  We consider the approximation of some optimal control problems for the
Navier-Stokes equation via a Dynamic Programming approach. These control
problems arise in many industrial applications and are very challenging from
the numerical point of view since the semi-discretization of the dynamics
corresponds to an evolutive system of ordinary differential equations in very
high dimension. The typical approach is based on the Pontryagin maximum
principle and leads to a two point boundary value problem. Here we present a
different approach based on the value function and the solution of a Bellman, a
challenging problem in high dimension. We mitigate the curse of dimensionality
via a recent multilinear approximation of the dynamics coupled with a dynamic
programming scheme on a tree structure. We discuss several aspects related to
the implementation of this new approach and we present some numerical examples
to illustrate the results on classical control problems studied in the
literature.
