Optimal Control of the Stochastic Navier-Stokes Equations

We consider an optimal control problem for incompressible random flows governed by the stochastic Navier-Stokes equations in a multidimensional domain. The control problem introduced in this talk is motivated by common strategies, such as tracking a desired velocity field and minimizing the enstrophy energy. Since existence and uniqueness results of mild solutions are provided only locally in time, the cost functional has to incorporate stopping times dependent on the controls. We state an existence and uniqueness result for the optimal control and using a stochastic maximum principle, we derive necessary optimality conditions. As a consequence, we get an explicit formula of the optimal control based on the adjoint equation, which is given by a backward SPDE. Moreover, we show that sufficient optimality conditions are satisfied. This enables us to obtain unique solutions of control problems constrained by the stochastic Navier-Stokes equations for two-dimensional as well as for three-dimensional domains.