The Navier-Stokes system of partial differential equations with Navier boundary condition that models the evolution of viscous, incompressible flows is studied as a controllable system and the optimal cost functional with respect to that control is invstigated. A dual dynamic programming approach is proposed and sufficient optimality conditions in terms of verification theorem are proved. The essential novelty of the paper is that all equations under consideration are treated in weak form. Dual feedback controls are proposed for the problem under study.