## Stability and Optimality of 2-D Mindlin-Timoshenko Plate System

Zhang Chunguo,, Fu Yuzhi, Liu Yubiao

Department of Mathematics, College of Science, Hangzhou Dianzi University, Hangzhou 310018

 基金资助: 国家自然科学基金.  61374096

 Fund supported: the NSFC.  61374096

Abstract

In this paper, 2-D Mindlin Timoshenko plate system with local boundary control is studied. By using the receding horizon control method, the infinite time domain optimality problem is transformed into the finite time domain optimality problem. With the help of the multiplier technique, a priori estimation is made for any finite time domain system, and the observability inequality is obtained, which proves that the energy of the system is uniformly exponentially decay. Furthermore, with the aid of dual system, by means of the variational principle and Bellman optimality principle, the suboptimal conditions of the system in infinite time domain are obtained, and it is proved that the optimal trajectory is also exponential decay.

Keywords： 2-D Mindlin Timoshenko plate ; Receding horizon control method ; Optimality ; Exponential decay

Zhang Chunguo, Fu Yuzhi, Liu Yubiao. Stability and Optimality of 2-D Mindlin-Timoshenko Plate System. Acta Mathematica Scientia[J], 2021, 41(5): 1465-1491 doi:

## 1 引言

$\Omega $${{\Bbb R}} ^{2} 中的有界开集, \partial\Omega = \Gamma = \Gamma_{0}\cup\Gamma_{1}(\Gamma_{0}\cap\Gamma_{1} = \emptyset) 满足Lipschitz边界条件, 且 \Gamma_{1} 是具有非空内部的闭集, \Gamma_{0}\neq\emptyset 是相对开的. 我们考虑下面具有边界控制的无限时域的二维Mindlin-Timosahenko系统 \begin{eqnarray} \left\{\begin{array}{lll} { } \rho_{1}\psi_{tt}-D(\psi_{xx}+\frac{1-\mu}{2}\psi_{yy} +\frac{1+\mu}{2}\phi_{xy}) +K(\psi+\omega_{x}) = 0, & (x, y, t)\in\Omega\times {{\Bbb R}} ^{+}, \\ { } \rho_{1}\phi_{tt}-D(\phi_{yy}+\frac{1-\mu}{2}\phi_{xx} +\frac{1+\mu}{2}\psi_{xy}) +K(\phi+\omega_{y}) = 0, &(x, y, t)\in\Omega\times {{\Bbb R}} ^{+}, \\ { } \rho_{2}\omega_{tt}-K[(\psi+\omega_{x})_{x}+(\phi+\omega_{y})_{y}] = 0, & (x, y, t)\in\Omega\times {{\Bbb R}} ^{+}, \end{array}\right. \end{eqnarray} 其中, \rho_{1} = \frac{\rho h^{3}}{2}, \rho_{2} = \rho h , \rho 是密度, h 是板的厚度, \mu\in(0, \frac{1}{2}) 是Poisson比, D = \frac{Eh^{3}}{12(1-\mu^{2})} ( E 杨氏模量)表示弹性模量, K = \frac{kEh}{2(1+\mu)} ( k 剪切校正, E 杨氏模量)表示剪切模量, 函数 \psi, \phi$$ \omega$依赖于$(x, y, t)\in\Omega\times {{\Bbb R}} ^{+}$表示板的全转角和板的横向位移.

$\begin{eqnarray} \left\{\begin{array}{lll} { } \psi = \phi = \omega = 0, & (x, y, t)\in\Gamma_{0}\times {{\Bbb R}} ^{+}, \\ { } D[\nu_{1}\psi_{x}+\mu\nu_{1}\phi_{y}+\frac{1-\mu}{2}(\psi_{y}+\phi_{x})\nu_{2}] = u_{1}, &(x, y, t)\in\Gamma_{1}\times {{\Bbb R}} ^{+}, \\ { } D[\nu_{2}\phi_{y}+\mu\nu_{2}\psi_{x}+\frac{1-\mu}{2}(\psi_{y}+\phi_{x})\nu_{1}] = u_{2}, & (x, y, t)\in\Gamma_{1}\times {{\Bbb R}} ^{+}, \\ { } K(\frac{\partial\omega}{\partial{\bf \nu}}+\nu_{1}\psi+\nu_{2}\phi) = u_{3}, & (x, y, t)\in\Gamma_{1}\times {{\Bbb R}} ^{+}, \\ { } (\psi(x, y, 0), \phi(x, y, 0), \omega(x, y, 0)) = (\psi_{01}, \phi_{01}, \omega_{01}), & (x, y)\in\Omega, \\ { } (\psi_{t}(x, y, 0), \phi_{t}(x, y, 0), \omega_{t}(x, y, 0)) = (\psi_{02}, \phi_{02}, \omega_{02}), & (x, y)\in\Omega, \end{array}\right. \end{eqnarray}$

$$$J_{\infty}({\cal Y}_{0}, U) = \int_{0}^{\infty}\ell({\cal Y}(t), U(t)){\rm d}t,$$$

$$$\ell({\cal Y}(t), U(t)) = \frac{1}{2}\|{\cal Y}(t)\|_{{\cal H}}^{2} +\frac{\beta}{2}\|U(t)\|_{{\cal U}}^{2},$$$

$\begin{eqnarray} \min\limits_{U\in L^{2}((t_{k}, t_{k}+T);{\cal U})} J_{T}({\cal Y}(t_{k}), U) = \min\limits_{U\in L^{2}((t_{k}, t_{k}+T);{\cal U})} \int_{t_{k}}^{t_{k}+T}\ell({\cal Y}(t), U(t)){\rm d}t, \end{eqnarray}$

$\begin{eqnarray} \left\{\begin{array}{lll} { } \rho_{1}\psi_{tt}-D(\psi_{xx}+\frac{1-\mu}{2}\psi_{yy} +\frac{1+\mu}{2}\phi_{xy}) +K(\psi+\omega_{x}) = 0, (x, y, t)\in\Omega\times (t_{k}, t_{k}+T), \\ { } \rho_{1}\phi_{tt}-D(\phi_{yy}+\frac{1-\mu}{2}\phi_{xx} +\frac{1+\mu}{2}\psi_{xy}) +K(\phi+\omega_{y}) = 0, (x, y, t)\in\Omega\times (t_{k}, t_{k}+T), \\ { } \rho_{2}\omega_{tt}-K[(\psi+\omega_{x})_{x}+(\phi+\omega_{y})_{y}] = 0, {\quad} (x, y, t)\in\Omega\times (t_{k}, t_{k}+T), \\ { } \psi = \phi = \omega = 0, {\quad} (x, y, t)\in\Gamma_{0}\times (t_{k}, t_{k}+T), \\ { } D[\nu_{1}\psi_{x}+\mu\nu_{1}\phi_{y}+\frac{1-\mu}{2}(\psi_{y}+\phi_{x})\nu_{2}] = u_{1}, {\quad} (x, y, t)\in\Gamma_{1}\times (t_{k}, t_{k}+T), \\ { } D[\nu_{2}\phi_{y}+\mu\nu_{2}\psi_{x}+\frac{1-\mu}{2}(\psi_{y}+\phi_{x})\nu_{1}] = u_{2}, {\quad} (x, y, t)\in\Gamma_{1}\times (t_{k}, t_{k}+T), \\ { } K(\frac{\partial\omega}{\partial{\bf \nu}}+\nu_{1}\psi+\nu_{2}\phi) = u_{3}, {\quad} (x, y, t)\in\Gamma_{1}\times (t_{k}, t_{k}+T), \\ { } (\psi(x, y, t_{k}), \phi(x, y, t_{k}), \omega(x, y, t_{k})) = (\psi_{t_{k}1}, \phi_{t_{k}1}, \omega_{t_{k}1}), {\quad}(x, y)\in\Omega, \\ { } (\psi_{t}(x, y, t_{k}), \phi_{t}(x, y, t_{k}), \omega_{t}(x, y, t_{k})) = (\psi_{t_{k}2}, \phi_{t_{k}2}, \omega_{t_{k}2}), {\quad}(x, y)\in\Omega. \end{array}\right. \end{eqnarray}$

