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

Abstract

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