数学物理学报, 2021, 41(4): 1088-1096 doi:

论文

一类扩散波方程的PDP反馈控制和稳定性分析

范东霞,, 赵东霞,, 史娜,, 王婷婷,

The PDP Feedback Control and Stability Analysis of a Diffusive Wave Equation

Fan Dongxia,, Zhao Dongxia,, Shi Na,, Wang Tingting,

通讯作者: 赵东霞, E-mail: zhaodongxia6@sina.com

收稿日期: 2020-09-24  

基金资助: 国家自然科学基金青年基金.  61603351
山西省自然科学基金面上项目.  201801D121027
国家自然科学基金青年基金.  201701-D221121
中北大学第十七届研究生科技立项.  20201747

Received: 2020-09-24  

Fund supported: the NSFC.  61603351
the NSF of Shanxi Province.  201801D121027
the NSF of Shanxi Province.  201701-D221121
the 17th Postgraduate Science and Technology Project of The North University of China.  20201747

作者简介 About authors

范东霞,E-mail:119256724@qq.com , E-mail:119256724@qq.com

史娜,E-mail:1835446397@qq.com , E-mail:1835446397@qq.com

王婷婷,E-mail:807790440@qq.com , E-mail:807790440@qq.com

Abstract

In this paper, a position and delayed position (PDP) feedback controller is established for the Hayami diffusive wave equation. The well-posedness of the closed system is studied firstly, and then, by the method of Lyapunov function, the exponential stability is obtained. Finally, compared with the previous results, the range of the control parameters is expanded in this paper, which shows the efficiency and feasibility of PDP feedback.

Keywords: Diffusive wave equation ; PDP feedback ; Lyapunov function ; Exponential stability

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本文引用格式

范东霞, 赵东霞, 史娜, 王婷婷. 一类扩散波方程的PDP反馈控制和稳定性分析. 数学物理学报[J], 2021, 41(4): 1088-1096 doi:

Fan Dongxia, Zhao Dongxia, Shi Na, Wang Tingting. The PDP Feedback Control and Stability Analysis of a Diffusive Wave Equation. Acta Mathematica Scientia[J], 2021, 41(4): 1088-1096 doi:

1 引言

自从1871年法国科学家圣维南提出了圣维南方程组, 圣维南方程组一直在水力及其他工程领域中有着广泛的应用, 许多学者致力于研究它的稳定性和各类渠道边界控制问题[1-6]. 但由于圣维南方程组是一组非线性的偏微分方程组, 无法得出解析解, 从而对其控制是非常困难的. 为此, 人们做出一系列的假设对其简化并线性化处理, 从而衍生了著名的Hayami模型

$ \begin{equation} {\frac{\partial{q}}{\partial{t}}} = {\alpha}{\frac{\partial^2{q}}{\partial{x^2}}}-{\beta}{\frac{\partial{q}}{\partial{x}}}, \end{equation} $

其中, $ x $表示空间位置(m), $ t $表示时间(s), $ q(x, t) $表示流量($ m^3/s $), 正常数$ {\alpha} $$ {\beta} $分别表示扩散率和波速. 该模型不必测量太多的渠道物理参数, 只需要两个参数来表征水流. 对于模型(1.1) 的控制和稳定问题已经有很多学者进行了研究[7-9], 其中文献[8] 和[9]通过如下分布式输入来控制波方程(1.1) 的输出稳定性

$ \begin{equation} \left\{\begin{array}{ll} { } {\frac{\partial{q}}{\partial{t}}} = {\alpha}{\frac{\partial^2{q}}{\partial{x^2}}}-{\beta}{\frac{\partial{q}}{\partial{x}}}+b(x)U(t), \;\; x\in (0, l), \\ { } q(0, {t}) = {\frac{\partial{q}}{\partial{x}}}(l, {t}) = 0, \;\;\;\; y(t) = {q(l, {t})}. \end{array}\right. \end{equation} $

其中, $ U(t) $代表输入控制, $ b(x)\in L^2(0, l) $是控制项的空间分布函数, $ y(t) $代表输出观测. 文献[8] 本质上是采用比例控制器$ U(t) = {\kappa{q(l, t)}} $去镇定系统, 其中$ {\kappa}\neq 0 $代表反馈增益, 利用算子半群理论和谱分析的方法讨论了系统的适定性, 谱的构成以及系统的指数稳定性. 进而, 考虑到实际中系统的控制器、传感器之间在通信时不可避免地会产生时滞, 因此文献[9] 讨论了输入控制中带有时滞的情形: $ U(t) = {\kappa{q(l, t-\tau)}} $, 其中, $ {\tau}>0 $表示时滞. 结果表明, 只要控制参数满足某个约束条件, 同样可以建立系统的指数稳定性.