## 2 弱解的先验估计

$\begin{eqnarray} \min\limits_{U\in L^{2}((0, T);{\cal U})} J_{T}({\cal Y}_{0}, U) = \min\limits_{U\in L^{2}((0, T);{\cal U})}\int_{0}^{T}\ell({\cal Y}(t), U(t)){\rm d}t, \end{eqnarray}$

$\begin{eqnarray} \left\{\begin{array}{lll} { } \rho_{1}\psi_{tt}-D(\psi_{xx}+\frac{1-\mu}{2}\psi_{yy} +\frac{1+\mu}{2}\phi_{xy}) +K(\psi+\omega_{x}) = 0, (x, y, t)\in\Omega\times (0, T), \\ { } \rho_{1}\phi_{tt}-D(\phi_{yy}+\frac{1-\mu}{2}\phi_{xx} +\frac{1+\mu}{2}\psi_{xy}) +K(\phi+\omega_{y}) = 0, (x, y, t)\in\Omega\times (0, T), \\ { } \rho_{2}\omega_{tt}-K[(\psi+\omega_{x})_{x}+(\phi+\omega_{y})_{y}] = 0, {\quad} (x, y, t)\in\Omega\times (0, T), \\ { } \psi = \phi = \omega = 0, {\quad} (x, y, t)\in\Gamma_{0}\times (0, T), \\ { } D[\nu_{1}\psi_{x}+\mu\nu_{1}\phi_{y}+\frac{1-\mu}{2}(\psi_{y}+\phi_{x})\nu_{2}] = u_{1}, {\quad} (x, y, t)\in\Gamma_{1}\times (0, T), \\ { } D[\nu_{2}\phi_{y}+\mu\nu_{2}\psi_{x}+\frac{1-\mu}{2}(\psi_{y}+\phi_{x})\nu_{1}] = u_{2}, {\quad}(x, y, t)\in\Gamma_{1}\times (0, T), \\ { } K(\frac{\partial\omega}{\partial{\bf \nu}}+\nu_{1}\psi+\nu_{2}\phi) = u_{3}, {\quad}(x, y, t)\in\Gamma_{1}\times (0, T), \\ { } (\psi(x, y, 0), \phi(x, y, 0), \omega(x, y, 0)) = (\psi_{01}, \phi_{01}, \omega_{01}), {\quad}(x, y)\in\Omega, \\ { } (\psi_{t}(x, y, 0), \phi_{t}(x, y, 0), \omega_{t}(x, y, 0)) = (\psi_{02}, \phi_{02}, \omega_{02}), {\quad} (x, y)\in\Omega. \end{array}\right. \end{eqnarray}$

$({\cal Y}, U)$满足系统(2.2).

$\begin{eqnarray} & &\int_{\Omega}\{\rho_{1}[\psi_{t}(t, x, y)\widehat{\psi}(t, x, y)+\phi_{t}(t, x, y)\widehat{\phi}(t, x, y)] +\rho_{2}\omega_{t}(t, x, y)\widehat{\omega}(t, x, y)\}{\rm d}x{\rm d}y\\ &&-\int_{\Omega}\{\rho_{1}[\psi_{02}\widehat{\psi}(0, x, y)+\phi_{02}\widehat{\phi}(0, x, y)] +\rho_{2}\omega_{02}\widehat{\omega}(0, x, y)\}{\rm d}x{\rm d}y\\ &&-\int_{0}^{t}\int_{\Omega}\{\rho_{1}[\psi_{t}(\tau, x, y)\widehat{\psi}_{t}(\tau, x, y)+\phi_{t}(\tau, x, y)\widehat{\phi}_{t}(\tau, x, y)] +\rho_{2}\omega_{t}(\tau, x, y)\widehat{\omega}_{t}(\tau, x, y)\}{\rm d}x{\rm d}y{\rm d}\tau\\ &&+\int_{0}^{t}a(\psi, \phi, \omega; \widehat{\psi}, \widehat{\phi}, \widehat{\omega}){\rm d}\tau -\int_{0}^{t}\int_{\Gamma_{1}}(\widehat{\psi}u_{1}+\widehat{\phi}u_{2}+\widehat{\omega}u_{3}){\rm d}\Gamma {\rm d}\tau = 0. \end{eqnarray}$

$\begin{eqnarray} &&\|\Phi\|_{C^{0}([0, T];W)} +\|\Phi_{t}\|_{C^{0}([0, T];H)}+\|\Phi_{tt}\|_{L^{2}([0, T];W^{\ast})}\\ &\leq &C(\|\Phi_{01}\|_{W} +\|\Phi_{02}\|_{H}+\|U\|_{L^{2}([0, T];{\cal U})}), \end{eqnarray}$

记$I(t) = a(\psi, \phi, \omega; \psi, \phi, \omega)+\int_{\Omega}[\rho_{1}(|\psi_{t}|^{2}+|\phi_{t}|^{2}) +\rho_{2}|\omega_{t}|^{2}]{\rm d}x{\rm d}y$, 则有

$\begin{eqnarray} \frac{{\rm d}I(t)}{{\rm d}t} & = &\frac{\partial a(\psi, \phi, \omega; \psi, \phi, \omega)}{\partial t} + 2\int_{\Omega}[\rho_{1}(\psi_{t}\psi_{tt}+\phi_{t}\phi_{tt}) +\rho_{2}\omega_{t}\omega_{tt}]{\rm d}x{\rm d}y\\ & = &2a(\psi, \phi, \omega; \psi_{t}, \phi_{t}, \omega_{t})+2\int_{\Omega}[\psi_{t}L_{1}\{\psi, \phi, \omega\} +\phi_{t}L_{2}\{\psi, \phi, \omega\}+\omega_{t}L_{3}\{\psi, \phi, \omega\}]{\rm d}x{\rm d}y\\ & = &2\int_{\Gamma_{1}}[\psi_{t}F_{1}\{\psi, \phi\} +\phi_{t}F_{2}\{\psi, \phi\}+\omega_{t}F_{3}\{\psi, \phi, \omega\}]{\rm d}\Gamma_{1}\\ &\leq& 2\varepsilon\int_{\Gamma_{1}}[|\psi_{t}|^{2}+|\phi_{t}|^{2}+|\omega_{t}|^{2}]{\rm d}\Gamma_{1} +\frac{1}{2\varepsilon}\int_{\Gamma_{1}}[|u_{1}|^{2}+|u_{2}|^{2}+|u_{3}|^{2}]{\rm d}\Gamma_{1}. \end{eqnarray}$

## 4 最优性条件

$\begin{eqnarray} \left\{\begin{array}{lll} { } \rho_{1}\psi_{tt}-D(\psi_{xx}+\frac{1-\mu}{2}\psi_{yy} +\frac{1+\mu}{2}\phi_{xy}) +K(\psi+\omega_{x}) = 0, (x, y, t)\in\Omega\times (0, T), \\ { } \rho_{1}\phi_{tt}-D(\phi_{yy}+\frac{1-\mu}{2}\phi_{xx} +\frac{1+\mu}{2}\psi_{xy}) +K(\phi+\omega_{y}) = 0, (x, y, t)\in\Omega\times (0, T), \\ { } \rho_{2}\omega_{tt}-K[(\psi+\omega_{x})_{x}+(\phi+\omega_{y})_{y}] = 0, \quad (x, y, t)\in\Omega\times (0, T), \\ { } \psi = \phi = \omega = 0, \quad (x, y, t)\in\Gamma_{0}\times (0, T), \\ { } D[\nu_{1}\psi_{x}+\mu\nu_{1}\phi_{y}+\frac{1-\mu}{2}(\psi_{y}+\phi_{x})\nu_{2}] = u_{1}, {\quad} (x, y, t)\in\Gamma_{1}\times (0, T), \\ { } D[\nu_{2}\phi_{y}+\mu\nu_{2}\psi_{x}+\frac{1-\mu}{2}(\psi_{y}+\phi_{x})\nu_{1}] = u_{2}, {\quad} (x, y, t)\in\Gamma_{1}\times (0, T), \\ { } K(\frac{\partial\omega}{\partial{\bf \nu}}+\nu_{1}\psi+\nu_{2}\phi) = u_{3}, \quad (x, y, t)\in\Gamma_{1}\times (0, T), \\ (\psi(x, y, 0), \phi(x, y, 0), \omega(x, y, 0)) = (0, 0, 0), {\quad} (x, y)\in\Omega, \\ (\psi_{t}(x, y, 0), \phi_{t}(x, y, 0), \omega_{t}(x, y, 0)) = (0, 0, 0), {\quad} (x, y)\in\Omega \end{array}\right. \end{eqnarray}$

