frac{1}{k}}\big),t\rightarrow \infty$.

首先, 考虑预解式

$\begin{matrix} i\lambda U-{\cal A}U=F,\quad\lambda\in \mathbb{R}.\end{matrix}$

作预解式(4.1)和$U$在空间${\cal H}$上的内积, 并取实部, 则有

$\begin{matrix}\label{4.1+}{\rm Re}\langle -{\cal A}U, U \rangle_{\cal H}={\rm Re}\langle F, U \rangle_{\cal H}\leq C\|U\|_{{\cal H}}\|F\|_{{\cal H}},\end{matrix}$

这里$F=(F_{1},F_{2},F_{3},F_{4},F_{5},F_{6})$. 预解式(4.1)可写为

$i\lambda V-f=F_{1},$
$i\lambda\rho f-{\alpha}V_{xx}+{\gamma\beta}P_{xx}={\rho}F_{2},$
$i\lambda \phi_{1}+\left(\xi^{2}+\eta\right)\phi_{1}(\xi,t)-f(L)\mu(\xi)=F_{3},$
$i\lambda P-g=F_{4},$
$i\lambda\mu g-{\beta}P_{xx}+{\gamma\beta}V_{xx}={\mu}F_{5},$
$i\lambda \phi_{2}+\left(\xi^{2}+\eta\right)\phi_{2}(\xi,t)-g(L)\mu(\xi)=F_{6}.$

为了方便后续的证明, 现引入一些泛函

$I_{V}=\rho q(L)|f(L)|^{2}+\alpha_{1}q(L)|V_{x}(L,t)|^{2},\quad I_{P}=\mu q(L)|g(L)|^{2}+\beta q(L)|(\gamma V_{x}-P_{x})(L,t)|^{2},$
${\cal N}^{2}=\int^{L}_{0}\rho |f|^{2} {\rm d}x +\int^{L}_{0}\mu|g|^{2} {\rm d}x +\int^{L}_{0}\alpha_{1}|V_{x}|^{2}{\rm d}x +\int^{L}_{0}\beta|\gamma V_{x}-P_{x}|^{2}{\rm d}x.$

引理4.1$F=(F_{1},F_{2},F_{3},F_{4},F_{5},F_{6})\in{\cal H}$, $\lambda\in \mathbb{R} $, ${U}=(V,f,\phi_{1},P,g,\phi_{2}) \in {\cal D}({\cal A})$满足 $i\lambda{U}-{\cal A}{U}={F}$. 那么对于$q\in C^{2}([L])$, $q(0)=0$

$I_{V}+I_{P}-\int^{L}_{0}\left[\rho q_{x}|f|^{2} + \mu q_{x}|g|^{2} +\alpha_{1} q_{x}|V_{x}|^{2}+\beta q_{x}|\gamma V_{x}-P_{x}|^{2}\right]{\rm d}x =-R_{1}-R_{2},$

其中,

$R_{1}={\rm Re}\int^{L}_{0}\left(2\mu q F_{4}\overline{P}_{x}+2\mu q \overline{F}_{3,x}g \right){\rm d}x,$
$R_{2}={\rm Re}\int^{L}_{0}\left(2\rho q F_{2}\overline{V}_{x}+2\rho q \overline{F}_{1,x}f\right) {\rm d}x.$

对等式(4.4)两边乘以 $q \overline{V}_{x}$, 并在 $[L]$上进行积分, 得

$\begin{matrix} \int^{L}_{0}\left(- i\lambda \rho f q\overline{V}_{x}+\alpha qV_{xx}\overline{V}_{x}-\gamma\beta qP_{xx}\overline{V}_{x} \right){\rm d}x=-\int^{L}_{0}\rho qF_{2}\overline{V}_{x} {\rm d}x. \end{matrix}$

注意到, 利用等式(4.3)的结果, 等式(4.9)中的第一项可以被重写为

$\begin{matrix}\int^{L}_{0} -i\lambda \rho f q\overline{V}_{x}{\rm d}x=\int^{L}_{0} \overline{(i\lambda V_{x})}\rho q f {\rm d}x=\int^{L}_{0}\rho qf\overline{(f_{x}+F_{1,x})}{\rm d}x. \end{matrix}$

对等式(4.7)两边乘以$q \overline{P}_{x}$, 且在 $[L]$上积分, 则有

$\begin{matrix} \int^{L}_{0}\left(- i\lambda\mu g q\overline{P}_{x}+\beta qP_{xx}\overline{P}_{x}-\gamma\beta qV_{xx}\overline{P}_{x}\right) {\rm d}x=-\int^{L}_{0}\mu qF_{5}\overline{P}_{x} {\rm d}x. \end{matrix}$

此时, 通过使用等式(4.6), 等式(4.11)中的第一项可进行如下变换

$\begin{matrix} \int^{L}_{0}- i\lambda \mu g q\overline{P}_{x}{\rm d}x=\int^{L}_{0} \overline{(i\lambda P_{x})}\mu q g {\rm d}x=\int^{L}_{0}\mu qg\overline{(g_{x}+F_{3,x})}{\rm d}x. \end{matrix}$

将等式(4.9)与等式(4.11)相加, 并使用等式(4.10)与(4.12)的结果, 可推得

$\begin{matrix}\label{4.5}& &\int^{L}_{0} \rho q \frac{\rm d}{{\rm d}x} |f|^{2} {\rm d}x +\int^{L}_{0} \alpha_{1} q \frac{\rm d}{{\rm d}x}|V_{x}|^{2} {\rm d}x +\int^{L}_{0} \mu q \frac{\rm d}{{\rm d}x}|g|^{2}{\rm d}x +\int^{L}_{0}\beta q \frac{\rm d}{{\rm d}x} |\gamma V_{x}-P_{x}|^{2} {\rm d}x \nonumber\\ & =&{\rm Re}\int^{L}_{0}\left(-2\mu q F_{4}\overline{P}_{x}-2\mu q \overline{F}_{3,x}g -2\rho q F_{2}\overline{V}_{x}-2\rho q f\overline{F}_{1,x}\right){\rm d}x. \end{matrix}$

之后, 对上式进行分部积分即得引理4.1. 证毕.

引理4.2 存在常数$C$使得

$\begin{matrix} {\cal N}^{2}\leq C\left(I_{V}+I_{P}+\|F\|^{2}_{{\cal H}}\right), \end{matrix}$

这里${\cal N}$, $I_{V}, I_{P}$ 是之前定义的泛函.

取定$q(x)=x, x\in [L]$. 由引理4.1的结果可知

$\begin{matrix} {\cal N}^{2}= I_{V}+I_{P}+R_{1}+R_{2}. \end{matrix}$

结合$R_{1},R_{2}$的具体表达, 则有

$\begin{matrix} |R_{1}|\leq C{\cal N}\|F\|_{{\cal H}}, \quad|R_{2}|\leq C{\cal N}\|F\|_{{\cal H}}. \end{matrix}$

应用估计式(4.16) 和Cauchy-Schwartz不等式, 可直接验证估计式(4.14) 是成立的. 证毕.

定理4.2$\rho({\cal A})$ 是算子 ${\cal A}$的预解集. 那么有$i \mathbb{R} \in\rho({\cal A})$.

首先, 定义集合${\cal M}=\left\{\beta>0:(-i\beta,i\beta)\subset \rho ({\cal A})\right\}. $ 由于第2节中已证明的结论$0\in \rho({\cal A})$, 可知集合${\cal M}\neq \emptyset$.显然, 若想证明所有的纯虚数$i \mathbb{R} $均属于$\rho({\cal A})$, 需证明$\sup\limits_{\beta>0}{{\cal M}}=+\infty$. 为此, 接下来将应用反证法的思想来证明 $\sup \limits_{\beta>0}{{\cal M}}< +\infty$这种情况不可能存在.

假设存在$\lambda>0$ 使得$\sup \limits_{\beta>0}{{\cal M}}=\lambda <+\infty$, 易得 $\lambda\notin {\cal M}$. 因此, 这里会存在 $\lambda_{n} \in {\cal M}$$\|\overline{F}_{n}\|_{{\cal H}}=1$$\overline{F}_{n}\in {\cal H}$, 使得$\left\|(i\lambda_{n} I-{\cal A})^{-1}\overline{F}_{n}\right\|_{{\cal H}}\rightarrow \infty$.