另一方面, 对于不稳定ODE系统, 文献[10-12] 采用PDP(position and delayed position) 控制器, 即瞬时位置和时滞位置的线性组合来设计控制器, 取得了一系列指数稳定性的研究成果. 受上述文献启发, 本文考虑PDP反馈

$ \begin{equation} U(t) = {k_1{q(l, t)}}+{k_2{q(l, t-\tau)}}, \;\;\;\;k_1, k_2\in {{\Bbb R}} , \;\;\tau>0 \end{equation} $

对系统(1.2) 稳定性的影响.

2 模型的建立与准备工作

将(1.3) 式代入(1.2) 式, 并给定初始值$ q_0 $$ u_0 $, 从而可得如下时滞PDE闭环系统

$ \begin{equation} \left\{\begin{array}{ll} { } \frac{\partial{q}}{\partial{t}} = \alpha{\frac{\partial^2{q}}{\partial{x^2}}}-{\beta}{\frac{\partial{q}}{\partial{x}}} +b(x)\big(k_1q(l, t)+k_2 q(l, t-\tau)\big), \;\; (x, t)\in(0, l)\times(0, \infty), \\ { } q(0, {t}) = {\frac{\partial{q}}{\partial{x}}}(l, {t}) = 0, \;\;t>0, \\ q(x, 0) = q_0(x), \;\;x\in(0, l), \\ q(l, \theta) = u_0(\theta), \;\;\theta\in(-\tau, 0). \end{array}\right. \end{equation} $

考虑到一阶双曲PDE初值问题

$ \begin{equation} \left\{\begin{array}{ll} { }\tau{\frac{\partial{u}}{\partial{t}}}(\rho, t)+{\frac{\partial{u}}{\partial{\rho}}}(\rho, t) = 0, \;\;(\rho, t)\in(0, 1)\times(0, \infty), \\ u(\rho, 0) = q(l, -\tau\rho) \end{array}\right. \end{equation} $

的解可以表示为

$ \begin{equation} u(\rho, t) = q(l, t-\tau\rho). \end{equation} $

于是可将时滞PDE系统(2.1) 改写为如下PDE-PDE耦合系统的形式

$ \begin{equation} \left\{\begin{array}{ll} { }\frac{\partial{q}}{\partial{t}} = \alpha{\frac{\partial^2{q}}{\partial{x^2}}}-{\beta}{\frac{\partial{q}}{\partial{x}}} +b(x)\big(k_1u(0, t)+k_2 u(1, t)\big), \;\; (x, t)\in(0, l)\times(0, \infty), \\ { }\tau{\frac{\partial{u}}{\partial{t}}}(\rho, t)+{\frac{\partial{u}}{\partial{\rho}}}(\rho, t) = 0, \;\;(\rho, t)\in(0, 1)\times(0, \infty), \\ { } q(0, {t}) = {\frac{\partial{q}}{\partial{x}}}(l, {t}) = 0, \;\;t>0, \\ q(x, 0) = q_0(x), \;\;x\in(0, l), \\ u(\rho, 0) = u_0(-\tau\rho), \;\;\rho\in(0, 1). \end{array}\right. \end{equation} $

为后续计算方便, 不妨令$ l = 1 $, 定义

$ \begin{equation} \phi(t) = (q(\cdot, t), u(\cdot, t)), \;\;\;\;\phi_0 = (q_0, u_0), \end{equation} $

那么系统(2.4) 转化为

$ \begin{equation} \left\{\begin{array}{ll} { } \dot{\phi}(t) = \Big(\alpha q''-\beta q'+b(x)\big(k_1u(0, t)+k_2 u(1, t)\big), -\frac{1}{\tau}u'\Big), \\ { } q(0, {t}) = {\frac{\partial{q}}{\partial{x}}}(1, {t}) = 0, \;\;t>0, \\ \phi(0) = \phi_0, \end{array}\right. \end{equation} $

其中, $ ' $表示对变量$ x $$ \rho $的一阶导数, $ '' $表示对变量$ x $$ \rho $的二阶导数.

