In this report, we are investigating the numerical approximation of an optimal control problem involving the evolution of a newtonian viscous incompressible fluid described by the Navier-Stokes equations. This PDE system is discretized using a low order finite element in space coupled with a Lagrange-Galerkin scheme for the nonlinear advection operator. We introduce a full discrete linearized scheme that is used to compute the gradient of a given cost function by ensuring its consistency. Using gradient based optimization algorithms, we are able to deal with two fluid flow control problems : Drag reduction around a moving cylinder, an identification of far-field velocity using the knowledge of the fluid load on a rectangular bluff body, for both fixed and moving configuration.