$\begin{eqnarray} \left\{\begin{array}{lll} { }\rho_{1}p_{tt}-D(p_{xx}+\frac{1-\mu}{2}p_{yy} +\frac{1+\mu}{2}q_{xy}) +K(p+r_{x}) = g_{1}, (x, y, t)\in\Omega\times (0, T), \\ { } \rho_{1}q_{tt}-D(q_{yy}+\frac{1-\mu}{2}q_{xx} +\frac{1+\mu}{2}p_{xy}) +K(q+r_{y}) = g_{2}, (x, y, t)\in\Omega\times (0, T), \\ { } \rho_{2}r_{tt}-K[(p+r_{x})_{x}+(q+r_{y})_{y}] = g_{3}, \quad (x, y, t)\in\Omega\times (0, T), \\ { } \psi = \phi = \omega = 0, \quad (x, y, t)\in\Gamma_{0}\times (0, T), \\ { } D[\nu_{1}p_{x}+\mu\nu_{1}q_{y}+\frac{1-\mu}{2}(p_{y}+q_{x})\nu_{2}] = 0, \quad (x, y, t)\in\Gamma_{1}\times (0, T), \\ { } D[\nu_{2}q_{y}+\mu\nu_{2}p_{x}+\frac{1-\mu}{2}(p_{y}+q_{x})\nu_{1}] = 0, \quad (x, y, t)\in\Gamma_{1}\times (0, T), \\ { } K(\frac{\partial r}{\partial{\bf \nu}}+\nu_{1}p+\nu_{2}q) = 0, \quad (x, y, t)\in\Gamma_{1}\times (0, T), \\ { } (p(x, y, T), q(x, y, T), r(x, y, T)) = (p_{0}(T), q_{0}(T), r_{0}(T)), \quad (x, y)\in\Omega, \\ { } (p_{t}(x, y, T), q_{t}(x, y, T), r_{t}(x, y, T)) = (p_{1}(T), q_{1}(T), r_{1}(T)), {\quad}(x, y)\in\Omega, \end{array}\right. \end{eqnarray}$

$\begin{eqnarray} & &\int_{0}^{T}\int_{\Gamma_{1}}(pu_{1}+qu_{2}+ru_{3}){\rm d}\Gamma_{1}{\rm d}t\\ & = &\int_{0}^{T}\langle(\psi, \phi, \omega), (g_{1}, g_{2}, g_{3})\rangle_{W\times W^{\ast}}{\rm d}t +\left((p_{0}(T), q_{0}(T), r_{0}(T)), (\psi_{t}(T), \phi_{t}(T), \omega_{t}(T))\right)_{H}\\ &&-\langle(\rho_{1}\psi(T), \rho_{1}\phi(T), \rho_{2}\omega(T)), (p_{1}(T), q_{1}(T), r_{1}(T))\rangle_{W\times W^{\ast}}. \end{eqnarray}$

对于$(p, q, r)\in C^{0}([0, T];W)\cap C^{1}([0, T];H)$, 并用$(p, q, r)$分别乘以系统(4.1)前三式, 在$[0, T]\times\Omega$上积分并相加得

$\begin{eqnarray} \left\{\begin{array}{lll} { } \rho_{1}\overline{\psi}_{tt}-D(\overline{\psi}_{xx}+\frac{1-\mu}{2}\overline{\psi}_{yy} +\frac{1+\mu}{2}\overline{\phi}_{xy}) +K(\overline{\psi}+\overline{\omega}_{x}) = 0, (x, y, t)\in\Omega\times (0, T), \\ { } \rho_{1}\overline{\phi}_{tt}-D(\overline{\phi}_{yy}+\frac{1-\mu}{2}\overline{\phi}_{xx} +\frac{1+\mu}{2}\overline{\psi}_{xy}) +K(\overline{\phi}+\overline{\omega}_{y}) = 0, (x, y, t)\in\Omega\times (0, T), \\ { } \rho_{2}\overline{\omega}_{tt}-K[(\overline{\psi}+\overline{\omega}_{x})_{x}+(\overline{\phi}+\overline{\omega}_{y})_{y}] = 0, {\quad} (x, y, t)\in\Omega\times (0, T), \\ { } \overline{\psi} = \overline{\phi} = \overline{\omega} = 0, \quad (x, y, t)\in\Gamma_{0}\times (0, T), \\ { } D[\nu_{1}\overline{\psi}_{x}+\mu\nu_{1}\overline{\phi}_{y}+\frac{1-\mu}{2}(\overline{\psi}_{y}+\overline{\phi}_{x})\nu_{2}] = \overline{u}_{1}, {\quad} (x, y, t)\in\Gamma_{1}\times (0, T), \\ { } D[\nu_{2}\overline{\phi}_{y}+\mu\nu_{2}\overline{\psi}_{x}+\frac{1-\mu}{2}(\overline{\psi}_{y}+\overline{\phi}_{x})\nu_{1}] = \overline{u}_{2}, {\quad}(x, y, t)\in\Gamma_{1}\times (0, T), \\ { } K(\frac{\partial\overline{\omega}}{\partial{\bf \nu}}+\nu_{1}\overline{\psi}+\nu_{2}\overline{\phi}) = \overline{u}_{3}, {\quad}(x, y, t)\in\Gamma_{1}\times (0, T), \\ { } (\overline{\psi}(x, y, 0), \overline{\phi}(x, y, 0), \overline{\omega}(x, y, 0)) = (\psi_{01}, \phi_{01}, \omega_{01}), {\quad} (x, y)\in\Omega, \\ (\overline{\psi}_{t}(x, y, 0), \overline{\phi}_{t}(x, y, 0), \overline{\omega}_{t}(x, y, 0)) = (\psi_{02}, \phi_{02}, \omega_{02}), {\quad}(x, y)\in\Omega \end{array}\right. \end{eqnarray}$

