On the exact controllability of a Galerkin scheme for 3D viscoelastic fluids with fractional Laplacian viscosity and anisotropic filtering

We study a mathematical model describing 3D viscoelastic fluids with memory, fractional viscosity, and regularized by means of a horizontal anisotropic filter. This regularization is obtained through the action of the inverse of the horizontal Helmholtz operator, and the system is considered in a fully-periodic space-domain $\Omega$. After introducing a controlled version of such a model, we take into account for it a suitable Galerkin approximation scheme. Exploiting the Hilbert uniqueness method we establish the exact controllability of the finite dimensional Galerkin system.