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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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