$\begin{eqnarray} \left\{\begin{array}{lll} { } \rho_{1}\overline{p}_{tt}-D(\overline{p}_{xx}+\frac{1-\mu}{2}\overline{p}_{yy} +\frac{1+\mu}{2}\overline{q}_{xy}) +K(\overline{p}+\overline{r}_{x})\\ { } = -[\rho_{1}\overline{\psi}_{tt}+D(\overline{\psi}_{xx}+\frac{1-\mu}{2}\overline{\psi}_{yy} +\frac{1+\mu}{2}\overline{\phi}_{xy}) -K(\overline{\psi}+\overline{\omega}_{x})], (x, y, t)\in\Omega\times (0, T), \\ { } \rho_{1}\overline{q}_{tt}-D(\overline{q}_{yy}+\frac{1-\mu}{2}\overline{q}_{xx} +\frac{1+\mu}{2}\overline{p}_{xy}) +K(\overline{q}+\overline{r}_{y})\\ { } = -[\rho_{1}\overline{\phi}_{tt}+D(\overline{\phi}_{yy}+\frac{1-\mu}{2}\overline{\phi}_{xx} +\frac{1+\mu}{2}\overline{\psi}_{xy}) -K(\overline{\phi}+\overline{\omega}_{y})], (x, y, t)\in\Omega\times (0, T), \\ { } \rho_{2}\overline{r}_{tt}-K[(\overline{p}+\overline{r}_{x})_{x}+(\overline{q}+\overline{r}_{y})_{y}]\\ { } = -(\rho_{2}\overline{\omega}_{tt}+K[(\overline{\psi}+\overline{\omega}_{x})_{x} +(\overline{\phi}+\overline{\omega}_{y})_{y}]), {\quad} (x, y, t)\in\Omega\times (0, T), \\ { } \overline{p} = \overline{q} = \overline{r} = 0, \quad (x, y, t)\in\Gamma_{0}\times (0, T), \\ { } D[\nu_{1}\overline{p}_{x}+\mu\nu_{1}\overline{q}_{y}+\frac{1-\mu}{2}(\overline{p}_{y}+\overline{q}_{x})\nu_{2}] = 0, \quad (x, y, t)\in\Gamma_{1}\times (0, T), \\ { } D[\nu_{2}\overline{q}_{y}+\mu\nu_{2}\overline{p}_{x}+\frac{1-\mu}{2}(\overline{p}_{y}+\overline{q}_{x})\nu_{1}] = 0, \quad (x, y, t)\in\Gamma_{1}\times (0, T), \\ { } K(\frac{\partial\overline{r}}{\partial{\bf \nu}}+\nu_{1}\overline{p}+\nu_{2}\overline{q}) = 0, \quad (x, y, t)\in\Gamma_{1}\times (0, T), \\ { } (\overline{p}(x, y, T), \overline{q}(x, y, T), \overline{\omega}(x, y, T)) = (0, 0, 0), \quad (x, y)\in\Omega, \\ { } (\overline{p}_{t}(x, y, T), \overline{q}_{t}(x, y, T), \overline{r}_{t}(x, y, T)) = -(\overline{\psi}_{t}(T), \overline{\phi}_{t}(T), \overline{\omega}_{t}(T)), {\quad} (x, y)\in\Omega. \end{array}\right. \end{eqnarray}$

$$$(\overline{p}, \overline{q}, \overline{r}) = -\beta(\overline{u}_{1}, \overline{u}_{2}, \overline{u}_{3}).$$$

由目标函数的表达式得

$\begin{eqnarray} J_{T}({\cal Y}_{0}, U) & = &\int_{0}^{T}(\frac{1}{2}\|{\cal Y}(t)\|_{{\cal H}}^{2}+\frac{\beta}{2}\|U(t)\|_{{\cal U}}^{2}){\rm d}t\\ & = &\frac{1}{2}\int_{0}^{T}a(\psi, \phi, \omega;\psi, \phi, \omega){\rm d}t +\frac{1}{2}\int_{0}^{T}\int_{\Omega}[\rho_{1}(|\psi_{t}|^{2}+|\phi_{t}|^{2})+\rho_{2}|\omega_{t}|^{2}]{\rm d}x{\rm d}y{\rm d}t\\ &&+\frac{\beta}{2}\int_{0}^{T}\int_{\Gamma_{1}}(|u_{1}|^{2}+|u_{2}|^{2}+|u_{3}|^{2}){\rm d}\Gamma_{1}{\rm d}t. \end{eqnarray}$

$\begin{eqnarray} \left\{\begin{array}{ll} { } \rho_{1}\delta\psi_{tt}-D((\delta\psi)_{xx}+\frac{1-\mu}{2}(\delta\psi)_{yy} +\frac{1+\mu}{2}(\delta\phi)_{xy}) +K(\delta\psi+(\delta\omega)_{x}) = 0, \\ \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; (x, y, t)\in\Omega\times (0, T), \\ { } \rho_{1}\delta\phi_{tt}-D((\delta\phi)_{yy}+\frac{1-\mu}{2}(\delta\phi)_{xx} +\frac{1+\mu}{2}(\delta\psi)_{xy}) +K(\delta\phi+(\delta\omega)_{y}) = 0, \\ \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; (x, y, t)\in\Omega\times (0, T), \\ { } \rho_{2}\delta\omega_{tt}-K[(\delta\psi+(\delta\omega)_{x})_{x}+(\delta\phi+(\delta\omega)_{y})_{y}] = 0, \quad(x, y, t)\in\Omega\times (0, T), \\ { } \delta\psi = \delta\phi = \delta\omega = 0, \quad (x, y, t)\in\Gamma_{0}\times (0, T), \\ { } D[\nu_{1}(\delta\psi)_{x}+\mu\nu_{1}(\delta\phi)_{y}+\frac{1-\mu}{2}((\delta\psi)_{y}+(\delta\phi)_{x})\nu_{2}] = \delta u_{1}, (x, y, t)\in\Gamma_{1}\times (0, T), \\ { } D[\nu_{2}(\delta\phi)_{y}+\mu\nu_{2}(\delta\psi)_{x}+\frac{1-\mu}{2}((\delta\psi)_{y}+(\delta\phi)_{x})\nu_{1}] = \delta u_{2}, (x, y, t)\in\Gamma_{1}\times (0, T), \\ { } K(\frac{\partial(\delta\omega)}{\partial{\bf \nu}}+\nu_{1}\delta\psi+\nu_{2}\delta\phi) = \delta u_{3}, \quad (x, y, t)\in\Gamma_{1}\times (0, T), \\ { } (\delta\psi(x, y, 0), \delta\phi(x, y, 0), \delta\omega(x, y, 0)) = (0, 0, 0), \quad (x, y)\in\Omega, \\ { } (\delta\psi_{t}(x, y, 0), \delta\phi_{t}(x, y, 0), \delta\omega_{t}(x, y, 0)) = (0, 0, 0), \quad (x, y)\in\Omega. \end{array}\right. \end{eqnarray}$

$\begin{eqnarray} & &-\int_{0}^{T}\int_{\Omega}[\delta\psi(\rho_{1}\psi_{tt}+L_{1}\{\psi, \phi, \omega\})+\delta\phi(\rho_{1}\phi_{tt}+L_{2}\{\psi, \phi, \omega\}) +\delta\omega(\rho_{2}\omega_{tt}+L_{3}\{\psi, \phi, \omega\})]{\rm d}t\\ &&+\int_{\Omega}[\rho_{1}(\delta\psi(T)\psi_{t}(T)+\delta\phi(T)\phi_{t}(T))+\rho_{2}\delta\omega(T)\omega_{t}(T)]{\rm d}x{\rm d}y\\ &&+\beta\int_{0}^{T}\int_{\Gamma_{1}}(u_{1}\delta u_{1}+u_{2}\delta u_{2}+u_{3}\delta u_{3}){\rm d}\Gamma_{1}{\rm d}t = 0. \end{eqnarray}$

$\begin{eqnarray} & &\int_{0}^{T}\int_{\Gamma_{1}}(p\delta u_{1}+q\delta u_{2}+r\delta u_{3}){\rm d}\Gamma_{1}{\rm d}t\\ & = &\int_{0}^{T}\langle(\delta\psi, \delta\phi, \delta\omega), (g_{1}, g_{2}, g_{3})\rangle_{W\times W^{\ast}}{\rm d}t +\left((p_{0}(T), q_{0}(T), r_{0}(T)), (\delta\psi_{t}(T), \delta\phi_{t}(T), \delta\omega_{t}(T))\right)_{H}\\ &&-\langle(\rho_{1}\delta\psi(T), \rho_{1}\delta\phi(T), \rho_{2}\delta\omega(T)), (p_{1}(T), q_{1}(T), r_{1}(T))\rangle_{W\times W^{\ast}}. \end{eqnarray}$

## 5 能观性与能量指数衰减的等价性

$\begin{eqnarray} C_{1}\|{\cal Y}_{0}\|_{{\cal H}}^{2}\leq\int_{0}^{T}\int_{\widetilde{\Gamma}} (|\widetilde{\psi}_{t}|^{2}+|\widetilde{\phi}_{t}|^{2}+|\widetilde{\omega}_{t}|^{2}){\rm d}\widetilde{\Gamma}{\rm d}t, \end{eqnarray}$

