The evolution of two incompressible, immiscible, isothermal fluids in a bounded domain containing porous media is described by the coupled Cahn-Hilliard-Brinkman (CHB) system. The CHB system consists of the Cahn-Hilliard equation describing the dynamics of the relative concentration of fluids and the Brinkman equation for velocity. In this work, we address the optimal control problem for a two-dimensional non-locall CHB system with a singular-type potential. The existence and regularity results are obtained by approximating the singular potential by a sequence of regular potentials and introducing a sequence of mobility terms to resolve the blow-up due to the singularity of the potential. Further, we prove the existence of a strong solution under higher regularity assumptions on the initial data and the uniqueness of the solution using the weak-strong uniqueness technique. By considering the external forcing term in the velocity equation as a control variable, we prove the existence of an optimal control for a tracking type cost functional. The differentiability properties of the control-to-state operator are studied to establish the first-order necessary optimality conditions. Moreover, the optimal control is characterised in terms of the adjoint variable.

Shell models of turbulence are representation of turbulence equations in Fourier domain. Various shell models are studied for their mathematical relevance and the numerical simulations which exhibit at most resemblance with turbulent flows. One of the mathematically well studied shell model of turbulence is called sabra shell model. This work concerns with two important issues related to shell model namely feedback stabilization and robust stabilization. We first address stabilization problem related to sabra shell model of turbulence and prove that the system can be stabilized via finite dimensional controller. Thus only finitely many modes of the shell model would suffice to stabilize the system. Later we study robust stabilization in the presence of the unknown disturbance and corresponding control problem by solving an infinite time horizon max-min control problem. We first prove the $H^ \infty$ stabilization of the associated linearized system and characterize the optimal control in terms of a feedback operator by solving an algebraic riccati equation. Using the same riccati operator we establish asymptotic stability of the nonlinear system.