Control Problems for the Lagrangian Averaged Navier-Stokes Equations

The Lagrangian-Averaged Navier-Stokes equations
(also known as Navier-Stokes-alpha equations), a
regularization of the Navier-Stokes equations, describe the mean motion of a viscous incompressible fluid.
We study time-dependent optimal control problems for the Lagrangian-Averaged Navier-Stokes equations on a three-dimensional periodic domain. The problem of 'turbulence control' via enstrophy reduction and the velocity matching problem are adressed. In both cases external forcing serves as control variable. It is shown that for both control problems optimal controls exist and that the optimal controls can be characterized
by a first-order necessary condition, involving the linearized adjoint Lagrangian-Averaged Navier-Stokes equations.
Numerical methods to compute the optimal controls
are outlined.