$\begin{eqnarray} C_{2}\|{\cal Y}_{0}\|_{{\cal H}}^{2}\leq\int_{0}^{T}\int_{\Omega} (|\widetilde{\psi}_{t}|^{2}+|\widetilde{\phi}_{t}|^{2}+|\widetilde{\omega}_{t}|^{2}){\rm d}x{\rm d}y{\rm d}t, \end{eqnarray}$

$\begin{eqnarray} \left\{\begin{array}{lll} { } \rho_{1}\widetilde{\psi}_{tt}-D(\widetilde{\psi}_{xx}+\frac{1-\mu}{2}\widetilde{\psi}_{yy} +\frac{1+\mu}{2}\widetilde{\phi}_{xy}) +K(\widetilde{\psi}+\widetilde{\omega}_{x}) = 0, \quad (x, y, t)\in\Omega\times (0, T), \\ { } \rho_{1}\widetilde{\phi}_{tt}-D(\widetilde{\phi}_{yy}+\frac{1-\mu}{2}\widetilde{\phi}_{xx} +\frac{1+\mu}{2}\widetilde{\psi}_{xy}) +K(\widetilde{\phi}+\widetilde{\omega}_{y}) = 0, \quad (x, y, t)\in\Omega\times (0, T), \\ { } \rho_{2}\widetilde{\omega}_{tt}-K[(\widetilde{\psi}+\widetilde{\omega}_{x})_{x}+(\widetilde{\phi}+\widetilde{\omega}_{y})_{y}] = 0, \quad (x, y, t)\in\Omega\times (0, T), \\ { } \widetilde{\psi} = \widetilde{\phi} = \widetilde{\omega} = 0, \quad (x, y, t)\in\Gamma_{0}\times (0, T), \\ { } D[\nu_{1}\widetilde{\psi}_{x}+\mu\nu_{1}\widetilde{\phi}_{y}+\frac{1-\mu}{2}(\widetilde{\psi}_{y}+\widetilde{\phi}_{x})\nu_{2}] = 0, \quad (x, y, t)\in\Gamma_{1}\times (0, T), \\ { } D[\nu_{2}\widetilde{\phi}_{y}+\mu\nu_{2}\widetilde{\psi}_{x}+\frac{1-\mu}{2}(\widetilde{\psi}_{y}+\widetilde{\phi}_{x})\nu_{1}] = 0, \quad (x, y, t)\in\Gamma_{1}\times (0, T), \\ { } K(\frac{\partial\widetilde{\omega}}{\partial{\bf \nu}}+\nu_{1}\widetilde{\psi}+\nu_{2}\widetilde{\phi}) = 0, \quad (x, y, t)\in\Gamma_{1}\times (0, T), \\ (\widetilde{\psi}(x, y, 0), \widetilde{\phi}(x, y, 0), \widetilde{\omega}(x, y, 0)) = (\psi_{01}, \phi_{01}, \omega_{01}), \quad (x, y)\in\Omega, \\ (\widetilde{\psi}_{t}(x, y, 0), \widetilde{\phi}_{t}(x, y, 0), \widetilde{\omega}_{t}(x, y, 0)) = (\psi_{02}, \phi_{02}, \omega_{02}), \quad (x, y)\in\Omega. \end{array}\right. \end{eqnarray}$

记$F(x, y) = \eta(x, y)((x-x_{0}), (y-y_{0}))$, 其中, 乘子$\eta(\cdot)\in C^{1}(\overline{\Omega})$且满足在$\overline{\Gamma}_{0}$上, $\eta = 0$; 在$\Gamma_{1}$上, $\eta = 1$. 对边界$\Gamma_{1}$作如下分割