定义$ \overline{U}_{n}=(i\lambda_{n}I-{\cal A})^{-1}\overline{F}_{n}$. 那么, 有 $i\lambda_{n}\overline{U}_{n}-{\cal A}\overline{U}_{n}=\overline{F}_{n}$.现记

$U_{n}=\frac{\overline{U}_{n}}{\|(i\lambda _{n}-{\cal A})^{-1}\overline{F}_{n}\|_{{\cal H}}}, \quad F_{n}=\frac{\overline{F}_{n}}{\|(i\lambda _{n}-{\cal A})^{-1}\overline{F}_{n}\|_{{\cal H}}}.$

显然, $U_{n}$$F_{n}$ 满足$\|U_{n}\|_{{\cal H}}=1$以及

$i\lambda_{n}U_{n}-{\cal A}U_{n}=F_{n}.$

因为$\|\overline{F}_{n}\|_{{\cal H}}=1$$\|(i\lambda_n I -{\cal A})^{-1}\overline{F}_{n}\|_{{\cal H}}\rightarrow \infty$, 所以有$F_{n}\rightarrow 0$. 在空间${\cal H}$上取其与 $U_{n}$ 的内积可得$ i\lambda_{n}\|U_{n}\|^{2}_{{\cal H}}-\langle {\cal A}U_{n},U_{n}\rangle _{{\cal H}}=\langle F_{n},U_{n}\rangle _{{\cal H}}. $取实部并使用$F_{n}\rightarrow 0$的事实, 可得$ -{\rm Re}\langle {\cal A}U_{n},U_{n}\rangle _{\cal H}={\rm Re}\langle F_{n},U_{n}\rangle _{{\cal H}}\rightarrow 0,$这也意味着

$\int^{\infty}_{-\infty}(\xi^{2}+\eta)\left(l_{1}|\phi_{1,n}|^{2}+l_{2}|\phi_{2,n}|^{2}\right){\rm d}\xi\rightarrow 0.$

由方程(4.5), (4.8)以及$F_{n}\rightarrow 0$的事实, 可得 $f_{n}(L),g_{n}(L)\rightarrow 0$. 使用边界条件(2.8), (2.9) 以及$\phi_{1,n}, \phi_{2,n}\rightarrow 0$, 可得 $V_{x,n}(L,t),P_{x,n}(L,t)\rightarrow 0$. 至此, 可得 $I_{V}+I_{P}\rightarrow 0$.通过使用引理4.2 和$F_{n}, \phi_{1,n}, \phi_{2,n}\rightarrow 0$可知$U_{n}\rightarrow 0$. 然而, 结论$U_{n}\rightarrow 0$ 与前述 $\|U_{n}\|_{{\cal H}}=1$是矛盾的. 因此, 假设$\sup \limits_{\beta>0}{{\cal M}}=\lambda <+\infty$ 是不成立的, 也即, $\sup\limits_{\beta>0}{{\cal M}}=+\infty$. 证毕.

在开始证明主要结果之前, 首先给出一个有用的不等式, 它描述了能量导数在$x=L$处的耗散项$\frac{\sin(a\pi)}{\pi}\int^{\infty}_{-\infty}(\xi^{2}+\eta)\left(l_{1}|\phi_{1}|^{2}+l_ |\phi_{2}|^{2}\right){\rm d}\xi$与项$l_{i}\frac{\sin(a\pi)}{\pi}\int^{\infty}_{-\infty}\mu(\xi)\phi_{i}(\xi,t){\rm d}\xi$之间的关系.

引理4.3$\mu(\xi), \phi_{i}(\xi,t), a, \eta $, $l_i\geq 0$是系统中出现的函数和系数, 那么

$\begin{matrix}\label{nuse2} \left[l_{i}\frac{\sin(a\pi)}{\pi}\int^{\infty}_{-\infty}\mu(\xi)\phi_{i}(\xi,t){\rm d}\xi\right]^{2}\leq M\frac{\sin(a\pi)}{\pi}\int^{\infty}_{-\infty}(\xi^{2}+\eta)\left(l_{1}|\phi_{1}|^{2}+l_{2}|\phi_{2}|^{2}\right){\rm d}\xi, \end{matrix}$

其中$M$ 是一个常数.

首先, 不等式(4.17)的左端可以重写为

$\begin{matrix} && \left[l_{i}\frac{\sin(a\pi)}{\pi}\int^{\infty}_{-\infty}\mu(\xi)\phi_{i}(\xi,t){\rm d}\xi\right]^{2} \\ &=& \left[l_{i}\frac{\sin(a\pi)}{\pi}\right]^{2}\left[\int^{\infty}_{-\infty}\frac{\mu(\xi)}{\sqrt{\xi^{2}+\eta+|\lambda|}}\phi_{i}(\xi,t){\sqrt{\xi^{2}+\eta+|\lambda|}}{\rm d}\xi\right]^{2} \end{matrix}$

由Hölder不等式和引理2.1, 直接得

$\begin{matrix} && \left[\int^{\infty}_{-\infty}\frac{\mu(\xi)}{\sqrt{\xi^{2}+\eta+|\lambda|}}\phi_{i}(\xi,t){\sqrt{\xi^{2}+\eta+|\lambda|}}{\rm d}\xi\right]^{2} \\ &\leq& \frac{\pi(\eta+|\lambda|)^{a-1}}{\sin(a\pi)}\left[\int^{\infty}_{-\infty}\phi^{2}_{i}(\xi,t)\left({\xi^{2}+\eta+|\lambda|}\right){\rm d}\xi\right]. \end{matrix}$

也即

$\left[l_{i}\frac{\sin(a\pi)}{\pi}\int^{\infty}_{-\infty}\mu(\xi)\phi_{i}(\xi,t){\rm d}\xi\right]^{2}\leq M_{i}\frac{\sin(a\pi)}{\pi}\int^{\infty}_{-\infty}(\xi^{2}+\eta)\left(l_{i}|\phi_{1}|^{2}\right){\rm d}\xi,$

这里$M_{i}=\eta^{a-1}l_i$, $i=1,2$. 最终, 取定$M=\max\{M_{1},M_{2}\}$, 即可推得该引理. 证毕.

接下来, 将根据定理4.1和引理4.2 -引理4.3的结果, 证明系统是多项式稳定的.

定理4.3 带有分数阶磁效应的压电梁系统(2.3)-(2.10)在无热效应时是多项式稳定的, 且满足

$\|{U}(t)\|_{{\cal H}}\leq \|{U}_{0}\|_{{\cal D}({\cal A})}\frac{C}{{t}^{\frac{1}{2-2a}}}.$

为得到此结果, 首先需要考虑$|f(L)|^{2},|g(L)|^{2}$$\|F\|_{{\cal H}}$的关系. 将方程(4.8)乘以${\mu(\xi)}/\left({\xi^{2}+\eta+|\lambda|}\right)$

$\begin{matrix} g(L)\frac{\mu^2(\xi)}{\xi^{2}+\eta+|\lambda|}=-\frac{F_{6}\mu(\xi)}{\xi^{2}+\eta+|\lambda|}+\frac{i\lambda\mu(\xi)}{\xi^{2}+\eta+|\lambda|}\phi_{2} +\frac{(\xi^{2}+\eta)\mu(\xi)}{\xi^{2}+\eta+|\lambda|}\phi_{2}. \end{matrix}$

上述等式两边同时取模, 有

$\begin{matrix} |g(L)|\frac{\mu^2(\xi)}{\xi^{2}+\eta+|\lambda|}\leq\frac{|F_{6}|\mu(\xi)}{\xi^{2}+\eta+|\lambda|}+\mu(\xi)|\phi_{2}|. \end{matrix}$

对上式在 $(-\infty,+\infty)$上进行积分, 并应用引理2.1的结果, 可推得

$\begin{matrix} |g(L)|\frac{\pi}{\sin(a\pi)}\left(\eta+|\lambda|\right)^{a-1}\leq C\|F\|+\left(\int^{\infty}_{-\infty}\frac{\mu^{2}(\xi)}{\xi^{2}+\eta}{\rm d}\xi\right)^{\frac{1}{2}} \left(\int^{\infty}_{-\infty}{(\xi^{2}+\eta)}|\phi_{2}|^{2}{\rm d}\xi\right)^{\frac{1}{2}}. \end{matrix}$

因此, 可得 $|g(L)|^{2}\leq C|\lambda|^{2-2a}\|F\|^{2}_{{\cal H}}+C|\lambda|^{2-2a}\|U\|_{{\cal H}}\|F\|_{{\cal H}}$. 使用类似的方法处理方程(4.5)可得$|f(L)|^{2}\leq C|\lambda|^{2-2a}\|F\|^{2}_{{\cal H}}+C|\lambda|^{2-2a}\|U\|_{{\cal H}}\|F\|_{{\cal H}}$.

