A Boundary Control Problem for Micropolar Fluids

An optimal boundary control problem for the micropolar fluid equations in 3D bounded domains, with mixed boundary conditions, is analyzed. By considering boundary controls for the velocity vector and the angular velocity of rotation of particles, the existence of optimal solutions is proved. The analyzed optimal boundary control problem includes the minimization of a Lebesgue norm between the velocities and some desired fields, as well as the resistance in the fluid due to the viscous friction. By using the Theorem of Lagrange multipliers, an optimality system is derived. A second-order sufficient condition is also given.