$M^{2} = \max\limits_{(x, y)\in\overline{\Omega}}|F(x, y)|^{2} = \max\limits_{(x, y)\in\overline{\Omega}}\{|x-x_{0}|^{2}+|y-y_{0}|^{2}\}, \quad 2M_{1} = \max\{\rho_{1}, \rho_{2}\}. $$F\cdot\nabla\widetilde{\psi}, F\cdot\nabla\widetilde{\phi}, F\cdot\nabla\widetilde{\omega} 分别乘以系统(5.3)前三式, 并在 [0, T]\times\Omega 上积分, 则有 分部积分得 \begin{eqnarray} &&\int_{0}^{T}\int_{\Gamma_{1}}(F\cdot\nu)[\rho_{1}(|\widetilde{\psi}_{t}|^{2}+|\widetilde{\phi}_{t}|^{2}) +\rho_{2}|\widetilde{\omega}_{t}|^{2}]{\rm d}\Gamma_{1}{\rm d}t\\ & = &2\int_{\Omega}[\rho_{1}(\widetilde{\psi}_{t}F\cdot\nabla\widetilde{\psi}+\widetilde{\phi}_{t}F\cdot\nabla\widetilde{\phi})+ \rho_{2}\widetilde{\omega}_{t}F\cdot\nabla\widetilde{\omega}]_{0}^{T}{\rm d}x{\rm d}y\\ &&+2\int_{0}^{T}\int_{\Omega}[\rho_{1}(|\widetilde{\psi}_{t}|^{2} +|\widetilde{\phi}_{t}|^{2})+\rho_{2}|\widetilde{\omega}_{t}|^{2}]{\rm d}x{\rm d}y{\rm d}t +2\int_{0}^{T}a(\widetilde{\psi}, \widetilde{\phi}, \widetilde{\omega};\widetilde{\psi}, \widetilde{\phi}, \widetilde{\omega}){\rm d}t\\ &&-2D\int_{0}^{T}\int_{\Omega}[|\widetilde{\psi}_{x}|^{2}+|\widetilde{\phi}_{y}|^{2} +2\mu\widetilde{\psi}_{x}\widetilde{\phi}_{y}+\frac{1-\mu}{2}|\widetilde{\psi}_{y}+\widetilde{\phi}_{x}|^{2}]{\rm d}x{\rm d}y{\rm d}t\\ &&-2K\int_{0}^{T}\int_{\Omega}[|\widetilde{\psi}+\widetilde{\omega}_{x}|^{2}+|\widetilde{\phi}+\widetilde{\omega}_{y}|^{2}]{\rm d}x{\rm d}y{\rm d}t. \end{eqnarray} 由Poincaré不等式得 \begin{eqnarray} &&\left|\int_{\Omega}[\rho_{1}(\widetilde{\psi}_{t}F\cdot\nabla\widetilde{\psi}+\widetilde{\phi}_{t}F\cdot\nabla\widetilde{\phi})+ \rho_{2}\widetilde{\omega}_{t}F\cdot\nabla\widetilde{\omega}]_{0}^{T}{\rm d}x{\rm d}y\right|\\ &\leq &c_{1}\left\{\int_{\Omega}[\rho_{1}(|\widetilde{\psi}_{t}|^{2} +|\widetilde{\phi}_{t}|^{2})+\rho_{2}|\widetilde{\omega}_{t}|^{2}]_{0}^{T}{\rm d}x{\rm d}y +a(\widetilde{\psi}, \widetilde{\phi}, \widetilde{\omega};\widetilde{\psi}, \widetilde{\phi}, \widetilde{\omega})|_{0}^{T}\right\}\\ &\leq &2c_{1}I(0). \end{eqnarray} \widetilde{\psi}, \widetilde{\phi}, \widetilde{\omega} 分别乘以系统(4.3)前三式, 并在 [0, T]\times\Omega 上积分并相加, 得 分部积分得 \begin{eqnarray} &&\int_{0}^{T}\int_{\Omega}[\rho_{1}(|\widetilde{\psi}_{t}|^{2} +|\widetilde{\phi}_{t}|^{2})+\rho_{2}|\widetilde{\omega}_{t}|^{2}]{\rm d}x{\rm d}y{\rm d}t -\int_{0}^{T}a(\widetilde{\psi}, \widetilde{\phi}, \widetilde{\omega};\widetilde{\psi}, \widetilde{\phi}, \widetilde{\omega}){\rm d}t\\ & = &\int_{\Omega}[\rho_{1}(\widetilde{\psi}_{t}\widetilde{\psi}+\widetilde{\phi}_{t}\widetilde{\phi}) +\rho_{2}(\widetilde{\omega}_{t}\widetilde{\omega}]_{0}^{T}{\rm d}x{\rm d}y. \end{eqnarray} 再由Poincaré不等式得 \begin{eqnarray} \left|\int_{0}^{T}\int_{\Omega}[\rho_{1}(|\widetilde{\psi}_{t}|^{2} +|\widetilde{\phi}_{t}|^{2})+\rho_{2}|\widetilde{\omega}_{t}|^{2}]{\rm d}x{\rm d}y{\rm d}t -\int_{0}^{T}a(\widetilde{\psi}, \widetilde{\phi}, \widetilde{\omega};\widetilde{\psi}, \widetilde{\phi}, \widetilde{\omega}){\rm d}t\right|\leq c_{0}I(0). \end{eqnarray} 由(5.4), (5.5)和(5.7)式得 \begin{eqnarray} MM_{1}\int_{0}^{T}\int_{\widetilde{\Gamma}}[|\widetilde{\psi}_{t}|^{2} +|\widetilde{\phi}_{t}|^{2}+|\widetilde{\omega}_{t}|^{2}]{\rm d}\widetilde{\Gamma}{\rm d}t &\geq&\frac{1}{2}\int_{0}^{T}\int_{\Gamma_{1}}(F\cdot\nu)[\rho_{1}(|\widetilde{\psi}_{t}|^{2} +|\widetilde{\phi}_{t}|^{2})+\rho_{2}|\widetilde{\omega}_{t}|^{2}]{\rm d}\Gamma_{1}{\rm d}t\\ &\geq&\int_{0}^{T}a(\widetilde{\psi}, \widetilde{\phi}, \widetilde{\omega}; \widetilde{\psi}, \widetilde{\phi}, \widetilde{\omega}){\rm d}t-2c_{1}I(0)-c_{0}I(0). \end{eqnarray} 又由(5.8)式得 \begin{eqnarray} \int_{0}^{T}a(\widetilde{\psi}, \widetilde{\phi}, \widetilde{\omega};\widetilde{\psi}, \widetilde{\phi}, \widetilde{\omega}){\rm d}t\geq \int_{0}^{T}\int_{\Omega}[\rho_{1}(|\widetilde{\psi}_{t}|^{2} +|\widetilde{\phi}_{t}|^{2})+\rho_{2}|\widetilde{\omega}_{t}|^{2}]{\rm d}x{\rm d}y{\rm d}t - c_{0}I(0). \end{eqnarray} 因此, 由(5.8)–(5.9)式得 \begin{eqnarray} &&MM_{1}\int_{0}^{T}\int_{\widetilde{\Gamma}}[|\widetilde{\psi}_{t}|^{2} +|\widetilde{\phi}_{t}|^{2}+|\widetilde{\omega}_{t}|^{2}]{\rm d}\widetilde{\Gamma}{\rm d}t\\ &\geq&\int_{0}^{T}\int_{\Omega}[\rho_{1}(|\widetilde{\psi}_{t}|^{2} +|\widetilde{\phi}_{t}|^{2})+\rho_{2}|\widetilde{\omega}_{t}|^{2}]{\rm d}x{\rm d}y{\rm d}t-2c_{1}I(0)-2c_{0}I(0). \end{eqnarray} 由(5.8)和(5.10)式可得 因此, 记 T_{1} = 4c_{1}+3c_{0}>0 , 当 T\geq T_{1} 时, 记 C_{1} = \frac{1}{2MM_{0}}(T-T_{1})\geq0 , 则有 故(5.1)式得证. 又由(5.7)式得 $$2\int_{0}^{T}\int_{\Omega}[\rho_{1}(|\widetilde{\psi}_{t}|^{2} +|\widetilde{\phi}_{t}|^{2})+\rho_{2}|\widetilde{\omega}_{t}|^{2}]{\rm d}x{\rm d}y{\rm d}t \geq TI(0)-c_{0}I(0).$$ 因此, 记 T_{2} = c_{0}>0$$ T\geq T_{2}$时, 记$C_{2} = \frac{1}{2M_{1}}(T-T_{2})\geq0$, 则由(5.11)式得

(Ⅰ) 充分性

$\begin{eqnarray} \left\{\begin{array}{lll} { } \rho_{1}\psi_{tt}-D(\psi_{xx}+\frac{1-\mu}{2}\psi_{yy} +\frac{1+\mu}{2}\phi_{xy}) +K(\psi+\omega_{x}) = 0, \quad (x, y, t)\in\Omega\times (0, T), \\ { } \rho_{1}\phi_{tt}-D(\phi_{yy}+\frac{1-\mu}{2}\phi_{xx} +\frac{1+\mu}{2}\psi_{xy}) +K(\phi+\omega_{y}) = 0, \quad (x, y, t)\in\Omega\times (0, T), \\ { } \rho_{2}\omega_{tt}-K[(\psi+\omega_{x})_{x}+(\phi+\omega_{y})_{y}] = 0, \quad (x, y, t)\in\Omega\times (0, T), \\ { } \psi = \phi = \omega = 0, \quad (x, y, t)\in\Gamma_{0}\times (0, T), \\ { } D[\nu_{1}\psi_{x}+\mu\nu_{1}\phi_{y}+\frac{1-\mu}{2}(\psi_{y}+\phi_{x})\nu_{2}] = -\psi_{t}, \quad (x, y, t)\in\Gamma_{1}\times (0, T), \\ { } D[\nu_{2}\phi_{y}+\mu\nu_{2}\psi_{x}+\frac{1-\mu}{2}(\psi_{y}+\phi_{x})\nu_{1}] = -\phi_{t}, \quad (x, y, t)\in\Gamma_{1}\times (0, T), \\ { } K(\frac{\partial\omega}{\partial{\bf \nu}}+\nu_{1}\psi+\nu_{2}\phi) = -\omega_{t}, \quad (x, y, t)\in\Gamma_{1}\times (0, T), \\ (\psi(x, y, 0), \phi(x, y, 0), \omega(x, y, 0)) = (\psi_{01}, \phi_{01}, \omega_{01}), \quad (x, y)\in\Omega, \\ (\psi_{t}(x, y, 0), \phi_{t}(x, y, 0), \omega_{t}(x, y, 0)) = (\psi_{02}, \phi_{02}, \omega_{02}), \quad (x, y)\in\Omega. \end{array}\right. \end{eqnarray}$

$\psi_{t}, \phi_{t}, \omega_{t}$分别乘以系统(5.12)前三式, 并在$[0, T_{1}]\times\Omega$上积分并相加, 利用边界条件得到

$$$2\int_{0}^{T_{1}}\int_{\Gamma_{1}}[|\psi_{t}|^{2}+|\phi_{t}|^{2}+|\omega_{t}|^{2}]{\rm d}\Gamma_{1}{\rm d}t = I(0)-I(T_{1}).$$$

