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Interior controllability of the thermoelastic plate equation. (English) Zbl 1243.93019

Summary: In this paper we prove the interior controllability of the thermoelastic plate equation \[ \begin{cases} w_{tt}+\Delta^2w+\alpha\Delta w=1_{\omega}u_{1}(t,x),& \text{in} \quad (0, \tau) \times \Omega,\\ \theta_t-\beta\Delta\theta-\alpha\Delta w_t=1_{\omega}u_{2}(t,x), & \text{in} \quad (0, \tau) \times \Omega,\\ \theta=w=\Delta w=0, & \text{on} \quad (0, \tau) x \partial \Omega, \end{cases} \] 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 \(\omega\) 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 \(\tau >0\).

MSC:

93B05 Controllability
93C25 Control/observation systems in abstract spaces
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Full Text: Euclid