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Interior Controllability of the Thermoelastic Plate Equation

Volume 12, Number 1 (2011), 46 - 59

Interior Controllability of the Thermoelastic Plate Equation

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Abstract

In this paper we prove the interior controllability of the Thermoelastic Plate Equation

 

$\left\{ \begin{array}{ll} w_{tt}+\Delta^2w+\alpha\Delta w=1_{\omega}u_{1}(t,x),& \mbox{in} \quad (0, \tau) \times \Omega,\\ \theta_t-\beta\Delta\theta-\alpha\Delta w_t=1_{\omega}u_{2}(t,x), & \mbox{in} \quad (0, \tau) \times \Omega,\\ \theta=w=\Delta w=0, & \mbox{on} \quad (0, \tau) \times \partial \Omega, \end{array} \right.$

 

where $\alpha\neq 0$, $\beta>0$, $\Omega$ is a sufficiently regular bounded domain in ${\mathbb R}^{N}$ $(N\geq 1)$, $\omega$ is an open nonempty subset of $\Omega$, $1_{\omega}$ denotes the characteristic function of the set ω and the distributed control $u_{i}\in L^{2}([0,\tau]; L^{2}(\Omega)), i=1,2.$. Specifically, we prove the following statement: For all $\tau >0$ the system is approximately controllable on $[0, \tau]$. Moreover, we exhibit a sequence of controls steering the system from an initial state to a final state in a prefixed time $\tau >0$ .