$\begin{eqnarray} \left\{\begin{array}{lll} { } \rho_{1}\widehat{\psi}_{tt}-D(\widehat{\psi}_{xx}+\frac{1-\mu}{2}\widehat{\psi}_{yy} +\frac{1+\mu}{2}\widehat{\phi}_{xy}) +K(\widehat{\psi}+\widehat{\omega}_{x}) = 0, \quad (x, y, t)\in\Omega\times (0, T), \\ { } \rho_{1}\widehat{\phi}_{tt}-D(\widehat{\phi}_{yy}+\frac{1-\mu}{2}\widehat{\phi}_{xx} +\frac{1+\mu}{2}\widehat{\psi}_{xy}) +K(\widehat{\phi}+\widehat{\omega}_{y}) = 0, \quad (x, y, t)\in\Omega\times (0, T), \\ { } \rho_{2}\widehat{\omega}_{tt}-K[(\widehat{\psi}+\widehat{\omega}_{x})_{x}+(\widehat{\phi}+\widehat{\omega}_{y})_{y}] = 0, \quad (x, y, t)\in\Omega\times (0, T), \\ { } \widehat{\psi} = \widehat{\phi} = \widehat{\omega} = 0, \quad (x, y, t)\in\Gamma_{0}\times (0, T), \\ { } D[\nu_{1}\widehat{\psi}_{x}+\mu\nu_{1}\widehat{\phi}_{y}+\frac{1-\mu}{2}(\widehat{\psi}_{y}+\widehat{\phi}_{x})\nu_{2}] = -\psi_{t}, \quad (x, y, t)\in\Gamma_{1}\times (0, T), \\ { } D[\nu_{2}\widehat{\phi}_{y}+\mu\nu_{2}\widehat{\psi}_{x}+\frac{1-\mu}{2}(\widehat{\psi}_{y}+\widehat{\phi}_{x})\nu_{1}] = -\phi_{t}, \quad (x, y, t)\in\Gamma_{1}\times (0, T), \\ { } K(\frac{\partial\widehat{\omega}}{\partial{\bf \nu}}+\nu_{1}\widehat{\psi}+\nu_{2}\widehat{\phi}) = -\omega_{t}, \quad (x, y, t)\in\Gamma_{1}\times (0, T), \\ { } (\widehat{\psi}(x, y, 0), \widehat{\phi}(x, y, 0), \widehat{\omega}(x, y, 0)) = (0, 0, 0), \quad (x, y)\in\Omega, \\ { } (\widehat{\psi}_{t}(x, y, 0), \widehat{\phi}_{t}(x, y, 0), \widehat{\omega}_{t}(x, y, 0)) = (0, 0, 0), \quad (x, y)\in\Omega. \end{array}\right. \end{eqnarray}$

$\begin{eqnarray} I(0)& = &\|(\Phi_{01}\Phi_{02})\|_{{\cal H}}^{2}\leq\frac{1}{C}\int_{0}^{T_{1}}\int_{\Gamma_{1}}(|\widetilde{\psi}_{t}|^{2}+|\widetilde{\phi}_{t}|^{2}+|\widetilde{\omega}_{t}|^{2}){\rm d}\Gamma_{1}{\rm d}t\\ &\leq&\frac{1}{C}\bigg[\int_{0}^{T_{1}}\int_{\Gamma_{1}}(|\psi_{t}|^{2}+|\phi_{t}|^{2}+|\omega_{t}|^{2}){\rm d}\Gamma_{1}{\rm d}t +\int_{0}^{T_{1}}\int_{\Gamma_{1}}(|\widehat{\psi}_{t}|^{2}+|\widehat{\phi}_{t}|^{2}+|\widehat{\omega}_{t}|^{2}){\rm d}\Gamma_{1}{\rm d}t\bigg]\\ &\leq& C' \int_{0}^{T_{1}}\int_{\Gamma_{1}}(|\psi_{t}|^{2}+|\phi_{t}|^{2}+|\omega_{t}|^{2}){\rm d}\Gamma_{1}{\rm d}t. \end{eqnarray}$

$\alpha = \frac{\ln(1+\frac{1}{C'})}{T_{1}}$, 则有

$$$I(T_{1})\leq {\rm e}^{-\alpha T_{1}}I(0).$$$

$M = 1+\frac{2}{C'}$, 则$I(t)\leq M{\rm e}^{-\alpha t}I(0)$, 即$\|(\Phi, \Phi_{t})\|_{{\cal H}}^{2}\leq M{\rm e}^{-\alpha t}\|(\Phi_{01}, \Phi_{02})\|_{{\cal H}}^{2}$.

(Ⅱ) 必要性

$\begin{eqnarray} \int_{0}^{T'}\int_{\Gamma_{1}}(|\psi_{t}|^{2}+|\phi_{t}|^{2}+|\omega_{t}|^{2}){\rm d}\Gamma_{1}{\rm d}t\geq \frac{1}{4}I(0). \end{eqnarray}$

$$$0\leq\frac{1}{2}\|(\widehat{\Phi}(T'), \widehat{\Phi}_{t}(T'))\|_{{\cal H}}^{2} = -\int_{0}^{T'}\int_{\Gamma_{1}}(\widehat{\psi}_{t}(\widehat{\psi}_{t}+\widetilde{\psi}_{t}) +\widehat{\phi}_{t}(\widehat{\phi}_{t}+\widetilde{\phi}_{t})+\widehat{\omega}_{t}(\widehat{\omega}_{t}+\widetilde{\omega}_{t})){\rm d}\Gamma_{1}{\rm d}t.$$$

$\begin{eqnarray} \int_{0}^{T'}\int_{\Gamma_{1}}(|\widehat{\psi}_{t}|^{2}+|\widehat{\phi}_{t}|^{2}+|\widehat{\omega}_{t}|^{2}){\rm d}\Gamma_{1}{\rm d}t \leq C''\int_{0}^{T'}\int_{\Gamma_{1}}(|\widetilde{\psi}_{t}|^{2}+|\widetilde{\phi}_{t}|^{2}+|\widetilde{\omega}_{t}|^{2}){\rm d}\Gamma_{1}{\rm d}t. \end{eqnarray}$

$\begin{eqnarray} & &\int_{0}^{T'}\int_{\Gamma_{1}}(|\widehat{\psi}_{t}|^{2}+|\widehat{\phi}_{t}|^{2}+|\widehat{\omega}_{t}|^{2}){\rm d}\Gamma_{1}{\rm d}t +\int_{0}^{T'}\int_{\Gamma_{1}}(|\widetilde{\psi}_{t}|^{2}+|\widetilde{\phi}_{t}|^{2}+|\widetilde{\omega}_{t}|^{2}){\rm d}\Gamma_{1}{\rm d}t\\ &\geq&\frac{1}{2}\int_{0}^{T'}\int_{\Gamma_{1}}(|\psi_{t}|^{2}+|\phi_{t}|^{2}+|\omega_{t}|^{2}){\rm d}\Gamma_{1}{\rm d}t. \end{eqnarray}$

## 6 次最优性和最优轨线指数稳定性

$\begin{eqnarray} V_{T}({\cal Y}_{0})\leq J_{T}({\cal Y}_{0};\widehat{U})\leq \gamma_{1}(T)\|{\cal Y}_{0}\|_{{\cal H}}^{2} , \end{eqnarray}$

$\begin{eqnarray} V_{T}({\cal Y}_{0})\geq \gamma_{2}(T)\|{\cal Y}_{0}\|_{{\cal H}}^{2} . \end{eqnarray}$

取$u_{1} = -\psi_{t}, u_{2} = -\phi_{t}, u_{3} = -\omega_{t}$, 那么

$\begin{eqnarray} I(t)\leq M{\rm e}^{-\alpha t}I(0), \quad \forall t\in[0, T]. \end{eqnarray}$

$\gamma_{1}(T) = \frac{M}{2\alpha}(1-{\rm e}^{-\alpha T})+\frac{\beta}{4}$, 由值函数的定义, 我们有

