Controllability with one scalar control of a system of interaction between the Navier–Stokes system and a damped beam equation

We consider the controllability of a fluid–structure interaction system, where the fluid is modeled by the Navier–Stokes system and where the structure is a damped beam located on a part of its boundary. The motion of the fluid is bidimensional, whereas the deformation of the structure is one-dimensional, and we use periodic boundary conditions in the horizontal direction. Our result is the local null-controllability of this free-boundary system by using only one scalar control acting on an arbitrary small part of the fluid domain. This improves a previous result obtained by the authors where three scalar controls were needed to achieve the local null-controllability. In order to show the result, we prove the final-state observability of a linear Stokes-beam interaction system in a cylindrical domain. This is done by using a Fourier decomposition, proving Carleman inequalities for the corresponding system for the low-frequency solutions and in the case where the observation domain is an horizontal strip. Then, we conclude this observability result by using a Lebeau–Robbiano strategy for the heat equation and a uniform exponential decay for the high-frequency solutions. Finally, the result on the nonlinear system can be obtained by a change of variables and a fixed-point argument.