设Hilbert状态空间为

$ \begin{equation} {\cal H} = {L^2(0, 1)}\times{L^2(0, 1)}, \end{equation} $

其中的内积定义为: $ \forall \phi_1 = (q_1, u_1), \;\phi_2 = (q_2, u_2)\in {\cal H} $, 有

$ \begin{equation} \langle\phi_1, \phi_2\rangle_{{\cal H}} = \int_0^1 q_1(x)\overline{q_2(x)}{\rm d}x+\tau\xi\int_0^1 u_1(\rho)\overline{u_2(\rho)}{\rm d}\rho, \end{equation} $

其中, $ \xi<\beta $是一个正常数.

定义线性算子$ {\cal A}:{\cal H}\longrightarrow{\cal H} $如下

$ \begin{equation} \left\{\begin{array}{ll} { } {\cal A}(q, u) = \Big(\alpha q''-\beta q'+b(x)\big(k_1 u(0)+k_2 u(1)\big), -\frac{1}{\tau}u'\Big), \\ { } D({\cal A}) = \{(q, u)\in{\cal H}\mid q\in H^2(0, 1), \;\;u\in H^1(0, 1), \;\;q(0) = q'(1) = 0\}. \end{array}\right. \end{equation} $

那么系统(2.6) 可以写成Hilbert状态空间$ {\cal H} $上的抽象发展方程的形式

$ \begin{equation} \left\{\begin{array}{ll} \dot{\phi}(t) = {\cal A}\phi(t), \; t>0, \\ \phi(0) = \phi_0. \end{array}\right. \end{equation} $

3 系统(2.10) 的适定性

定理3.1  由(2.9) 式定义的算子$ {\cal A} $生成$ {\cal H} $上的一个$ C_0 $半群.

   结合(2.8) 和(2.9) 式以及分部积分公式可得

$ \begin{eqnarray} \langle{\cal A}{\phi}, {\phi}\rangle_{{\cal H}} & = &-\alpha\int_{0}^{1}{q'}^2(x){\rm d}x-\frac{\beta}{2}u^2(0)+(k_1u(0)+k_2u(1))\int_{0}^{1}b(x)q(x){\rm d}x \\ &&-\frac{\xi}{2}[u^2(1)-u^2(0)]\\ & = &-\alpha\int_{0}^{1}{q'}^2(x){\rm d}x+\frac{\xi-\beta}{2}u^2(0)- \frac{\xi}{2}u^2(1)+k_1u(0)\int_{0}^{1}b(x)q(x){\rm d}{x}\\ &&+k_2u(1)\int_{0}^{1}b(x)q(x){\rm d}x\\ &\leq&-\alpha\int_{0}^{1}{q'}^2(x){\rm d}x+\frac{\left|{k_1}\right|A_1}{2}{u^2(0)} +\frac{\left|{k_1}\right|\|b\|^2}{2A_1}\int_{0}^{1}{q^2(x)}{\rm d}x +\frac{\left|{k_2}\right|A_2}{2}{u^2(1)}\\ &&+\frac{\left|{k_2}\right|\|b\|^2}{2A_2}\int_{0}^{1}{q^2(x)}{\rm d}x +\frac{\xi-\beta}{2}u^2(0)-\frac{\xi}{2}u^2(1)\\ & = &-\alpha\int_{0}^{1}{q'}^2(x){\rm d}x+\frac{(\left|{k_1}\right|A_2 +\left|{k_2}\right|A_1)\|b\|^2}{2A_1 A_2}\int_{0}^{1}{q^2(x)}{\rm d}x \\ &&+\frac{\xi-\beta+\left|{k_1}\right|A_1}{2}u^2(0)+\frac{\left|{k_2}\right|A_2-\xi}{2}u^2(1), \end{eqnarray} $

其中, $ A_1 $$ A_2 $为正常数, 且满足$ A_1 = \frac{\beta-\xi}{\left|{k_1}\right|}, \;\; A_2 = \frac{\xi}{\left|{k_2}\right|} $, 我们可以选择$ \xi<\beta $, 于是可得

