Title: An Optimal Control Problem for the Steady Nonhomogeneous Asymmetric Fluids,
Abstract: We study an optimal boundary control problem for the two-dimensional
stationary micropolar fluids system with variable density. We control the
system by considering boundary controls, for the velocity vector and angular
velocity of rotation of particles, on parts of the boundary of the flow domain.
On the remaining part of the boundary, we consider mixed boundary conditions
for the vector velocity (Dirichlet and Navier conditions) and Dirichlet
boundary conditions for the angular velocity. We analyze the existence of a
weak solution obtaining the fluid density as a scalar function of the stream
function. We prove the existence of an optimal solution and, by using the
Lagrange multipliers theorem, we state first-order optimality conditions. We
also derive, through a penalty method, some optimality conditions satisfied by
the optimal controls.