Optimal control of hemivariational inequalities for nonstationary Navier–Stokes equations

ABSTRACT

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.

Keywords:

• Navier–Stokes
• nonmonotone boundary conditions
• optimal control
• equilibrium problems
• hemivariational inequalities

Acknowledgements

The authors are grateful to the referees for their valuable comments which lead to the improvement of this paper. Professor Ram Mohapatra has been supported by a grant from the Mohapatra Family Foundation.

Disclosure statement

No potential conflict of interest was reported by the author(s).