Boundary Control for Optimal Mixing by Stokes Flows

We discuss the optimal boundary control problem for mixing an inhomogeneous distribution of a passive scalar field in an unsteady Stokes flow. The problem is motivated by mixing the fluids within a cavity or vessel at low Reynolds numbers by moving the walls or stirring at the boundary. It is natural to consider the velocity field which is induced by a control input tangentially acting on the boundary of the domain through the Navier slip boundary conditions. Our main objective is to design an optimal Navier slip boundary control that optimizes mixing at a given final time. This essentially leads to a finite time optimal control problem of a bilinear system. In the current work, we consider a general open bounded and connected domain Ω⊂Rd,d=2,3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset \mathbb {R}^{d}, d=2,3$$\end{document}. We employ the Sobolev norm for the dual space (H1(Ω))′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(H^{1}(\Omega ))'$$\end{document} of H1(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^{1}( \Omega )$$\end{document} to quantify mixing of the scalar field in terms of the property of weak convergence. A rigorous proof of the existence of an optimal control is presented and the first-order necessary conditions for optimality are derived.