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The evolution of two incompressible, immiscible, isothermal fluids in abounded domain and a porous media is described by the coupledCahn-Hilliard-Brinkman (CHB) system. The CHB system consists of theCahn-Hilliard equation describing the dynamics of the relative concentration offluids and the Brinkman equation for velocity. This work addresses the optimalcontrol problem for a two-dimensional nonlocal CHB system with a singular-typepotential. The existence and regularity results are obtained by approximatingthe singular potential by a sequence of regular potentials and introducing asequence of mobility terms to resolve the blow-up due to the singularity of thepotential. Further, we prove the existence of a strong solution under higherregularity assumptions on the initial data and the uniqueness of the solutionusing the weak-strong uniqueness technique. By considering the external forcingterm in the velocity equation as a control, we prove the existence of anoptimal control for a tracking type cost functional. The differentiabilityproperties of the control-to-state operator are studied to establish thefirst-order necessary optimality conditions. Moreover, the optimal control ischaracterised in terms of the adjoint variable.

Regularity and Optimal Control of Non Local Cahn Hilliard Brinkman system with Singular Potential