另一方面, 需考虑 $\alpha_{1}|V_{x}(L)|^{2}, \beta|(\gamma V_{x}-P_{x})(L)|^{2}$$\|F\|_{{\cal H}}$的关系. 利用边界条件(2.8)和(2.9), 计算可得

$\alpha_{1}V_{x}(L)=-l_{1}\frac{\sin(a\pi)}{\pi}\int^{\infty}_{-\infty}\mu(\xi)\phi_{1}(\xi,t){\rm d}\xi-\gamma l_{2}\frac{\sin(a\pi)}{\pi}\int^{\infty}_{-\infty}\mu(\xi)\phi_{2}(\xi,t){\rm d}\xi,$
$\beta(\gamma V_{x}-P_{x})(L)=-l_{2}\frac{\sin(a\pi)}{\pi}\int^{\infty}_{-\infty}\mu(\xi)\phi_{2}(\xi,t){\rm d}\xi.$

利用引理4.3和不等式(4.2), 可得

$\begin{matrix} \alpha_{1}|V_{x}(L)|^{2}\leq C\|U\|_{{\cal H}}\|F\|_{{\cal H}},\quad \beta|(\gamma V_{x}-P_{x})(L)|^{2}\leq C\|U\|_{{\cal H}}\|F\|_{{\cal H}}. \end{matrix}$

应用${\cal H}$空间中范数的定义, 并结合引理 4.2, 有

$\|{U}\|^{2}_{{\cal H}}\leq C|\lambda|^{2-2a}\|F\|^{2}_{{\cal H}}+C|\lambda|^{2-2a}\|U\|_{{\cal H}}\|F\|_{{\cal H}},$

对于$|\lambda|>1$均成立. 即

$\|{U}\|^{2}_{{\cal H}}\leq C|\lambda|^{4-4a}\|F\|^{2}_{{\cal H}}. $

最终, 结合定理 4.1, 该定理得证. 证毕.

5 有热效应的压电梁系统的适定性

本节考虑带有傅里叶律热效应的压电梁系统的适定性. 利用定理2.1, 系统(1.8)-(1.10)可以被重写为一个增广模型

$\rho V_{tt}-\alpha V_{xx}+\gamma\beta P_{xx}+\delta\theta_{x}=0,\ \ (x,t)\in (0,L)\times (0,T), $
$\partial_{t}\phi_{1}(\xi,t)+\left(\xi^{2}+\eta\right)\phi_{1}(\xi,t)-V_{t}(L,t)\mu(\xi)=0, \ \ (\xi,t)\in (-\infty,+\infty)\times (0,+\infty),$
$\mu P_{tt}-\beta P_{xx}+\gamma\beta V_{xx}=0, \ \ (x,t)\in(0,L)\times (0,T),$
$\partial_{t}\phi_{2}(\xi,t)+\left(\xi^{2}+\eta\right)\phi_{2}(\xi,t)-P_{t}(L,t)\mu(\xi)=0,\ \ \ (\xi,t)\in (-\infty,+\infty)\times (0,+\infty),$
$c\theta_{x}-\kappa\theta_{xx}+\delta V_{xt}=0, \ \ \ (x,t)\in(0,L)\times (0,T),$

其边界条件为

$\theta_x(0,t)=\theta(L,t)=0,\ \ t\in (0,T),$
$V(0,t)=0,\quad\alpha V_{x}(L,t)-\gamma\beta P_{x}(L,t)=-l_{1}\frac{\sin(a\pi)}{\pi}\int^{\infty}_{-\infty}\mu(\xi)\phi_{1}(\xi,t){\rm d}\xi,\ \ t\in (0,T),$
$P(0,t)=0,\quad\beta P_{x}(L,t)-\gamma\beta V_{x}(L,t)=-l_{2}\frac{\sin(a\pi)}{\pi}\int^{\infty}_{-\infty}\mu(\xi)\phi_{2}(\xi,t){\rm d}\xi,\ \ t\in (0,T),$

初始条件为

$\begin{matrix}\label{5.17}\left(V(x,0), V_{t}(x,0), \phi_{1}(0), P(x,0), P_{t}(x,0),\phi_{2}(0),\theta(x,0)\right)=\left(V_{0},V_{1},\phi_{01},P_{0},P_{1},\phi_{02},\theta_{0}\right).\end{matrix}$

这里$x \in (0,L)$.

函数所在的空间定义为

${\cal H}_{1}:=\left[H^{1}_{*}(0,L)\times {L}^{2}(0,L)\times{L}^{2}(-\infty,+\infty)\right]^2\times H^1_c(0,L),$

这里 $ {{H}}^{1}_{*}(0,L)=\left\{f\in{{H}}^{1}(0,L):f(0)=0\right\}$, $H^1_c(0,L)=\left\{f\in{{H}}^{1}(0,L):f_x(0)=f(L)=0\right\}$, 相关的内积定义为

$\begin{matrix}\langle{U}_{1},{U}_{2}\rangle_{{\cal H}_{1}}&=&\int^{L}_{0}\left[\rho f_{1}\overline{f}_{2}+\mu g_{1}\overline{g}_{2}+\alpha_{1}V_{1,x}\overline{V}_{2,x}+\beta(\gamma V_{1,x}-P_{1,x})\overline{(\gamma V_{2,x}-P_{2,x})}+c\theta_{1}\overline{\theta}_{2}\right]{\rm d}x \\&&+\frac{\sin(a\pi)}{\pi}\int^{+\infty}_{-\infty}\left(l_{1}\phi_{1,1}\overline{\phi}_{1,2}+l_{2}\phi_{2,1}\overline{\phi}_{2,2}\right){\rm d}x,\end{matrix}$

这里${{U}}_{i}=\left(V_{i},f_{i}, \phi_{1,i}, P_{i}, g_{i},\phi_{2,i},\theta_{i}\right)\in {\cal {H}}_{1},i=1,2$.设算子 ${\cal A}_{1}:{\cal D}({\cal A}_{1})\subset {\cal H}_{1}\rightarrow {\cal H}_{1}$

$\begin{matrix}{\cal A}_{1}\left[\begin{array}{cc} V \\ f \\ \phi_{1} \\ P \\ g\\ \phi_{2} \\ \theta \end{array}\right]=\left[\begin{array}{cc}f \\ \frac{\alpha}{\rho}V_{xx} -\frac{\gamma\beta}{\rho}P_{xx}-\frac{\delta}{\rho} \theta_{x} \\ -\left(\xi^{2}+\eta\right)\phi_{1}(\xi,t)+f(L,t)\mu(\xi)\\g \\ -\frac{\gamma\beta}{\mu}V_{xx} + \frac{\beta}{\mu}P_{xx} \\ -\left(\xi^{2}+\eta\right)\phi_{2}(\xi,t)+g(L,t)\mu(\xi)\\ \frac{k}{c}\theta_{xx}-\frac{\delta}{c} f_{x}\end{array}\right].\end{matrix}$

这里

$\begin{matrix}{\cal D}({\cal A}_{1}):&=&\Big\{{{U}}\in {\cal H}; V,P \in {H}^{2}(0,L)\cap H^{1}_{*}(0,L), f,g\in H^{1}_{*}(0,L), \theta\in H^2(0,L)\cap H^{1}_c(0,L), \\ &&|\xi|\phi_{1},|\xi|\phi_{2}\in L^{2}(-\infty,+\infty), -\left(\xi^{2}+\eta\right)\phi_{1}(\xi,t)+f(L,t)\mu(\xi)\in L^{2}(-\infty,+\infty),\\&&-\left(\xi^{2}+\eta\right)\phi_{2}(\xi,t)+g(L,t)\mu(\xi)\in L^{2}(-\infty,+\infty)\Big\}.\end{matrix}$

那么, 系统(5.1)-(5.9)可以转化为一阶微分系统

$\begin{matrix} \left\{\begin{array}{ll}{{U}}_{t}={{\cal A}_{1}}{{U}},\\{{U}}(0)={{U}}_{0},\end{array}\right.\end{matrix}$

这里 ${{U}}_{0}=\left(V_{0},V_{1}, \phi_{1,0}, P_{0}, P_{1}, \phi_{2,0},\theta_{0}\right)^{T}$.

