Boundary null-controllability for the beam equation with classical structural damping

Let $\Delta$ be the Dirichlet Laplacian on the interval $(0,\pi)$, and let $T>0$. We prove a well-posedness results for the structurally damped beam equation $$u_{tt}+\Delta^2 u-\rho \Delta u_t=0, x\in (0,\pi),t>0$$ with various boundary conditions including $$u(0,t)=u_{xx}(0,t)=0; u(\pi,t)=f(t),u_{xx}(\pi,t)=0,$$ and $f\in H_0^2(0,T)$ and appropriate initial conditions. Viewing $f$ as a control, we prove null controllability for all $\rho \leq 2$. For $\rho >2$, we show null controllability for arbitrary $T>0$ holds for almost all $\rho$, but fails for a dense subset of $(2,\infty)$. An analagous result is proven for Neumann control.