  We introduce reduced order methods as an efficient strategy to solve
parametrized non-linear and time dependent optimal flow control problems
governed by partial differential equations. Indeed, the optimal control
problems require a huge computational effort in order to be solved, most of all
in physical and/or geometrical parametrized settings. Reduced order methods are
a reliable and suitable approach, increasingly gaining popularity, to achieve
rapid and accurate optimal solutions in several fields, such as in biomedical
and environmental sciences. In this work, we employ a POD-Galerkin reduction
approach over a parametrized optimality system, derived from the
Karush-Kuhn-Tucker conditions. The methodology presented is tested on two
boundary control problems, governed respectively by (i) time dependent Stokes
equations and (ii) steady non-linear Navier-Stokes equations.
