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

Let $\mathrm{\Delta }$ 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 $$u_{tt}+{\mathrm{\Delta }}^{2}u-\rho \mathrm{\Delta }{u}_t=0,x\in (0,\pi ),t>0$$ 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 $\rho \le 2$. For $\rho >2$, we show null controllability for arbitrary $T>0$ holds for almost all $\rho$, but fails for a dense subset of $(2,\mathrm{\infty })$. An analagous result is proven for Neumann control.