Controllability of the Semilinear Beam Equation

Abstract

In this paper, we prove the approximate controllability of the following semilinear beam equation:

$$ \left\{ \begin{array}{lll} \displaystyle{\partial^{2} y(t,x) \over \partial t^{2}} & = & 2\beta\Delta\displaystyle\frac{\partial y(t,x)}{\partial t}- \Delta^{2}y(t,x)+ u(t,x) + f(t,y,y_{t},u),\; \mbox{in}\; (0,\tau)\times\Omega, \\ y(t,x) & = & \Delta y(t,x)= 0 , \ \ \mbox{on}\; (0,\tau)\times\partial\Omega, \\ y(0,x) & = & y_{0}(x), \ \ y_{t}(x)=v_{0}(x), x \in \Omega, \end{array} \right. $$

in the states space $Z_{1}=D(\Delta)\times L^{2}(\Omega)$ with the graph norm, where β > 1, Ω is a sufficiently regular bounded domain in IR ^N, the distributed control u belongs to L ²([0,τ];U) (U = L ²(Ω)), and the nonlinear function $f:[0,\tau]\times I\!\!R\times I\!\!R\times I\!\!R\longrightarrow I\!\!R$ is smooth enough and there are a,c ∈ IR such that $a<\lambda_{1}^{2}$ and

$$ \displaystyle\sup\limits_{(t,y,v,u)\in Q_{\tau}}\mid f(t,y,v,u) - ay -cu\mid<\infty, $$

where Q _τ  = [0,τ]×IR×IR×IR. We prove that for all τ > 0, this system is approximately controllable on [0,τ].