## African Diaspora Journal of Mathematics

### 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 $\tau >0$.

#### Article information

Source
Afr. Diaspora J. Math. (N.S.), Volume 14, Number 1 (2012), 30-38.

Dates
First available in Project Euclid: 18 July 2013

Mathematical Reviews number (MathSciNet)
MR3080394

Zentralblatt MATH identifier
1274.93035

Subjects
Primary: 93B05; 93C25

#### Citation

Leiva , H.; Pereira , W. Interior Controllability of the Linear Beam Equation. Afr. Diaspora J. Math. (N.S.) 14 (2012), no. 1, 30--38. https://projecteuclid.org/euclid.adjm/1374153553

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