Local Controllability of a One-Dimensional Beam Equation

We prove that the beam equation with clamped ends is locally controllable in a $H^{5+\epsilon} \times H^{3+\epsilon} ((0,1),\mathbb{R})$-neighborhood of a particular trajectory of the free system, with $\epsilon>0$ and with control functions in $H^{1}_{0}((0,T),\mathbb{R})$. Ball,. Marsden, and Slemrod already proved that this equation is not controllable in $H^{2}_{0} \times L^{2}((0,1),\mathbb{R})$ with control functions in $L^{r}_{\text{loc}}(\mathbb{R},\mathbb{R})$, $r>1$. This article justifies that their negative result is due to a choice of functional spaces which does not allow controllability. Our proof uses moment theory and the Nash-Moser theorem.