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

Abstract

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