An approximating approach for boundary control of optimal mixing via Navier-Stokes flows.

• An approximating control is addressed for optimal mixing via fluid flows. • Passive and active scalars governed by the transport equation will be discussed. • Navier slip boundary control is employed with low regularity. • Sobolev norm for (H 1 (Ω)) ′ is adopted for quantifying mixing. • Uniqueness of the optimal solution is obtained. The present work focuses on an approximating control design for optimal mixing of a non-dissipative scalar via Navier-Stokes flows in an open bounded and connected domain Ω ⊂ R 2. The objective is to achieve optimal mixing at a given final time T > 0 , via the active control of the flow velocity through the Navier slip boundary control, where Sobolev norm for the dual space (H 1 (Ω)) ′ of H 1 (Ω) is adopted for quantifying mixing. Both passive and active scalars governed by the transport equation will be investigated. Our current approach will lead to a more transparent optimality system for characterizing the optimal solution compared to our previous work [12]. This is achieved by first introducing a small diffusivity to the transport equation and then establishing a rigorous analysis of convergence of the approximating control problem to the original one as the diffusivity converges to zero. Moreover, uniqueness of the optimal solution is obtained. [ABSTRACT FROM AUTHOR]