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In this paper, we formulate a distributed optimal control problem related tothe evolution of two isothermal, incompressible, immiscible fluids in a twodimensional bounded domain. The distributed optimal control problem is framedas 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's minimum principle and provesecond order necessary and sufficient conditions of optimality for the problem.

Second Order Optimality Conditions for Optimal Control Problems Governed by 2D Nonlocal Cahn Hilliard Navier Stokes Equations