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Carleman estimates for the Euler-Bernoulli plate operator. (English) Zbl 0963.93039
The aim of this paper is to solve a problem of boundary observability for the following system consisting of the Euler-Bernoulli plate coupled with the heat equation \[ \begin{aligned} w_{tt}+ \Delta^2w= \alpha\Delta\theta\quad &\text{in }]0,T[\times \Omega,\\ \theta_t- \Delta\theta= \beta\Delta w\quad &\text{in }]0,T[\times \Omega,\tag{1}\\ w= \partial_\nu w=\Delta w=\theta= 0\quad &\text{on }[0,T]\times \partial\Omega,\end{aligned} \] where \(\alpha,\beta\in \mathbb{R}\), \(T>0\) and \(\nu= (\nu_0,\dots, \nu_n)\) is the unit outer normal vector to \([0,T]\times \partial\Omega\).
First of all the author recalls a Carleman estimate with singular weight for the heat equation: \[ \begin{aligned} \theta_t(t, x)- \Delta\theta(t, x)= f(t,x)\quad &\text{in }]0,T[\times \Omega,\\ \theta= 0\quad &\text{on }[0, T]\times \partial\Omega,\\ \theta(0,x)= \theta_0(x)\quad &\text{in }\Omega,\end{aligned} \] with \((f, \theta_0)\in L^1([0, T]\times \Omega)\times H^1_0(\Omega)\). Thus, for the solution of the above problem there exists \(\lambda_0\) so that for each \(\lambda> \lambda_0\) there exists \(\tau_\lambda\) so that for \(\tau> \tau_\lambda\) the following estimate holds uniformly in \(\lambda\), \(\tau\): \[ C_1\lambda\|\widetilde\tau^{-{1\over 2}} e^{\tau\phi}\theta\|^2_{2,\widetilde\tau}\leq \|e^{\tau\phi} f\|^2+ \int_{[0,T]\times \partial\Omega} \widetilde\tau e^{2\tau\phi} \partial_\nu\psi\mid\partial_\nu \theta|^2 d\sigma, \] where \(\widetilde\tau= \lambda\tau ge^{\lambda\psi}\), \(C_1>0\) and \(\phi(t, x):= g(t)(e^{\lambda\psi(x)}- 2e^{\lambda\Phi})\), with \(\Phi= \|\psi\|_{L^\infty(\Omega)}\), \(\nabla\psi(x)\neq 0\) for all \(x\in\Omega\), \(\nabla^2\psi(x)\xi\cdot \xi\geq \gamma|\xi|^2\) for all \(\xi\).
Next, the author considers the following problem: \[ \begin{aligned} w_{tt}+ \Delta^2w= f\quad &\text{in }]0,T[\times \Omega,\\ w= \partial_\nu w= \Delta w=0\quad &\text{on }[0,T]\times \partial\Omega,\end{aligned} \]
\[ (w(0, x), w_t(0,x))\in (H^4(\Omega)\cap H^2_0(\Omega))\times H^2(\Omega), \] with \(f\in L^2([0, T]\times \Omega)\). If \(w\) is the solution of this problem and \(\phi\) is given as above, then there exists \(\lambda_0\) so that for each \(\lambda>\lambda_0\) there exists \(\tau_\lambda> 0\) so that for \(\tau> \tau_\lambda\) the following estimate holds uniformly in \(\lambda, \tau\): \[ C\|e^{\tau\phi} w\|^2_{2,\widetilde\tau}\leq \int_{[0,T]\times \partial\Omega} e^{2\tau\phi} \partial_\nu \psi|\partial_\nu\Delta w|^2 d\sigma+ \|e^{\tau\phi} \widetilde\tau^{-{1\over 2}} f\|^2, \] with \(\widetilde\tau= \lambda\tau ge^{\lambda\psi}\) and \(C>0\).
Finally, putting together the previous estimates one obtains an observability estimate for the coupled system. More precisely, if \((w,\theta)\) is a solution of the system (1), then there exists \(\lambda_0\) so that for each \(\lambda> \lambda_0\) there exists \(\tau_\lambda> 0\) so that for \(\tau> \tau_\lambda\) the following estimate holds uniformly in \(\lambda,\tau\): \[ \|e^{\tau\psi} w\|^2_{2,\widetilde\tau}+ \|e^{2\phi} \widetilde\tau^{-{1\over 2}} \theta\|^2_{2,\widetilde\tau}\leq C\int_{[0,T]\times \partial\Omega} (e^{2\tau\phi} \partial_\nu\psi|\partial_\nu\Delta w|^2+ \widetilde\tau e^{2\tau\phi}\partial_\nu \psi|\partial_\nu\theta|^2) d\sigma. \]

MSC:
93C20 Control/observation systems governed by partial differential equations
93B07 Observability
74F05 Thermal effects in solid mechanics
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