$ \begin{eqnarray} Re\langle{\cal A}{\phi}, {\phi}\rangle_{{\cal H}}&\leq&-\alpha\int_{0}^{1}{q'}^2(x){\rm d}x+\frac{({\left|{k_{1}}\right|}\cdot\frac{\xi}{\left|k_{2}\right|}+ \left|k_{2}\right|\cdot\frac{\beta-\xi}{\left|k_{1}\right|})\|b\|^2}{2\cdot\frac{\xi}{\left|{k_{2}}\right|}\cdot\frac{\beta-\xi}{\left|{k_{1}}\right|}}\int_{0}^{1}q^2(x){\rm d}x\\ & = &-\alpha\int_{0}^{1}{q'}^2(x){\rm d}x+\frac{(\xi k_1^2+(\beta-\xi)k_2^2){\|b\|^2}}{2\xi\cdot(\beta-\xi)}\int_{0}^{1}q^2(x){\rm d}x\\ &\leq&\frac{(\xi k_1^2+(\beta-\xi)k_2^2){\|b\|^2}}{2\xi\cdot(\beta-\xi)}\int_{0}^{1}q^2(x){\rm d}x. \end{eqnarray} $

显然, $ {\cal A} $是闭稠定的线性算子. 通过计算可得, $ {\cal A} $的伴随算子$ {\cal A}^{\ast} $定义如下

$ \begin{equation} \left\{\begin{array}{ll} { } {\cal A}^{\ast}(q, u) = \Big(\alpha q''+\beta q', \frac{1}{\tau}u'\Big), \\ { } D({\cal A}^{\ast}) = \Big\{(q, u)\in{\cal H}\mid q\in H^2(0, 1), \;\;u\in H^1(0, 1), \;\;q(0) = 0, \;\\ { } \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\alpha q'(1)+\beta q(1) = 0, \;\;u(0)+\frac{k_1}{k_2}u(1) = 0, \;\;\xi u(1) = k_{2}\int_{0}^{1}b(x)q(x){\rm d}x\Big\}. \end{array}\right. \end{equation} $

$ (q, u)\in D({\cal A}^{\ast}) $, 结合(2.8) 和(3.3) 式以及分部积分公式可得

$ \begin{eqnarray} \langle{\cal A}^{\ast}(q, u), (q, u)\rangle_{{\cal H}} & = &\alpha\int_{0}^{1}q''q{\rm d}x+\beta\int_{0}^{1}q'q{\rm d}x+\tau\xi\int_{0}^{1}\frac{u'}{\tau}ud{\rho}\\ & = &\alpha q'(1)q(1)-\alpha\int_{0}^{1}{q'}^{2}(x){\rm d}x+\frac{\beta}{2}q^{2}(1)+\frac{\xi}{2}[u^2(1)-u^2(0)]\\ & = &\alpha q'(1)q(1)-\alpha\int_{0}^{1}{q'}^{2}(x){\rm d}x+\frac{\beta}{2}q^{2}(1) +\frac{\xi}{2}\cdot\frac{k_{2}^2-k_1^2}{\xi^{2}}\Big(\int_{0}^{1}b(x)q(x){\rm d}x\Big)^2\\ & = &-\frac{\beta}{2}q^2(1)-\alpha\int_{0}^{1}{q'}^{2}(x){\rm d}x+\frac{k_{2}^2-k_1^2}{2\xi} \Big(\int_{0}^{1}b(x)q(x){\rm d}x\Big)^2\\ &\leq&\frac{(k_{2}^2-k_1^2)||b||^2}{2\xi}\int_{0}^{1}q^2(x){\rm d}x. \end{eqnarray} $

$ \begin{equation} {\eta}_{1} = \frac{(\xi k_1^2+(\beta-\xi)k_2^2){\|b\|^2}}{2\xi\cdot(\beta-\xi)}, \;\;\;\; {\eta}_{2} = \frac{(k_{2}^2-k_1^2)||b||^2}{2\xi}, \end{equation} $

$ \eta = \max\{{\eta}_{1}, {\eta}_{2}\} $, 则

由文献[9] 中的定理2得, 算子$ {\cal A} $生成$ {\cal H} $上的一个$ C_{0} $半群$ S(t) $满足$ \|S(t)\|_{{\cal \psi}({\cal H})}\leq {\rm e}^{\eta t} $.

