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Interior Controllability of the Linear Beam Equation

Volume 14, Number 1 (2012), 30 - 38

Interior Controllability of the Linear Beam Equation

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Abstract

In this paper we prove the interior controllability of the Linear Beam Equation
$$
\left\{%
\begin{array}{ll}
u_{tt}-2\beta\Delta u_t + \Delta^2 u= 1_{\omega}u(t,x),& \mbox{in} \quad (0, \tau) \times \Omega,\\
u = \Delta u = 0, & \mbox{on} \quad (0, \tau) \times \partial \Omega,
\end{array}%
\right.
$$
where $\beta>1$, $\Omega$ 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 $u\in L^{2}([0,\tau]; L^{2}(\Omega)).$ 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 $\tau >0$.