Navier-Stokes dynamical shape control : from state derivative to Min-Max principle

This report deals with recent progress in the study of shape optimization problems in case of a moving domain. We may restrict ourself to the case of newtonian viscous incompressible fluids described by the Navier-Stokes equations. We suggest three strategies in order to solve an optimal control problem involving the shape variable, respectively based on, the state derivative with respect to the shape and its associated adjoint state, the Min-Max principal coupled with a function space parametrization, the Min-Max principal coupled with a function space embedding