Controllability of the Semilinear Beam Equation

In this paper, we prove the approximate controllability of the following semilinear beam equation: $$ left{ begin{array}{lll} displaystyle{partial^{2} y(t,x) over partial t^{2}} & = & 2betaDeltadisplaystylefrac{partial y(t,x)}{partial t}- Delta^{2}y(t,x)+ u(t,x) + f(t,y,y_{t},u),; mbox{in}; (0,tau)timesOmega, y(t,x) & = & Delta y(t,x)= 0 , mbox{on}; (0,tau)timespartialOmega, y(0,x) & = & y_{0}(x), y_{t}(x)=v_{0}(x), x in Omega, end{array} right. $$ 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 $f:[0,tau]times I!!Rtimes I!!Rtimes I!!Rlongrightarrow I!!R$ is smooth enough and there are a,c?∈?IR such that $a and $$ displaystylesuplimits_{(t,y,v,u)in Q_{tau}}mid f(t,y,v,u) - ay -cumid where Q _τ ?=?[0,τ]×IR×IR×IR. We prove that for all τ?>?0, this system is approximately controllable on [0,τ].