The thesis is organized in two main parts. In the first part, the distributed optimal control problem of the non-smooth Cahn-Hilliard-Stokes system is considered, assuming that the homogeneous free energy density in the Cahn-Hilliard equations corresponds to the double-obstacle potential. The analysis is performed at continuous level and by a finite dimensional approach. Numerical experiments are displayed.

In the second part, the distributed optimal control problem of the smooth Cahn-Hilliard-Navier-Stokes system is studied. In this case the homogeneous free energy density is equal to the double-well potential. This problem is analyzed considering infinite dimensional settings and a discrete approach. Significant numerical experiments are proposed.