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We study the problem of transporting a distribution of fluid particles in aStokes flow to a desired final distribution in a fixed, finite time bycontrolling the torques of a pair of microrotors at fixed positions in theflow. Our approach is based on a finite dimensional approximation of theLiouville operator, the infinitesimal generator of the semi-group of PerronFrobenius operators, which describes the density transport dynamics associatedwith the microrotor flow fields. Using this operator, we express the transportproblem as an optimal control problem in terms of the moments of the densityfunction of the particle distribution. The finite time optimal control problemis then solved using differential dynamic programming, an iterative trajectoryoptimization method. We apply this framework to the microrotor driven flow onfour related problems: transport using rotors in an unbounded flow, transportnear to an infinite plane wall, transport within a circular domain, and thesimultaneous transport of two particle distributions to a common finaldistribution in an unbounded flow. These examples demonstrate the effectivenessof the proposed framework and also allow us to better understand the effects ofboundaries on the ability to achieve a desired fluid transport using arotor-driven flow.

Controlled density transport by microrotors in a Stokes flow using linear transfer operators