$\begin{eqnarray} \left\{\begin{array}{lll} { } \rho_{1}\widehat{\psi}_{tt}-D(\widehat{\psi}_{xx}+\frac{1-\mu}{2}\widehat{\psi}_{yy} +\frac{1+\mu}{2}\widehat{\phi}_{xy}) +K(\widehat{\psi}+\widehat{\omega}_{x}) = 0, \quad (x, y, t)\in\Omega\times (0, T), \\ { } \rho_{1}\widehat{\phi}_{tt}-D(\widehat{\phi}_{yy}+\frac{1-\mu}{2}\widehat{\phi}_{xx} +\frac{1+\mu}{2}\widehat{\psi}_{xy}) +K(\widehat{\phi}+\widehat{\omega}_{y}) = 0, \quad (x, y, t)\in\Omega\times (0, T), \\ { } \rho_{2}\widehat{\omega}_{tt}-K[(\widehat{\psi}+\widehat{\omega}_{x})_{x}+(\widehat{\phi}+\widehat{\omega}_{y})_{y}] = 0, \quad (x, y, t)\in\Omega\times (0, T), \\ { } \widehat{\psi} = \widehat{\phi} = \widehat{\omega} = 0, \quad (x, y, t)\in\Gamma_{0}\times (0, T), \\ { } D[\nu_{1}\widehat{\psi}_{x}+\mu\nu_{1}\widehat{\phi}_{y}+\frac{1-\mu}{2}(\widehat{\psi}_{y}+\widehat{\phi}_{x})\nu_{2}] = u_{1}, \quad (x, y, t)\in\Gamma_{1}\times (0, T), \\ { } D[\nu_{2}\widehat{\phi}_{y}+\mu\nu_{2}\widehat{\psi}_{x}+\frac{1-\mu}{2}(\widehat{\psi}_{y}+\widehat{\phi}_{x})\nu_{1}] = u_{2}, \quad (x, y, t)\in\Gamma_{1}\times (0, T), \\ { } K(\frac{\partial\widehat{\omega}}{\partial{\bf \nu}}+\nu_{1}\widehat{\psi}+\nu_{2}\widehat{\phi}) = u_{3}, \quad (x, y, t)\in\Gamma_{1}\times (0, T), \\ { } (\widehat{\psi}(x, y, 0), \widehat{\phi}(x, y, 0), \widehat{\omega}(x, y, 0)) = (0, 0, 0), \quad (x, y)\in\Omega, \\ { } (\widehat{\psi}_{t}(x, y, 0), \widehat{\phi}_{t}(x, y, 0), \widehat{\omega}_{t}(x, y, 0)) = (0, 0, 0), \quad (x, y)\in\Omega. \end{array}\right. \end{eqnarray}$

$\begin{eqnarray} V_{T}({\cal Y}_{T}^{\ast}(\delta, {\cal Y}_{0}, 0)) \leq\int_{\delta}^{\widehat{t}}\ell({\cal Y}_{T}^{\ast}(t, {\cal Y}_{0}, 0), {\cal U}_{T}^{\ast}(t, {\cal Y}_{0}, 0)){\rm d}t +\gamma_{1}(T+\delta-\widehat{t})\left\|{\cal Y}_{T}^{\ast}(\widehat{t}, {\cal Y}_{0}, 0)\right\|_{{\cal H}}^{2} \end{eqnarray}$

$\begin{eqnarray} \int_{\widetilde{t}}^{T}\ell({\cal Y}_{T}^{\ast}(t, {\cal Y}_{0}, 0), {\cal U}_{T}^{\ast}(t, {\cal Y}_{0}, 0)){\rm d}t \leq\gamma_{1}(T-\widetilde{t})\left\|{\cal Y}_{T}^{\ast}(\widetilde{t}, {\cal Y}_{0}, 0)\right\|_{{\cal H}}^{2}. \end{eqnarray}$

取$\xi = \frac{1}{2}\min\{1, \beta\}$, 对于任意${\cal Y}_{0}\in{\cal H} $$\overline{t}\in[0, T] , 由(1.4)式得 \begin{eqnarray} \ell({\cal Y}_{T}^{\ast}(t, {\cal Y}_{0}, 0), U_{T}^{\ast}(t, {\cal Y}_{0}, 0))\geq\xi [\|{\cal Y}_{T}^{\ast}(t, {\cal Y}_{0}, 0)\|_{{\cal H}}^{2}+\|U_{T}^{\ast}(t, {\cal Y}_{0}, 0)\|_{{\cal U}}^{2}]. \end{eqnarray} 因此, 我们有 又由(2.4)式得 因此, 对于任意 \overline{t}\in[0, T] , 都有 {\cal Y}_{T}^{\ast}(\overline{t}, {\cal Y}_{0}, 0)\in{\cal H} . 对于任意 \widehat{t}\in[\delta, T] , 我们有 {\cal Y}_{T}^{\ast}(\cdot, {\cal Y}^{\ast}(\delta), \delta) 是系统在区间 [\delta, T+\delta] 的最优解, 根据贝尔曼最优性原理, 我们有 因此, (6.5)式成立. 接下来证明(6.6)式. 对于任意 \widetilde{t}\in[0, T] , 我们有 因此, 对所有 \widetilde{t}\in[0, T] 都有 证毕. 定理6.1 对于任意初始值 {\cal Y}_{0}\in{\cal H} , 以及样本时间 \delta>0 和预测时间 T>\delta , 存在只与 T 有关的函数 使得以下估计式 $$V_{T}({\cal Y}_{T}^{\ast}(\delta, {\cal Y}_{0}, 0)) \leq\sigma_{1}(T)\int_{\delta}^{T}\ell({\cal Y}_{T}^{\ast}(t, {\cal Y}_{0}, 0), U_{T}^{\ast}(t, {\cal Y}_{0}, 0)){\rm d}t$$ $$\int_{\delta}^{T}\ell({\cal Y}_{T}^{\ast}(t, {\cal Y}_{0}, 0), U_{T}^{\ast}(t, {\cal Y}_{0}, 0)){\rm d}t \leq\sigma_{2}(T)\int_{0}^{\delta}\ell({\cal Y}_{T}^{\ast}(t, {\cal Y}_{0}, 0), U_{T}^{\ast}(t, {\cal Y}_{0}, 0)){\rm d}t$$ 成立. 由于 {\cal Y}_{T}^{\ast}(\cdot, {\cal Y}_{0}, 0)\in C([0, T];{\cal H}) , 故一定存在 t_{1}\in[\delta, T], t_{2}\in[0, \delta] , 使得 由(6.5)和(6.7)式可得 即(6.8)成立. 同理, 由(6.6)和(6.7)式得 因此(6.9)式成立. 证毕. 定理6.2 对于任意初始值 {\cal Y}_{0}\in{\cal H} 以及样本时间 \delta>0 , 存在 \widetilde{T}>\delta$$ \kappa\in(0, 1)$, 当$T\geq\widetilde{T}$时, 使得

$$$V_{T}({\cal Y}_{T}^{\ast}(\delta, {\cal Y}_{0}, 0))\leq V_{T}({\cal Y}_{0}) -\kappa\int_{0}^{\delta}\ell({\cal Y}_{T}^{\ast}(t, {\cal Y}_{0}, 0), U_{T}^{\ast}(t, {\cal Y}_{0}, 0)){\rm d}t.$$$

由定理6.1得

$$$V_{T}({\cal Y}_{T}^{\ast}(t_{k}, {\cal Y}(t_{k-1}), t_{k-1})) \leq\mu V_{T}({\cal Y}_{T}^{\ast}(t_{k-1}, {\cal Y}(t_{k-1}), t_{k-1})).$$$

$\gamma = \frac{\gamma_{1}(T)}{\gamma_{2}(T)}$, 则对$k\in{\Bbb N}^{+}$

$\begin{eqnarray} \|{\cal Y}_{T}^{\ast}(t_{k}, {\cal Y}(t_{k-1}), t_{k-1})\|_{{\cal H}}^{2} \leq \gamma {\rm e}^{-\eta t_{k}}\|{\cal Y}_{0}\|_{{\cal H}}^{2}. \end{eqnarray}$

$M = \gamma C(1+\frac{2\gamma_{1}(T)}{\beta})\frac{1}{1-\frac{\kappa\gamma_{2}(\delta)}{\gamma_{1}(T)}}\|{\cal Y}_{0}\|_{{\cal H}}^{2}$, 故有$\|{\cal Y}_{T}^{\ast}(t, {\cal Y}_{0}, 0)\|_{{\cal H}}^{2} \leq M {\rm e}^{-\eta t}.$证毕.

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