系统(5.1)-(5.9)的能量定义为

$\begin{matrix}E_{1}(t)&=&\frac{1}{2}\int^{L}_{0}\left[\rho\left|V_{t}\right|^{2}+\alpha_{1}\left|V_{x}\right|^{2}+\mu\left|P_{t}\right|^{2}+\beta\left|\gamma V_{x}-P_{x}\right|^{2}+c|\theta|^{2}\right] {\rm d}x\\&&+\frac{\sin(a\pi)}{2\pi}\int^{\infty}_{-\infty}\left(l_{1}|\phi_{1}|^{2}+l_{2}|\phi_{2}|^{2}\right){\rm d}\xi.\end{matrix}$

将方程(5.1), (5.3), (5.7)分别乘以 $ V_{t}$, $P_{t}$, $\theta$, 并在$(0,L)$上积分. 方程(5.2), (5.4) 分别乘以 $l_{1}\frac{\sin(a\pi)}{\pi}\phi_{1}$, $l_{2}\frac{\sin(a\pi)}{\pi}\phi_{2}$, 并在 $\mathbb{R}$积分. 将所得的积分结果相加, 整理即可得

$\begin{matrix}\label{5.21}\frac{\rm d}{{\rm d}t}E_{1}(t)=-\frac{\sin(a\pi)}{\pi}\int^{\infty}_{-\infty}(\xi^{2}+\eta)\left(l_{1}|\phi_{1}|^{2}+l_{2}|\phi_{2}|^{2}\right){\rm d}\xi-k\int^{L}_{0}|\theta_{x}|^{2} {\rm d}x.\end{matrix}$

定理5.1${U}_{0}\in {\cal D}({\cal A}_{1})$. 那么系统(5.10)存在唯一解${U}(t)={\cal S}_{{\cal A}_{1}}(t){U}_{0}$ 使得

${U}\in{C}\left([0,\infty);{\cal D}({\cal A}_{1})\right)\cap{C}^{1} \left([0,\infty); {\cal H}_{1}\right).$

证明主要利用Lumer-Phillips定理来说明算子 ${\cal A}_{1}$$C_{0}-$ 半群$\{{\cal S}_{{\cal A}_{1}}(t)\}_{t\geq0}$的无穷小生成元, 从而得到系统存在唯一解. 由等式(5.12)可知算子 ${\cal A}_{1}$是耗散、封闭且稠密的.因此, 仅需说明 $0\in \rho({\cal A})$. 也即, 对任意的 ${F}=\left(F_{1},F_{2},F_{3},F_{4},F_{5},F_{6},F_{7}\right)\in {\cal H}_{1}$, 需证明存在 ${{U}}=\left(V,f, \phi_{1}, P, g, \phi_{2},\theta\right)$ 使得 $-{\cal A}_1{U}={F}$. 等价地, 需要考虑下述系统的解的存在唯一性

$-f=F_{1}$
$-{\alpha}V_{xx}+{\gamma\beta}P_{xx}+\delta\theta_{x}={\rho}F_{2},$
$\left(\xi^{2}+\eta\right)\phi_{1}(\xi,t)-f(L)\mu(\xi)=F_{3},$
$-g=F_{4},$
$-{\beta}P_{xx}+{\gamma\beta}V_{xx}={\mu}F_{5},$
$\left(\xi^{2}+\eta\right)\phi_{2}(\xi,t)-g(L)\mu(\xi)=F_{6},$
$-k\theta_{xx}+\delta f_{x}=cF_{7}.$

由方程(5.13)和(5.16)知$f,g\in H^{1}_{*}(0,L)$

$\begin{matrix} f=-F_{1},\quad & \quad g=-F_{4}.\end{matrix}$

结合(5.20), (5.15)和(5.18)式, 有

$\begin{matrix} \phi_{1}=\frac{-F_{1}(L)\mu(\xi)+F_{3}}{\xi^{2}+\eta},\quad \quad\phi_{2}=\frac{-F_{4}(L)\mu(\xi)+F_{6}}{\xi^{2}+\eta}.\end{matrix}$

由等式(5.20), (5.19)和边界条件 (5.6), 可推得

$\begin{matrix} \theta=\frac{1}{k}\int^{L}_{x}\int^{y}_{0}\left(\delta F_{1,x}+cF_{7}\right)(\xi){\rm d}\xi dy. \end{matrix}$

由引理2.1, 得$\phi_{i}\in L^{2}(R)$. 至此, 仅需说明系统

$\begin{matrix} \left\{ \begin{array}{ll} {\alpha}V_{xx}-{\gamma\beta}P_{xx}=-\rho F_{2}+\frac{\delta}{k}\int^{x}_{0}\left(-\delta F_{1,x}-cF_{7}\right)(\xi){\rm d}\xi,\\ \beta P_{xx}-\gamma\beta V_{xx}=-\mu F_{5},\\ \alpha V_{x}(L,t)-\gamma\beta P_{x}(L,t)=-l_{1}\frac{\sin(a\pi)}{\pi}\int^{\infty}_{-\infty}\mu(\xi)\frac{-F_{1}(L)\mu(\xi)+F_{3}(\xi)}{\xi^{2}+\eta}{\rm d}\xi,\\[3mm] \beta P_{x}(L,t)-\gamma\beta V_{x}(L,t)=-l_{2}\frac{\sin(a\pi)}{\pi}\int^{\infty}_{-\infty}\mu(\xi)\frac{-F_{4}(L)\mu(\xi)+F_{6}(\xi)}{\xi^{2}+\eta}{\rm d}\xi \end{array} \right. \end{matrix}$

存在唯一解即可. 利用Lax-Milgram定理, 可推得该系统存在唯一解$(V,P,\theta)\in {H}^{1}_{*}(0,L)\times {H}^{1}_{*}(0,L)\times H^1_c(0,L)$. 结合 (5.22), (5.20) 和 (5.21)式, 得$0\in \rho({\cal A})$. 证毕.

6 有热效应的压电梁系统的指数稳定性

本节通过构造四个扰动泛函来证明具有傅里叶律热效应的压电梁系统是指数稳定的.

引理6.1

$I_{1}=\int^{L}_{0}\left( \rho V_{t}V+\mu P_{t}P \right){\rm d}x.$

对于任意$\eta_{1}>0$, 有

$\begin{matrix} \frac{\rm d}{{\rm d}t}I_{1}&\leq& -\left(\alpha_{1}-(1+\gamma^{2})\eta_{1}\right)\int^{L}_{0}|V_{x}|^{2}{\rm d}x-\left(\beta-\eta_{1}\right)\int^{L}_{0}|\gamma V_{x}-P_{x}|^{2}{\rm d}x +\frac{\delta^2 C_p}{2\eta_{1}}\int^{L}_{0}|\theta_{x}|^{2}{\rm d}x\\& &+\frac{ML}{\eta_{1}}\frac{\sin(a\pi)}{2\pi}\int^{\infty}_{-\infty}(\xi^{2}+\eta)\left(l_{1}|\phi_{1}|^{2}+l_{2}|\phi_{2}|^{2}\right){\rm d}\xi+\rho\int^{L}_{0}|V_{t}|^{2}{\rm d}x+\mu\int^{L}_{0}|P_{t}|^{2}{\rm d}x, \end{matrix}$

这里$C_p$是正常数.

