<p>We consider an optimal control problem for the Navier–Stokes system with Tresca boundary conditions. With such boundary conditions, the weak formulation of the system is a variational inequality. We approximate this system and the optimal control problem by regularizing the boundary conditions leading to a variational equality. We show that for the approximate system, there exists an optimal control and we derive the first optimality condition by using an adjoint system. We also prove that the approximate optimal controls converge towards an optimal control for the Navier–Stokes system with Tresca boundary conditions. Finally we show that as the threshold of the Tresca law goes to infinity, the corresponding optimal controls converge towards an optimal control for the Navier–Stokes system with the Dirichlet boundary condition.</p>