Statistical properties of controlled fluid flows with applications to control of mixing

In this paper, we study statistical properties of fluid flows that are actively controlled. Statistical properties such as Lagrangian and Eulerian time-averages are important flow quantities in fluid flows, particularly during mixing processes. Due to the assumption of incompressibility, the transformations in the state space can be described by a sequence of measure preserving transformations on a measure space. The classical Birkhoff's pointwise ergodic theorem does not necessarily apply in the context of sequences of transformations. We call B-regular a sequence for which this theorem holds. Motivated by mixing control concepts, we define three notions of asymptotic equivalence for sequences of transformations. We show an example in which Birkhoff's pointwise ergodic theorem does not hold even when a ‘strong’ asymptotic equivalence to a B-regular sequence is assumed. Under a ‘very strong’ asymptotic equivalence condition, we prove B-regularity. In the context of optimize-then-stabilize strategy for mixing control, we also prove that very strong asymptotic equivalence to a mixing sequence implies mixing. The mean ergodic theorem and the Poincare’ recurrence theorem are also proven for sequences of transformations under suitable asymptotic equivalence assumptions.