4 系统(2.1) 的Lyapunov稳定性分析

在这一部分, 我们致力于构造合适的Lyapunov函数并寻求调控参数$ k_1 $, $ k_2 $的取值范围, 从而使得闭环系统(2.1) 达到指数稳定.

构造系统(2.1) 的Lyapunov函数为

$ \begin{equation} E(t) = {E}_1(t)+{E}_2(t), \end{equation} $

其中

$ \begin{equation} {E}_1(t) = \frac{1}{2}\Big(\int_{0}^{1}{q^2(x, t)}{\rm d}x+{\tau}{\xi}\int_{0}^{1}{q^2(1, t-\tau\rho)}{\rm d}{\rho}\Big), \end{equation} $

$ \begin{equation} E_2(t) = \nu\tau\int_{0}^{1}{\rm e}^{-2\gamma\tau\rho}q^2(1, t-\tau\rho){\rm d}{\rho}, \end{equation} $

其中, $ \nu $是待定的正常数, $ \gamma $是任意的正常数. 显然, 存在两个正常数$ L_1 $$ L_2 $, 使得对于任意的$ t>0 $, 有

$ \begin{equation} {L_1}{{E}_1(t)}\leq E(t)\leq{L_2}{{E}_1(t)}. \end{equation} $

定理4.1  对任给的正常数$ \gamma $以及系统参数$ \alpha, \beta, \xi, \tau, b(x) $, 如果调控参数$ k_1 $, $ k_2 $满足

$ \begin{equation} |k_2|\beta-(|k_1|+|k_2|)\xi>0 \end{equation} $

$ \begin{equation} \frac{\alpha\pi^2[\xi+(\beta-\xi){\rm e}^{-2\gamma\tau}]}{2||b||^2}>(|k_1|+|k_2|)(|k_2|+|k_1|{\rm e}^{-2\gamma\tau}). \end{equation} $

则闭环系统(2.1) 是指数稳定的, 即$ \exists\;\zeta>0 $使得Lyapunov能量函数满足

其中, $ \zeta = \frac{\varepsilon}{L_2} $, $ L_2 $$ \varepsilon $分别由(4.4) 和(4.15) 式给出.

   结合系统(2.1) 和(2.4), 可计算得$ \dot{E}_1(t) $$ \dot{E}_2(t) $如下

$ \begin{eqnarray} \dot{E}_1(t)& = &\int_{0}^{1}{q(x, t)\frac{\partial{q(x, t)}}{\partial{t}}}{\rm d}x+{\tau}{\xi} \int_{0}^{1}{u(\rho, t)\frac{\partial{u(\rho, t)}}{\partial{t}}}{\rm d}{\rho}\\ & = &-\alpha\int_{0}^{1}{q_x}^2(x, t){\rm d}x-\frac{\beta}{2}q^2(1, t)+k_1q(1, t)\int_{0}^{1}q(x, t)b(x){\rm d}x\\ &&+k_2q(1, t-\tau)\int_{0}^{1}q(x, t)b(x){\rm d}x-\frac{\xi}{2}q^2(1, t-\tau)+\frac{\xi}{2}q^2(1, t), \end{eqnarray} $

$ \begin{eqnarray} \dot{E}_2(t)& = &2\nu\tau\int_{0}^{1}{\rm e}^{-2\gamma\tau\rho}q(1, t-\tau\rho)q_t(1, t-\tau\rho){\rm d}{\rho}\\ & = &2\nu\tau\int_{0}^{1}{\rm e}^{-2\gamma\tau\rho}q(1, t-\tau\rho)(-\frac{1}{\tau})q_{\rho}(1, t-\tau\rho){\rm d}{\rho}\\ & = &-2\nu\int_{0}^{1}{\rm e}^{-2\gamma\tau\rho}d\big(\frac{1}{2}q^2(1, t-\tau\rho)\big)\\ & = &-\nu{\rm e}^{-2\gamma\tau}q^2(1, t-\tau)+{\nu}q^2(1, t)-2\nu\gamma\tau\int_{0}^{1}{\rm e}^{-2\gamma\tau\rho}q^2(1, t-\tau\rho){\rm d}{\rho}. \end{eqnarray} $

