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  Stokes or 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 the problem of 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.   A rigorous  proof of the existence of an  optimal controller  and the first-order necessary conditions for optimality will be presented.

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