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

TL;DR

This paper investigates the boundary null-controllability of a structurally damped beam equation, establishing conditions under which control can steer solutions to zero, depending on damping parameter and boundary conditions.

Contribution

It provides new controllability results for the damped beam equation with various boundary conditions, including precise parameter ranges for controllability and failure cases.

Findings

Null controllability holds for all damping coefficients $ ho \,\leq 2$.

For $ ho > 2$, controllability holds for almost all $ ho$, but not for a dense subset.

Results extend to Neumann boundary control cases.

Abstract

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.