On the optimal control of steady fluid structure interaction systems

Fluid-structure interaction (FSI) systems consist of one or more solid structures that deform by interacting with a surrounding fluid flow and are commonly studied in many engineering and biomedical fields. Usually those kind of problems are solved in a direct approach, however it is also interesting to study the inverse problem, where the goal is to find the optimal value of some control parameters, such that the FSI problem solution is close to a desired one. In this work the optimal control problem is formulated with the Lagrange multipliers and adjoint variables formalism. In order to recover the symmetry of the state-adjoint system an auxiliary displacement field is introduced and used to extend the velocity field to the structure domain. As a consequence, the adjoint interface forces are balanced automatically. The optimality system is derived from the first order necessary condition by taking the Fréchet derivatives of the augmented Lagrangian with respect to all the variables involved. The optimal solution is obtained through a gradient-based algorithm applied to the optimality system. In order to support the proposed approach numerical test with distributed control, boundary control and parameter estimation are performed.