Navier-Stokes type models with control conditions given by inclusions

We analyze a control problem for a general class of coupled systems of stationary Navier-Stokes type equations in the incompressible case, with homogeneous Dirichlet condition on the boundary of a bounded domain in , N ≤ 3, and non-homogeneous terms of reaction type. Such a control problem may describe the flow of a viscous incompressible fluid in multidisperse porous media with a controllability condition imposed on the coefficients of the coupled systems and expressed by means of a continuous functional depending on the velocities and pressures. The controllability conditions are not necessarily given by equalities, but more generally are formulated by inclusions. A lower and upper solution technique is used for the exact and approximate solvability of the control problem, which requires the existence, uniqueness and continuous dependence of the solution on the coefficients.