由方程(5.1)和 (5.3) 可得 $\rho V_{tt}=\alpha V_{xx}-\gamma\beta P_{xx}-\delta\theta_{x}$, $\mu P_{tt}=\beta P_{xx}-\gamma\beta V_{xx}$. 然后, 将这些等式代入到$\frac{\rm d}{{\rm d}t}I_{1}$中, 则有

$\begin{matrix} \frac{\rm d}{{\rm d}t}I_{1}=\int^{L}_{0}\left(\left(\alpha V_{xx}-\gamma\beta P_{xx}\right)V-\delta\theta_{x}V+\left(\beta P_{xx}-\gamma\beta V_{xx}\right)P+\rho|V_{t}|^{2}+\mu|P_{t}|^{2}\right) {\rm d}x. \end{matrix}$

对上式进行分部积分并使用边界条件(5.6)-(5.8), 则有

$\begin{matrix} \frac{\rm d}{{\rm d}t}I_{1}&=&\int^{L}_{0}\left[\rho|V_{t}|^{2}+\mu|P_{t}|^{2}-\alpha_{1}|V_{x}|^{2}-\beta|\gamma V_{x}-P_{x}|^{2}-\delta\theta_{x}V\right]{\rm d}x\\& &-l_{1}\partial^{a,\eta}_{t}V(L,t)V(L,t)-l_{2}\partial^{a,\eta}_{t}P(L,t)P(L,t). \end{matrix}$

应用Hölder不等式和Young不等式 $ab\leq\frac{a^2}{4\varepsilon _{i}}+ \varepsilon _{i}b^2$, 其中$a,b\in \mathbb{R}$, $\varepsilon _{i}>0$$(i=1,2,3)$, 并结合不等式(4.17), 则有

$\begin{matrix} \left|-l_{1}\partial^{a,\eta}_{t}V(L,t)V(L,t)\right|&\leq &\frac{1}{4\varepsilon_1}\left[l_{1}\partial^{a,\eta}_{t}V(L,t)\right]^2+\varepsilon_1|V(L,t)|^2\\ &\leq &\frac{M}{4\varepsilon_1}\frac{\sin(a\pi)}{\pi}\int^{\infty}_{-\infty}(\xi^{2}+\eta)\left(l_{1}|\phi_{1}|^{2}+l_{2}|\phi_{2}|^{2}\right){\rm d}\xi+\varepsilon_1 L\int^L_0 |V_x|^2{\rm d}x. \\ \end{matrix}$

运用相同方法处理另一边界项, 有

$\begin{matrix} \left|-l_{2}\partial^{a,\eta}_{t}P(L,t)P(L,t)\right|\leq \frac{M}{4\varepsilon_2}\frac{\sin(a\pi)}{\pi}\int^{\infty}_{-\infty}(\xi^{2}+ \eta)\left(l_{1}|\phi_{1}|^{2}+l_{2}|\phi_{2}|^{2}\right){\rm d}\xi+\varepsilon_2 L\int^L_0 |P_x|^2{\rm d}x. \end{matrix}$

对积分项$\int^{L}_{0}|P_{x}|^{2}{\rm d}x$进行如下处理

$\begin{matrix}\label{6.5} \int^{L}_{0}|P_{x}|^{2}{\rm d}x =\int^{L}_{0}\left|\gamma V_{x}-P_{x}-\gamma V_{x}\right|^{2}{\rm d}x \leq 2\int^{L}_{0}\left|\gamma V_{x}-P_{x}\right|^{2}{\rm d}x+2\gamma^{2}\int^{L}_{0}\left|V_{x}\right|^{2}{\rm d}x. \end{matrix}$

另一方面, 由于$\theta(L,t)=0$$\theta(x,t)=\theta(L,t)-\int^L_x \theta_{y}(y,t){\rm d}y=-\int^L_x \theta_{y}(y,t){\rm d}y$. 那么, 结合Hölder不等式, 可得

$\begin{matrix} \int^L_0 {\theta}^2 {\rm d}x&=&\int^L_0 \bigg( \int^L_x \theta_{y}(y,t){\rm d}y \bigg)^2 {\rm d}x \leq C\int^L_0 \bigg( \int^L_x \theta^2_{y}(y,t){\rm d}y \bigg)\bigg( \int^L_x 1^2{\rm d}y\bigg) {\rm d}x\\ &=&C\int^L_0 (L-x)\bigg( \int^L_x \theta^2_{y}(y,t){\rm d}y \bigg) {\rm d}x. \end{matrix}$

$x\in (0,L)$, 故有$0< L-x< L$, 也即

$\begin{matrix}\label{poin} \int^L_0 {\theta}^2 {\rm d}x\leq CL \int^L_0 \bigg( \int^L_0 \theta^2_{x}(x,t){\rm d}x \bigg) {\rm d}x =C_p\int^L_0 \theta_{x}^2 {\rm d}x, \end{matrix}$

这里$C_p=CL^2>0$是一个常数. 结合不等式(6.3), (6.4)和(6.5), 应用Young不等式和不等式(6.6), 则不等式(6.2)可以进一步表示为

$\begin{matrix} \frac{\rm d}{{\rm d}t}I_{1}&\leq& -\left(\alpha_{1}-\varepsilon_{1}L-2\gamma^{2}\varepsilon_{2}L-\varepsilon_{3}C_p\right)\int^{L}_{0}|V_{x}|^{2}{\rm d}x-\left(\beta-2\varepsilon_{2}L\right)\int^{L}_{0}|\gamma V_{x}-P_{x}|^{2}{\rm d}x\\&& +\frac{\delta^{2}}{4\varepsilon_{3}}\int^{L}_{0}|\theta_{x}|^{2}{\rm d}x +\left(\frac{M}{4\varepsilon_{1}}+\frac{M}{4\varepsilon_{2}}\right)\frac{\sin(a\pi)}{\pi}\int^{\infty}_{-\infty}(\xi^{2}+\eta)\left(l_{1}|\phi_{1}|^{2}+l_{2}|\phi_{2}|^{2}\right){\rm d}\xi\\ && +\rho\int^{L}_{0}|V_{t}|^{2}{\rm d}x+\mu\int^{L}_{0}|P_{t}|^{2}{\rm d}x. \end{matrix}$

上式对任意$\varepsilon _{i}>0$均成立. 通过任取一个新常数$\eta_1>0$, 并选择 $\varepsilon _{i}>0$ 使得 $\varepsilon _{1}L=\varepsilon _{2}L=\varepsilon _{3}C_p=\frac{\eta_{1}}{2}$, 该引理得证. 证毕.

引理6.2

$I_{2}=\rho c \int^{L}_{0}\theta(x,t)\int^{L}_{x} V_{t}(y,t){\rm d}y {\rm d}x.$

对任意$\eta_{2}>0$, 有

$\begin{matrix} \frac{\rm d}{{\rm d}t}I_{2}&\leq& \eta_{2} \int^{L}_{0}|V_{x}|^{2}{\rm d}x+\eta_{2}\int^{L}_{0}|\gamma V_{x}-P_{x}|^{2}{\rm d}x-\left(\delta\rho-\frac{1}{2\gamma^{2}\beta^{2}}\eta_{2}\right)\rho\int^{L}_{0}|V_{t}|^{2}{\rm d}x\\& &+\left(C_{p}c\delta+\frac{\gamma^{2}\beta^{2}\rho^{2}k^{2}}{2\eta_{2}}+\frac{c^{2}C_{p}\alpha^{2}_{1}}{2\eta_{2}}+\frac{c^2 L C_p}{2 \eta_2}\right)\int^{L}_{0}|\theta_{x}|^{2}{\rm d}x\\ & &+\eta_2 M \frac{\sin(a\pi)}{2\pi}\int^{\infty}_{-\infty}(\xi^{2}+\eta)\left(l_{1}|\phi_{1}|^{2}+l_{2}|\phi_{2}|^{2}\right){\rm d}\xi, \end{matrix}$

这里$C_{p}$ 是正常数.