于是, 利用Cauchy-Schwartz不等式和Young's不等式可得

$ \begin{eqnarray} \dot{E}(t)& = &\dot{E}_1(t)+\dot{E}_2(t)\\ &\leq&{-\alpha}\int_{0}^{1}{q_x}^{2}(x, t){\rm d}x-\frac{\beta}{2}q^2(1, t) +\frac{\left|{k_1}\right|A}{2}q^2(1, t)+\frac{\left|{k_1}\right|\|b\|^2}{2A}\int_{0}^{1}q^2(x, t){\rm d}x\\ &&+\frac{\left|{k_2}\right|A}{2}{{q^2(1, t-\tau)}}+\frac{\left|{k_2}\right|\|b\|^2}{2A}\int_{0}^{1}q^2(x, t){\rm d}x-\frac{\xi}{2}q^2(1, t-\tau)+\frac{\xi}{2}q^2(1, t)\\ &&-\nu{\rm e}^{-2\gamma\tau}q^2(1, t-\tau)+{\nu}q^2(1, t)-2\nu\gamma\tau\int_{0}^{1}{\rm e}^{-2\gamma\tau\rho}q^2(1, t-\tau\rho){\rm d}{\rho}. \end{eqnarray} $

进而, 根据Wirtinger's不等式

$ \begin{equation} \int_{0}^{1}\varphi^2(x, t){\rm d}x\leq{\frac{4}{\pi^2}}\int_{0}^{1}{\varphi_x}^{2}(x, t){\rm d}x, \;\;\forall{\varphi}\in\{{f}\in{H^1(0, 1)}:{f}(0) = 0\}. \end{equation} $

整理可得

$ \begin{eqnarray} \dot{E}(t) &\leq&\big(-\frac{\alpha\pi^2}{4}+\frac{(\left|{k_1}\right|+|k_2|)\|b\|^2}{2A}\big)\int_{0}^{1}q^{2}(x, t){\rm d}x+\big(-\frac{\beta}{2}+\frac{\xi}{2} +\frac{\left|{k_1}\right|A}{2}+\nu\big)q^2(1, t)\\ &&+\big(\frac{\left|{k_2}\right|A}{2}-\frac{\xi}{2}-\nu {\rm e}^{-2\gamma\tau}\big)q^2(1, t-\tau) -2\nu\gamma\tau{\rm e}^{-2\gamma\tau}\int_{0}^{1}q^2(1, t-\tau\rho){\rm d}{\rho}. \end{eqnarray} $

选取参数$ A, \nu $使得

$ \begin{equation} -\frac{\beta}{2}+\frac{\xi}{2}+\frac{\left|{k_1}\right|A}{2}+\nu = 0, \;\; \frac{\left|{k_2}\right|A}{2}-\frac{\xi}{2}-\nu {\rm e}^{-2\gamma\tau} = 0, \end{equation} $

$ \begin{equation} A = \frac{\xi+(\beta-\xi){\rm e}^{-2\gamma\tau}}{|k_2|+|k_1|{\rm e}^{-2\gamma\tau}}, \;\; \nu = \frac{|k_2|\beta-(|k_1|+|k_2|)\xi}{2(|k_2|+|k_1|{\rm e}^{-2\gamma\tau})}, \end{equation} $

于是有

$ \begin{eqnarray} \dot{E}(t) &\leq&\big(-\frac{\alpha\pi^2}{4}+\frac{(|k_1|+|k_2|)\|b\|^2}{2A}\big)\int_{0}^{1}q^{2}(x, t){\rm d}x -2\nu\gamma\tau{\rm e}^{-2\gamma\tau}\int_{0}^{1}q^2(1, t-\tau\rho){\rm d}{\rho}\\ & = &\frac{1}{2}(-\varepsilon_1 \int_{0}^{1}q^{2}(x, t){\rm d}x-\varepsilon_2\tau\xi\int_{0}^{1}q^2(1, t-\tau\rho){\rm d}{\rho})\\ &\leq& -\varepsilon E_1(t), \end{eqnarray} $

其中

$ \begin{equation} \varepsilon = \min\{\varepsilon_1, \varepsilon_2\}, \;\; \varepsilon_1 = \frac{\alpha\pi^2}{2}-\frac{(|k_1|+|k_2|)\|b\|^2}{A}, \;\; \varepsilon_2 = \frac{4\nu\gamma{\rm e}^{-2\gamma\tau}}{\xi}. \end{equation} $

参数$ A, \nu $由(4.13) 式给出. 结合条件(4.5) 和(4.6) 可知, $ \varepsilon>0. $于是, 结合(4.4)和(4.14) 式可得: 存在正常数$ \zeta = \frac{\varepsilon}{L_2}>0 $使得$ \dot{E}(t)\leq-\zeta{E}(t) $, 因此系统(2.1) 指数稳定.

