Null-controllability for the beam equation with structural damping. Part 2: Integration by parts for fractional Laplacians and boundary control

Abstract

Let $\hat{I}”$ be the Neumann Laplacian on the interval $(0,\ddot{I}€)$, and let $T>0$. An integration by parts formula is proven for the spectral fractional Laplacian, $(-\hat{I}”{)}^{\hat{I}}\pm$, for $\hat{I}\pm \in (0,1)$. As an application, we prove well-posedness results for the structurally damped beam equation $$u_{tt}+\hat{I}{”}^{2}u+\ddot{I}(-\hat{I}”{)}^{\hat{I}}\pm u_t=0,x\in (0,\ddot{I}€),t>0$$ with various boundary conditions including $$u_x(0,t)=u_{xxx}(0,t)=0;u_x(\ddot{I}€,t)=f(t),u_{xxx}(\ddot{I}€,t)=0,$$ and $f\in L^2(0,T)$ and appropriate initial conditions. Viewing $f$ as a control, we prove null-controllability. Analagous results are proven for higher order controls, and for the Dirichlet Laplacian.