$\frac{\rm d}{{\rm d}t}I_{2}$中使用等式 $\rho V_{tt}=\alpha V_{xx}-\gamma\beta P_{xx}-\delta\theta_{x}$, $c\theta_{t}=k_{xx}-\delta V_{xt}$, 则有

$\begin{matrix} \frac{\rm d}{{\rm d}t}I_{2}=\rho\int^{L}_{0}\left(k\theta_{xx}-\delta V_{xt}\right)\int^{L}_{x} V_{t}{\rm d}y{\rm d}x+c\int^{L}_{0}\theta\int^{L}_{x}\left(\alpha V_{xx}-\gamma\beta P_{xx}-\delta\theta_{x}\right){\rm d}y{\rm d}x. \end{matrix}$

利用边界条件(5.6)-(5.8)可得

$\begin{matrix} \frac{\rm d}{{\rm d}t}I_{2}&=&\rho\int^{L}_{0}k\theta_{x}V_{t}{\rm d}x-\delta\rho\int^{L}_{0}|V_{t}|^{2}{\rm d}x-c\int^{L}_{0}\theta\left(\alpha V_{x}-\gamma\beta P_{x}\right){\rm d}x+c\delta\int^{L}_{0}|\theta|^{2}{\rm d}x\\& &+c\left[(\alpha V_x-\gamma\beta P_x)(L,t)\right]\int^{L}_{0}\theta(x,t){\rm d}x. \end{matrix}$

利用不等式(6.6)和Young不等式 $ab\leq\frac{a}{4\varepsilon _{i}}+ \varepsilon _{i}b$, 其中$a,b\in \mathbb{R} $, $\varepsilon _{i}>0$$(i=2,3,4)$, 并结合引理4.3, 可得, 对任意$\varepsilon _{i}>0$, 均有

$\begin{matrix} \frac{\rm d}{{\rm d}t}I_{2}&\leq& 2\varepsilon_{3}\alpha^{2}_{1} \int^{L}_{0}|V_{x}|^{2}{\rm d}x+2\varepsilon_{2}\gamma^{2}\beta^{2}\int^{L}_{0}|\gamma V_{x}-P_{x}|^{2}{\rm d}x-\left(\delta\rho-\varepsilon_{2}\right)\int^{L}_{0}|V_{t}|^{2}{\rm d}x\\& &+\left(C_{p}c\delta+\frac{\rho^{2}k^{2}}{4\varepsilon_{2}}+\frac{c^{2}C_{p}}{4\varepsilon_{3}}+\frac{c^2 LC_p}{4\varepsilon_4}\right)\int^{L}_{0}|\theta_{x}|^{2}{\rm d}x\\ & &+\varepsilon_4M\frac{\sin(a\pi)}{\pi}\int^{\infty}_{-\infty}(\xi^{2}+\eta)\left(l_{1}|\phi_{1}|^{2}+l_{2}|\phi_{2}|^{2}\right){\rm d}\xi. \end{matrix}$

接下来, 任取常数$\eta_{2}>0$ 并选择 $\varepsilon _{i}>0$使得 $\varepsilon _{2}=\frac{\eta_{2}}{2\gamma^{2}\beta^{2}}$, $\varepsilon _{3}=\frac{\eta_{2}}{2\alpha^{2}_{1}}$, $\varepsilon_4=\frac{\eta_2}{2}$, 即可证得该引理. 证毕.

引理6.3

$I_{3}=\rho\int^{L}_{0} V_{t}V {\rm d}x+\gamma\mu \int^{L}_{0}P_{t}V {\rm d}x.$

对任意$\eta_{3}>0$, 有

$\begin{matrix} \frac{\rm d}{{\rm d}t}I_{3}&\leq& -\left(\alpha_{1}-\frac{\alpha_{1}}{4\eta_{3}}\right)\int^{L}_{0}|V_{x}|^{2}{\rm d}x+(\rho+\eta_{3})\int^{L}_{0}|V_{t}|^{2}{\rm d}x+\frac{\gamma^{2}\mu^{2}}{4\eta_{3}}\int^{L}_{0}|P_{t}|^{2}{\rm d}x\\& &+\frac{4ML\eta_{3}}{\alpha_{1}}\frac{\sin(a\pi)}{2\pi}\int^{\infty}_{-\infty}(\xi^{2}+\eta)\left(l_{1}|\phi_{1}|^{2}+l_{2}|\phi_{2}|^{2}\right){\rm d}\xi+\frac{4C_{p}\delta^{2}\eta_{3}}{\alpha_{1}}\int^{L}_{0}|\theta_{x}|^{2}{\rm d}x, \end{matrix}$

其中, $C_{p}$是正常数.

由方程(5.1) 和 (5.3)可得 $\rho V_{tt}=\alpha V_{xx}-\gamma\beta P_{xx}-\delta\theta_{x}$, $\alpha=\alpha_{1}+\gamma^{2}\beta$ 以及 $\mu P_{tt}=\beta P_{xx}-\gamma\beta V_{xx}$.$\frac{\rm d}{{\rm d}t}I_{3}$中使用这些等式即可得

$\begin{matrix} \frac{\rm d}{{\rm d}t}I_{3}=\rho\int^{L}_{0}|V_{t}|^{2}{\rm d}x-\alpha_{1}\int^{L}_{0}|V_{x}|^{2}{\rm d}x +\alpha_{1}V_x(L,t)V(L,t)+\int^{L}_{0}\delta\theta V_{x}{\rm d}x +\int^{L}_{0}\gamma\mu P_{t}V_{t}{\rm d}x. \end{matrix}$

由不等式(6.6)和Young不等式, 可得

$\begin{matrix} \frac{\rm d}{{\rm d}t}I_{3}&\leq& -\left(\alpha_{1}-\varepsilon_{3}L-\varepsilon_{4}\right)\int^{L}_{0}|V_{x}|^{2}{\rm d}x+(\rho+\varepsilon_{5})\int^{L}_{0}|V_{t}|^{2}{\rm d}x+\frac{\gamma^{2}\mu^{2}}{4\varepsilon_{5}}\int^{L}_{0}|P_{t}|^{2}{\rm d}x\\& &+\frac{M}{2\varepsilon_{3}}\frac{\sin(a\pi)}{2\pi}\int^{\infty}_{-\infty}(\xi^{2}+\eta)\left(l_{1}|\phi_{1}|^{2}+l_{2}|\phi_{2}|^{2}\right){\rm d}\xi+\frac{C_{p}\delta^{2}}{4\varepsilon_{4}}\int^{L}_{0}|\theta_{x}|^{2}{\rm d}x, \end{matrix}$

对任意$\varepsilon _{i}>0$均成立. 通过任取一个常数 $\eta_{3}>0$, 并选择合适的$\varepsilon _{i}>0$ 使得 $\varepsilon _{3}L=\frac{\alpha_{1}}{8\eta_{3}}$, $\varepsilon _{4}=\frac{\alpha_{1}}{8\eta_{3}}$, $\varepsilon_{5}=\eta_{3}$, 可推得该引理中的不等式是成立的. 证毕.

