Optimal control of hemivariational inequalities for nonstationary Navier-Stokes equations

Optimal control of nonstationary Navier-Stokes equations is studied with nonlinear boundary conditions described by the Clarke subdifferential. Precisely, we aim at minimizing a general functional for a control problem whose state is a solution to a boundary value problem depending on the control itself. Accordingly, the lower level problem is expressed by a hemivariational inequality associated with a nonconvex nonsmooth locally Lipschitz superpotential. The existence of solutions to our problem is then shown via a convergence scheme based on mixed equilibria and a stability result with respect to variations on the control for the dynamic state control system associated with the main control problem.