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In this work, we address some optimal control problems related to theevolution of two isothermal, incompressible, immisible fluids in a twodimensional bounded domain. A distributed optimal control problem is formulatedas the minimization of a suitable cost functional subject to the controllednonlocal Cahn-Hilliard-Navier-Stokes equations. We describe the first ordernecessary conditions of optimality via Pontryagin minimum principle and provesecond order necessary and sufficient conditions of optimality for the problem.

Pontryagin's maximum principle and second order optimality condition for  optimal control problems for the nonlocal Cahn-Hilliard-Navier-Stokes systems  in two dimensions