Maximum Principle and Data Assimilation Problem for the Optimal Control Problems Governed by 2D Nonlocal Cahn-Hillard-Navier-Stokes Equations

We study some optimal control problems associated to the evolution of two isothermal, incompressible, immisible fluids in a two-dimensional bounded domain. The Cahn- Hilliard-Navier-Stokes model consists of a Navier-Stokes equation governing the fluid velocity field coupled with a convective Cahn-Hilliard equation for the relative concentration of one of the fluids. A distributed optimal control problem is formulated as the minimization of a cost functional subject to the controlled nonlocal Cahn-Hilliard- Navier-Stokes equations. We establish the first-order necessary conditions of optimality by proving the Pontryagin's maximum principle for optimal control of such system via the seminal Ekeland variational principle. The optimal control is characterized using the adjoint variable. We also study an another control problem which is similar to data assimilation problems in meteorology of obtaining unknown initial data. Considering the same underlying system as above we establish the optimal initial data in terms of the corresponding adjoint variable.