Null-controllability for the beam equation with structural damping. Part 1. Distributed control

Let $\Delta$ be the Dirichlet Laplacian on the interval $(0,\pi)$. The null controllability properties of the equation $$u_{tt}+\Delta^2 u+\rho (\Delta)^\alpha u_t=F(x,t)$$ are studied. Let $T>0$, and assume initial conditions $(u^0,u^1)\in Dom(\Delta)\times L^2(0,\pi)$. We first prove finite dimensional null control results: suppose $F(x,t)=f^1(t)h^1(x)+f^2(t)h^2(x)$ with $h^1,h^2$ given functions. For $\alpha \in [0,3/2)$, we prove that there exist $h^1,h^2\in L^2(0,\pi)$ such that for any $(u^0,u^1)$, there exist $L^2$ null controls $(f^1,f^2).$ For $\alpha< 1$ and $\rho <2$, we prove null controllability with $f^2=0$ and $h^1$ belonging to a large class of functions. For $\alpha\in [3/2,2)$, we prove spectral and null controllability both generally fail, but two dimensional weak controllability holds. Our second set of results pertains to $F(x,t)=\chi_\Omega(x)f(x,t)$, with $\Omega$ any open subset of $(0,\pi)$. For any $\alpha \in [0,3/2),$ we prove there exists a null control $f\in L^2(\Omega\times(0,T))$ To prove our main results, we use the Fourier method to rewrite the control problems as moment problems. These are then solved by constructing biorthogonal sets to the associated exponential families. These constructions seem to be non-standard and may be of independent interest.