注4.1  在对(4.9) 式进行放缩时, 可运用不同的方法, 从而得到如下定理.

定理4.2  对任给的常数$ c_1>0 $, $ c_2>1 $, 如果调控参数$ k_1 $, $ k_2 $满足

$ \begin{equation} (2c_1+1){k_1}^2+c_2{k_2}^2+(2c_1+c_2+1)|k_1k_2|<\frac{{\alpha}^2{\pi}^2}{4\|b\|^2}, \end{equation} $

则闭环系统(2.1) 是指数稳定的, 即$ \exists\; \hat{\zeta}>0 $使得Lyapunov能量函数满足

其中, $ \hat{\zeta} = \frac{\hat{\varepsilon}}{L_2} $, $ L_2 $$ \hat{\varepsilon} $分别由(4.4) 和(4.24) 式给出.

   由$ q(1, t) = \int_{0}^{1}q_x(x, t){\rm d}x $可得

$ \begin{equation} q^2(1, t)\leq{\int_{0}^{1}{q_x}^2(x, t){\rm d}x}. \end{equation} $

于是结合(4.9) 式、(4.10) 式和(4.17) 式, 整理可得

$ \begin{eqnarray} \dot{E}(t)&\leq&\frac{\pi^2 |k_1|A^2+(\pi^2\xi+2\pi^2\nu -2\pi^2\alpha)A+4||b||^2(|k_1|+|k_2|)}{2\pi^2A}\int_{0}^{1}{q_x}^2(x, t){\rm d}x\\ &&-{\frac{\beta}{2}}q^2(1, t)+(\frac{{\left|{k_2}\right|}A-\xi}{2}-\nu{\rm e}^{-2\gamma\tau})q^2(1, t-\tau) -2\nu\gamma\tau{\rm e}^{-2\gamma\tau}\int_{0}^{1}q^2(1, t-\tau){\rm d}{\rho}.{\qquad} \end{eqnarray} $

$ \nu = c_1{\left|{k_1}\right|}A $, $ \xi = c_2{\left|{k_2}\right|}A $, 其中, $ c_1>0 $, $ c_2>1 $, 于是(4.17) 式转化为

$ \begin{eqnarray} \dot{E}(t)&\leq&\frac{\pi^2\big[(2c_1+1){\left|{k_1}\right|}+c_2{\left|{k_2}\right|}\big]A^2-2\pi^2\alpha{A} +4\|b\|^2({\left|{k_1}\right|}+{\left|{k_2}\right|})}{2\pi^2A}\int_{0}^{1}{q_x}^{2}(x, t){\rm d}x\\ &&-2c_1{\left|{k_1}\right|}A\gamma\tau{\rm e}^{-2\gamma\tau}\int_{0}^{1}q^2(1, t-\tau){\rm d}{\rho}. \end{eqnarray} $

$ \begin{equation} g(A) = \pi^2\big[(2c_1+1){\left|{k_1}\right|}+c_2{\left|{k_2}\right|}\big]A^2-2\pi^2\alpha{A} +4\|b\|^2({\left|{k_1}\right|}+{\left|{k_2}\right|}), \end{equation} $

这是一个关于$ A $的二次多项式, 根据条件(4.16) 可知, 其判别式

$ \begin{equation} \Delta = 4\pi^4\alpha^2-16\pi^2||b||^2\big[(2c_1+1){\left|{k_1}\right|}+c_2{\left|{k_2}\right|}\big](|k_1|+|k_2|)>0. \end{equation} $

