Georgia Institute of Technology College of Sciences

We discuss the problem of optimal mixing of an inhomogeneous distribution of a scalar field via an active control of the flow velocity, governed by the Stokes or the Navier-Stokes equations, in a two dimensional open bounded and connected domain.  We consider the velocity field steered by a control input that acts tangentially on the boundary of the domain through the  Navier slip boundary conditions. This is motivated by mixing  within a cavity or vessel  by moving the walls or stirring at the boundaries. Our main objective is to design an optimal Navier slip boundary control  that optimizes mixing at a given final time. Non-dissipative scalars, both passive and active, governed by the transport equation will be discussed.  In the absence of diffusion, transport and mixing occur due to pure advection.  This essentially leads to a nonlinear control problem of a semi-dissipative system. We shall provide a rigorous proof of the existence of an optimal controller, derive the first-order necessary conditions for optimality, and present some preliminary results on the numerical implementation.