Sufficient $\varepsilon$-Optlmality Conditions for Navier-Stokes Flow. Numerical Algorithm

The sufficient ε-optimality conditions for optimal control problem governed by Navier-Stokes system of partial differential equations with Navier boundary condition is invstigated. An approximate dual dynamic programming approach is proposed and ε-sufficient optimality conditions in terms of the approximate verification theorem are proved. Essential novelty of the paper is that all inequalities in the approximate dual dynamic programming are treated in a weak form. A numerical algorithm for approximate minimum is proposed.