引理6.4

$I_{4}=\rho\int^{L}_{0} V_{t}(\gamma V-P){\rm d}x+\gamma\mu \int^{L}_{0}P_{t}(\gamma V-P){\rm d}x.$

对任意的$\eta_{4}>0$, 有

$\begin{matrix} \frac{\rm d}{{\rm d}t}I_{4}&\leq& \frac{\alpha_{1}^{2}}{\eta_{4}}\int^{L}_{0}|V_{x}|^{2}{\rm d}x+\frac{3\eta_{4}}{4}\int^{L}_{0}|\gamma V_{x}-P_{x}|^{2}{\rm d}x +\frac{\delta^{2}C_{p}}{\eta_{4}}\int^{L}_{0}|\theta_{x}|^{2}{\rm d}x\\ &&-\left(\gamma\mu-\frac{\eta_{4}}{2}\right)\int^{L}_{0}|P_{t}|^{2}{\rm d}x+\frac{\alpha_{1}^{2}M}{\eta_{4}}\frac{\sin(a\pi)}{2\pi}\int^{\infty}_{-\infty}(\xi^{2}+\eta)\left(l_{1}|\phi_{1}|^{2}+l_{2}|\phi_{2}|^{2}\right){\rm d}\xi\\&&+\left[\gamma\rho+\frac{\left(\rho^{2}+\gamma^{4}\mu^{2}\right)}{\eta_{4}}\right]\int^{L}_{0}|V_{t}|^{2}{\rm d}x. \end{matrix}$

其中, $C_{p}$是正常数.

利用方程(5.1), 并进行分部积分, 可得

$\begin{matrix} \frac{\rm d}{{\rm d}t}I_{4}&=&\int^{L}_{0}\bigg[\rho V_{t}(\gamma V-P)_t+\gamma\mu P_{t}(\gamma V-P)_{t}\bigg]{\rm d}x\\ &&-\int^{L}_{0}\bigg[\alpha_{1}V_{x}(\gamma V-P)_{x}-\delta\theta(\gamma V-P)_{x}\bigg]{\rm d}x+\int^{L}_{0}\gamma(\gamma\beta V_{xx}-\beta P_{xx})(\gamma V-P){\rm d}x\\&&+\gamma\int^{L}_{0}\mu P_{tt}(\gamma V-P){\rm d}x+\alpha_{1} V_{x}(L,t)(\gamma V-P)(L,t). \end{matrix}$

由方程(5.3)可得$\mu P_{tt}=\beta P_{xx}-\gamma\beta V_{xx}$. 那么, 上述不等式的第三项和第四项可转换为

$\int^{L}_{0}\gamma(\gamma\beta V_{xx}-\beta P_{xx})(\gamma V-P){\rm d}x+\gamma\int^{L}_{0}\mu P_{tt}(\gamma V-P){\rm d}x=0.$

整理以上结果得

$\begin{matrix} \frac{\rm d}{{\rm d}t}I_{4}&=&\gamma\rho\int^{L}_{0}|V_{t}|^{2}{\rm d}x-\gamma\mu\int^{L}_{0}|P_{t}|^{2}{\rm d}x-\int^{L}_{0}\rho V_{t}P_{t}{\rm d}x+\int^{L}_{0}\gamma^{2}\mu P_{t}V_{t}{\rm d}x\\& &-\alpha_{1}\int^{L}_{0}V_{x}(\gamma V-P)_{x}{\rm d}x+\delta\int^{L}_{0}\theta(\gamma V-p)_{x}{\rm d}x+\alpha_{1}V_{x}(L,t)(\gamma V-P)(L,t). \end{matrix}$

利用不等式(6.6)和Young不等式, 可得, 对任意$\varepsilon _{i}>0$, 均有

$\begin{matrix} \frac{\rm d}{{\rm d}t}I_{4}&\leq& \frac{\alpha_{1}^{2}}{4\varepsilon_{6}}\int^{L}_{0}|V_{x}|^{2}{\rm d}x+(\varepsilon_{6}+\varepsilon_{7}+\varepsilon_{8})\int^{L}_{0}|\gamma V_{x}-P_{x}|^{2}{\rm d}x +\frac{\delta^{2}C_{p}}{4\varepsilon_{7}}\int^{L}_{0}|\theta_{x}|^{2}{\rm d}x\\& &+\frac{\alpha_{1}^{2}M}{\eta_{4}}\frac{\sin(a\pi)}{2\pi}\int^{\infty}_{-\infty}(\xi^{2}+\eta)\left(l_{1}|\phi_{1}|^{2}+l_{2}|\phi_{2}|^{2}\right){\rm d}\xi\\ & &+\left(\gamma\rho+\frac{\rho^{2}}{4\varepsilon_{4}}\frac{\gamma^{4}\mu^{2}}{4\varepsilon_{5}}\right)\int^{L}_{0}|V_{t}|^{2}{\rm d}x -\left(\gamma\mu-\varepsilon_{4}-\varepsilon_{5} \right)\int^{L}_{0}|P_{t}|^{2}{\rm d}x. \end{matrix}$

通过定义一个新常数$\eta_{4}$, 且选择$\varepsilon _i=\frac{\eta_{4}}{4}$$(i={4,5,6,7,8})$, 即可推得不等式(6.7). 证毕.

引理6.5$N$ 是一个足够大的常数, 泛函${\cal L}(t)=N E(t)+N_{1} I_{1}+N_{2}I_{2}+N_{3} I_{3}+N_{4}I_{4}$. 那么存在正常数$m_{1}$$m_{2}$, 使得$m_{1}E(t)\leq {\cal L}(t)\leq m_{2}E(t). $这里$E(t)$是带有热效应的压电梁系统的总能量, $I_{i}\, (i=1,2,3,4)$ 是引理6.1-6.4所定义的泛函, $N_{i} (i = 1, 2, 3, 4)$ 均为常数.

显然, 由泛函${\cal L}(t)$的定义可知

$|{\cal L}(t)-N E(t)|=|N_{1} I_{1}+N_{2}I_{2}+N_{3} I_{3}+N_{4}I_{4}|.$

使用引理6.1-6.4的结果, 可得, 存在常数$C > 0$ 满足$|{\cal L}(t)-N E(t)|\leq C E(t).$也即,$(N-C)E(t)\leq {\cal L}(t)\leq (N+C)E(t).$证毕.

定理6.1$(V, V_{t}, \phi_{1}, P, P_{t}, \phi_{2}, \theta)$ 是系统(5.1)-(5.9)的任意一个解. 那么存在 $M > 0$ 和仅依赖于初值条件的$\omega > 0$使得$E(t)\leq E(0)e^{-\omega t}.$

$\eta,l_1,l_2>0, a\in(0,1)$

$\begin{matrix} -\frac{\sin(a\pi)}{2\pi}\int^{\infty}_{-\infty}(\xi^{2}+\eta)\left(l_{1}|\phi_{1}|^{2}+l_{2}|\phi_{2}|^{2}\right){\rm d}\xi\leq -\eta \frac{\sin(a\pi)}{2\pi}\int^{\infty}_{-\infty}\left(l_{1}|\phi_{1}|^{2}+l_{2}|\phi_{2}|^{2}\right){\rm d}\xi. \end{matrix}$

然后, 将引理6.1-6.4的结果代入到${\cal L}(t)$的定义中, 并利用上式结果, 可得

$\begin{matrix} \frac{\rm d}{{\rm d}t}{\cal L}(t)& \leq &-\left[N-N_{1}\frac{ML}{\eta_{1}}-N_2 \eta_2 M-N_{3}\frac{4ML\eta_{3}}{\alpha_{1}}-N_{4}\frac{\alpha_{1}^{2}M}{\eta_{4}}\right]\\&&\times \eta \frac{\sin(a\pi)}{2\pi}\int^{\infty}_{-\infty}\left[l_{1}|\phi_{1}|^{2}+l_{2}|\phi_{2}|^{2}\right]{\rm d}\xi \\ & &-\left[Nk-N_{1}\frac{\delta^{2}C_p}{2\eta_{1}}-N_{2}C_{\eta_{2}}-N_{3}\frac{4C_{p}\delta^{2}\eta_{3}}{\alpha_{1}}-N_{4}\frac{C_{p}\delta^{2}}{\eta_{4}}\right]\int^{L}_{0}|\theta_{x}|^{2}{\rm d}x \\ & &-\left[N_{2}\delta\rho-N_{2}\frac{\eta_{2}}{2\gamma^{2}\beta^{2}}-N_{1}\rho-N_{3}\rho-N_{3}\eta_{3}-N_{4}\gamma\rho-N_{4}\frac{\rho^{2}+\gamma^{4}\mu^{2}}{\eta_{4}}\right]\int^{L}_{0}|V_{t}|^{2}{\rm d}x \\ & &-\left[N_{4}\gamma\mu-N_{4}\frac{\eta_{4}}{2}-N_{1}\mu-N_{3}\frac{\gamma^{2}\mu^{2}}{4\eta_{4}}\right]\int^{L}_{0}|P_{t}|^{2}{\rm d}x \\ &&-\left[N_{1}\alpha_{1}-N_{1}(1+\gamma^{2})\eta_{1}-N_{2}\eta_{2}+N_{3}\alpha_{1}-N_{3}\frac{\alpha_{1}}{4\eta_{3}}-N_{4}\frac{\alpha^{2}_{1}}{\eta_{4}}\right]\int^{L}_{0}|V_{x}|^{2}{\rm d}x \\& &-\left[N_{1}\beta-N_{1}\eta_{1}-N_{2}\eta_{2}-N_{4}\frac{3\eta_{4}}{4}\right]\int^{L}_{0}|\gamma V_{x}-P_{x}|^{2}{\rm d}x.\label{ly1} \end{matrix}$

