Interior Controllability of the Linear Beam Equation

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 $\tau >0$.