那么当正常数$ A $满足

$ \begin{equation} A\in\Big(\frac{2\pi^2\alpha-\sqrt{\Delta}}{2\pi^2 \big[(2c_1+1){\left|{k_1}\right|}+c_2{\left|{k_2}\right|}\big]}, \frac{2\pi^2\alpha+\sqrt{\Delta}}{2\pi^2 \big[(2c_1+1){\left|{k_1}\right|}+c_2{\left|{k_2}\right|}\big]}\Big) \end{equation} $

时, 有$ g(A)<0. $于是结合(4.10) 式可将(4.19) 式改写为

$ \begin{eqnarray} \dot{E}(t)&\leq&\frac{\pi^2\big[(2c_1+1){\left|{k_1}\right|}+c_2{\left|{k_2}\right|}\big]A^2-2\pi^2\alpha{A} +4\|b\|^2({\left|{k_1}\right|}+{\left|{k_2}\right|})}{8A}\int_{0}^{1}q^2(x, t){\rm d}x\\ &&-2c_1{\left|{k_1}\right|}A\gamma\tau{\rm e}^{-2\gamma\tau}\int_{0}^{1}q^2(1, t-\tau){\rm d}{\rho}. \end{eqnarray} $

$ \begin{eqnarray} \hat{\varepsilon}_1& = &-\frac{\pi^2\big[(2c_1+1){\left|{k_1}\right|}+c_2{\left|{k_2}\right|}\big]A^2-2\pi^2\alpha{A} +4\|b\|^2({\left|{k_1}\right|}+{\left|{k_2}\right|})}{4A}, \\ \hat{\varepsilon}_2& = &\frac{4c_1{\left|{k_1}\right|}\gamma{A}{\rm e}^{-2\gamma\tau}}{\xi}, \;\;\;\; \hat{\varepsilon} = \min\{\hat{\varepsilon}_1, \hat{\varepsilon}_2\}>0, \end{eqnarray} $

则得

$ \begin{equation} \dot{E}(t)\leq -\hat{\varepsilon}{E}_1(t). \end{equation} $

进而结合(4.4) 式可得$ \dot{E}(t)\leq -\frac{\hat{\varepsilon}}{L_2} E(t), $因此系统(2.1) 指数稳定.

注4.2  当$ k_2 = 0 $时, 控制项$ U(t) = {k_1{q(1, t)}}+{k_2{q(1, t-\tau)}} $变为单纯状态反馈$ U(t) = {k_1{q(1, t)}} $, 稳定性条件(4.16) 式退化为$ \left|{k_1}\right|<\frac{\pi\alpha}{2\|b\|\sqrt{2c_1+1}}. $与文献[8]得到的参数结果$ \left|{\kappa}\right|<\frac{\sqrt{2}\alpha}{\|b\|} $相比较, 可得: 当$ 0<c_1<\frac{\pi^2-8}{16} $

显然, 本文得到的参数范围更广.

注4.3  当$ k_1 = 0 $时, 控制$ U(t) = {k_1{q(1, t)}}+{k_2{q(1, t-\tau)}} $变为纯时滞状态反馈$ U(t) = {k_2{q(1, t-\tau)}} $, 稳定性条件(4.16) 式退化为$ \left|{k_2}\right|<\frac{\pi\alpha}{2\|b\|\sqrt{c_2}}. $与文献[9] 得到的参数结果$ \left|{\kappa}\right|<\frac{\pi\alpha}{2\|b\|\sqrt{2c_1+c_2}} $作比较, 显然

即本文得到的参数范围更广.

注4.4  在定理$ 4.1 $中, 调控参数$ k_1 $$ k_2 $满足的不等式组(4.5)–(4.6)有解. 例如: 令

我们可以取到$ k_1 = 0.5, \; k_2 = 0.9 $满足不等式组(4.5)–(4.6). 在定理$ 4.2 $中, 参数$ \alpha, \; b(x) $取法同前, $ c_1 = 0.5, \; c_2 = 1.2 $, 则可取$ k_1 = 0.2, \; k_2 = 0.3 $满足不等式$ (4.16) $.

5 结论

本文选取了不同于文献[8] 和文献[9] 的PDP反馈控制, 描述了所得时滞PDE闭环系统的适定性, 并对系统进行了稳定性分析, 通过Lyapunov方法得出了系统稳定参数所满足的条件, 通过与文献[8] 和文献[9] 的比较结果可知, 我们得到的参数范围更为广泛.

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