这里 $C_{\eta_{2}}=C_{p}c\delta+\frac{\gamma^{2}\beta^{2}\rho^{2}k^{2}}{2\eta_{2}}+\frac{c^{2}C_{p}\alpha^{2}_{1}}{2\eta_{2}}+\frac{c^2 C_p L}{2\eta_2}$.

接下来, 取定常数$\frac{N_{3}}{2}=\eta_{3}$, $3N_{4}\eta_{4}=4\beta$, $N_{1}\eta_{1}=\beta$, $N_{2}\eta_{2}=\beta$.$\frac{3 \alpha^{2}_{1}N_{4}}{4\beta}$ 简记为 $C_{N_{4}}$, 那么 (6.8) 式可被重写为

$\begin{matrix} \frac{\rm d}{{\rm d}t}{\cal L}(t)& \leq& -\lambda_{1} \eta\frac{\sin(a\pi)}{2\pi}\int^{\infty}_{-\infty}\left(l_{1}|\phi_{1}|^{2}+l_{2}|\phi_{2}|^{2}\right){\rm d}\xi-\lambda_{2}\int^{L}_{0}|\theta_{x}|^{2}{\rm d}x-\lambda_{3}\int^{L}_{0}|V_{t}|^{2}{\rm d}x\\ & &-\lambda_{4}\int^{L}_{0}|P_{t}|^{2}{\rm d}x-\lambda_{5}\int^{L}_{0}|V_{x}|^{2}{\rm d}x- \lambda_{6}\int^{L}_{0}|\gamma V_{x}-P_{x}|^{2}{\rm d}x. \end{matrix}$

这里

$\begin{matrix} & &\lambda_{1}:=N-N_{1}\frac{ML}{\eta_{1}}-N_2 \eta_2 M-N_{3}\frac{4ML\eta_{3}}{\alpha_{1}}-N_{4}\frac{\alpha_{1}^{2}M}{\eta_{4}},\\ & &\lambda_{2}:=Nk-N_{1}\frac{\delta^{2}C_p}{2\eta_{1}}-N_{2}C_{\eta_{2}}-N_{3}\frac{4C_{p}\delta^{2}\eta_{3}}{\alpha_{1}}-N_{4}\frac{C_{p}\delta^{2}}{\eta_{4}},\\ &&\lambda_{3}:=N_{2}\delta\rho-\frac{1}{2\gamma^{2}\beta}-N_{1}\rho-N_{3}\rho-N_{3}\eta_{3}-N_{4}\gamma\rho-\frac{3N^{2}_{4}}{4\beta}\left(\rho^{2}+\gamma^{4}\mu^{2}\right),\\& &\lambda_{4}:=N_{4}\gamma\mu-\frac{2\beta}{3}-N_{1}\mu-\frac{\gamma^{2}\mu^{2}}{8},\\ & &\lambda_{5}:=N_{1}\alpha_{1}-(2+\gamma^{2})\beta+N_{3}\alpha_{1}-\frac{\alpha_{1}}{2}-C_{N_{4}}N_{4},\\ & &\lambda_{6}:=N_{1}\beta-3\beta. \end{matrix}$

至此, 需选择合适的$N$$N_{i}\ (i=1,2,3,4)$ 使得 $\lambda_{i}\ (i=1,\cdots,6)$ 均为正数. 当$\lambda_{6}>0$时, $N_{1}$ 需满足

$ N_{1}>3.$

为实现$\lambda_{4}>0$, $N_{4}$需满足

$\begin{matrix} N_{4}>\frac{2\beta}{3\gamma\mu}+\mu N_{1}+\frac{\gamma\mu}{8}. \end{matrix}$

使用相似的计算技巧, 推得$N_{3}$$N_{2}$ 需满足

$\begin{matrix} N_{3}&>&\frac{(2+\gamma^{2})\beta}{\alpha_{1}}+\frac{1}{2}+\frac{C_{N_{4}}N_{4}}{\alpha_{1}}-N_{1},\\ N_{2}&>&\frac{1}{2\gamma^{2}\beta\delta\rho}+\frac{ N_{1}}{\delta}+\frac{ N_{3}}{\delta}+\frac{ N_{3}^{2}}{2\delta\rho}+\frac{\gamma N_{4}}{\delta}+\frac{3 N_{4}^{2}}{4\beta\delta\rho}\left(\rho^{2}+\gamma^{4}\mu^{2}\right). \end{matrix}$

一旦 $N_i, i={1,2,3,4}$ 被固定, 则可以选择 $N$ 充分大, 用以确保 $\lambda_{1},\lambda_{2}>0$. 最终可得, 存在常数 $N_{0}:=2\min\{{\lambda_{1}\eta}, \frac{\lambda_{2}}{c}, \frac{\lambda_{3}}{\rho}, \frac{\lambda_{4}}{\mu}, \frac{\lambda_{5}}{\alpha_{1}}, \frac{\lambda_{6}}{\beta}\}$ 使得

$\frac{\rm d}{{\rm d}t}{\cal L}(t)\leq -N_{0}E(t), \quad t\geq 0. $

由引理6.5易得存在$M > 0$$\omega > 0$ 使得$E(t)\leq E(0)e^{-\omega t}.$证毕.

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DOI:10.1007/s00033-019-1224-x      [本文引用: 1]

Mustafa M I.

Optimal energy decay result for nonlinear abstract viscoelastic dissipative systems

Z Angew Math Phys, 2021, 72(2): Paper No. 67

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Cardozo C L, Jorge Silva M A, Ma T F, Muñoz Rivera J E.

Stability of Timoshenko systems with thermal coupling on the bending moment

Math Nachr, 2019, 292(12): 2537-2555

DOI:10.1002/mana.201800546      [本文引用: 1]

The Timoshenko system is a distinguished coupled pair of differential equations arising in mathematical elasticity. In the case of constant coefficients, if a damping is added in only one of its equations, it is well-known that exponential stability holds if and only if the wave speeds of both equations are equal. In the present paper we study both non-homogeneous and homogeneous thermoelastic problems where the model's coefficients are non-constant and constants, respectively. Our main stability results are proved by means of a unified approach that combines local estimates of the resolvent equation in the semigroup framework with a recent control-observability analysis for static systems. Therefore, our results complement all those on the linear case provided in [22], by extending the methodology employed in [4] to the case of Timoshenko systems with thermal coupling on the bending moment.

Feng B, Yang X G.

Long-time dynamics for a nonlinear Timoshenko system with delay

Appl Anal, 2017, 96(4): 606-625

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Stabilization of a thermoelastic laminated beam with past history

Appl Math Optim, 2019, 80(1): 103-133

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Long-time behavior for a thermoelastic microbeam problem with time delay and the Coleman-Gurtin thermal law

Acta Math Sci, 2021, 41B(2): 609-632

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Boundedness and exponential stabilization in a parabolic-elliptic Keller-Segel model with signal-dependent motilities for local sensing chemotaxis

Acta Math Sci, 2022, 42B(3): 825-846

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欧阳成, 汪维刚, 莫嘉琪.

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Acta Math Sci, 2020, 40A(2): 452-459

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