  In this report we present the approximation of an infinite horizon optimal
control problem for the evolutive Navier-Stokes system. The method is based on
a model reduction technique, using a POD approximation, coupled with a
Hamilton-Jacobi equation which characterizes the value function of the
corresponding control problem for the reduced system. Although the
approximation schemes available for the HJB are shown to be convergent for any
dimension, in practice we need to restrict the dimension to rather small
numbers and this limitation affects the accuracy of the POD approximation. We
will present numerical tests for the control of the time-dependent
Navier-Stokes system in two-dimensional spatial domains to illustrate our
approach and to show the effectiveness of the method.
