A switched system approach to optimize mixing of fluids

In this paper, we discuss the problem of efficient fluid mixing which is tackled by means of (approximate) dynamic programming from a switched systems perspective. In current practice, typically pre-determined periodic mixing protocols are used. As we will show in this paper, feedback control can be used to improve mixing significantly. To make this control problem tractable, temporal and spatial discretization is used by means of the cell-mapping method on the original infinite-dimensional fluid models. This translates the original control problem into the design of a (sub)optimal switching law that determines discrete mixing actions for a discrete-time switched linear system. Exploiting this switched systems perspective, a novel feedback law for mixing fluids is proposed inspired by suboptimal rollout policies in dynamic programming contexts. By design, this feedback law for mixing guarantees a performance improvement over any given (open-loop) periodic mixing protocol. This new design methodology is validated by means of simulations for the benchmark journal bearing flow showing improved performance with respect to periodic mixing strategies.