The fluid-structure interaction (FSI) problem has been extensively studied, and many papers and books are available in the literature on the subject. In this work, we consider some optimal FSI pressure boundary control applications by using a membrane model derived from the Koiter shell equations where the thickness of the solid wall can be neglected and the computational cost of the numerical problem reduced. We study the inverse problem with the aim of achieving a certain objective by changing some design parameters (e.g. forces, boundary conditions or geometrical domain shapes) by using an optimal control approach based on Lagrange multipliers and adjoint variables. In particular, a pressure boundary optimal control is presented in this work. The optimality system is derived from the first-order optimality condition by taking the Fréchet derivatives of the Lagrangian with respect to all the variables involved. This system is solved by using a finite element code with mesh-moving capabilities. In order to support the proposed approach, we perform numerical tests where the pressure on a fluid domain boundary controls the displacement that occurs in a well-defined region of the solid domain
